I wholeheartedly believe that giving context to the history and slowly guiding students through the mindset of mathematicians is objectively better than spoon-feeding them equations.
My Algebra teacher in college used to tell us stories like that and I remember him telling us this one too. He later went on to become the minister of education in my country.
We had this in every chapter's explanation in our books, a large one ~one page paragraph explaining the history or the person behind the concept, unfortunately it was never used
As someone who's really bad with math, these visuals have helped me realize a lot of what I didn't understand with basic algebra and trig functions from school as a kid.
Schools convert man to a learning engine so that he will grasp everything that makes humans worse than google . But human brain is much more than just a fact storage device.
"Only by giving up maths' connection to reality could it guide us to a deeper truth about how the universe works." Bravo! A thoroughly professional presentation from algebraic dependence on visual geometry through Mediterranean ego vignettes segueing into physics, with remarkable insights along the way, culminating in the quote above.
All throughout grade school and college I struggled to understand the "why" portion of math beyond plug and chug. Usually professors couldn't give me an adequate explanation. Completing the square was one term that never really clicked for me. The first 3 minutes of this video are pure genius. So simple and understandable. This makes math so much more digestible.
3blue1brown did a video series on calculus in an equally visual way that helped me understand it better than any teacher in high school ever could. This has been the pattern of my learning after school in general tbh, the internet has been the most valuable resource in my learning journey.
If you'd like a book that focuses on visuals, I would recommend "Proof without words" - my professor recommended it to me recently & gotta say, it's a fun gift for those who like picture books.
History of mathematics should be taught as early as in middle school, and this video tells exactly the reason why it would immensely help students appreciate what they are taught.
History of everything should be taught.. otherwise the new student must do what literary fiction does.. cause suspension of disbelief. In other words.. believe in magic..
It should, but it isn't and it won't. Schools are more concerned about your kid's dress code violations and football than to teach them anything useful. And if they are close to flunking, there's always the Army to set them straight ( in all manners).
Borrrrriiiinnnnnggggggg. History is boring to teens, they do not care about the past, more about the future, or even more the present, for most anyway.
For the entirety of my higher education, I've been told to "complete the square," but 6 teachers and 4 professors have never explained this further than restating the equation. In one extremely brief visual and explanation, you've managed to answer a question I'd long since forgotten. I don't know how to describe my astonishment, nor my gratitude for your content.
No kidding! I recall that math was usually taught by coaches reading from a book. I had the same reaction as you to the "complete the square" part of the video. I could have been so much smarter had I only had math teachers with this gentleman's style...
Mathematics needs a James Burke. This video is totally on that path... talking about secrets, jobs, politics, challenges, motivation, compromises and re-derivation of formulas.
@@allmotion_video_channel5434 whether would it make you smarter depends on what aspect are you talking about. if we are only talking about taking test and exam, explaining the equation to you wouldn't make you smarter in doing in math classes. It mostly depends on how much practise questions you've done. The same philosophy can also apply to college. Surely, however, explaining the meaning behind mathematical equations and practices would help you have a better understanding of the world. Though it have little use in real life since mathematics are mainly used in real world as a tool to solve, not a tool to understand.
@@jimmyli319 interesting outlook on the subject you have. I am a flight instructor and have seen how different people learn in different ways. For me, I like to understand the underlying theory. That helps me judge whether the results of a “solution”makes sense. I also have beginning engineers that I work with that will just plug numbers in to a program like “MATLAB” but do not understand the fundamentals well enough to judge whether the “answer” is reasonable enough to be correct. But, in your context, it is valid that one does not need to know how an internal combustion engine works just to drive a car…
I was one of the worst math graduates in my highschool class but recently I had a spark of love for maths and reteach myself everything. This video is nothing short of amazing. Its just mindblowing!!
Next brilliant video: "Your Daily Equation #2: Time Dilation" by Brian Greene. Only pythagoras and basic algebra needed. But, for bonus, you can try to find the unit circle that links time dilation with speed
Imagine minding your own business as a mathematician and suddenly someone challenges you to MATH DUEL, that can make you lose your job. Man, the older times were really intense for mathematicians.
Man, change "depressed quadratic" to an obscure magic spell and you literally get a fantasy duel story, complete with a sage old mentor, an underdog protagonist, an enchantment and a boastful proud villain wtf
Stories of wizards and stuff are probably inspired directly by mathematicians. So of all those T-shirts comparing some profession to wizardry (you see it for engineers or IT a lot), the most legitimate claim is for the mathematician imo.
That is what its supposed to be in the first place we are doing a top down calculation to simplify it to our understanding. The complex structure is still the answer regardless of whether it is solved (simplified) or not so we are not trying to solve it but make it understandable for ourselves, which means making it elementary
Would it be too random to declare my intend to recommend my fellow science-youtuber-fans some... well... more science-youtuber? I mean, in my mind, it just makes sense, but many call me B0t, so... your choice...
Complex numbers are dual to real numbers. Perpendicularity or orthogonality = DUALITY! Column vectors are dual to row vectors -- group theory. Electro is dual to magnetic -- Maxwell's equations. The electric field is perpendicular (dual) to the magnetic field -- probability waves. Positive charge is dual to negative charge -- electric fields. North poles are dual to south poles -- magnetic fields. Electro-magnetic energy or photons are dual. Points are dual to lines -- the principle of duality in geometry. Group theory:- the image is a copy, equivalent or dual to the factor or quotient group. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference). Homo is dual to hetero, same is dual to different. Injective is dual to surjective synthesizes bijective or isomorphism. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger. Duality (energy) creates reality. Action is dual to reaction -- Sir Isaac Newton (the duality of force). Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force. Monads are units of force -- Gottfried Wilhelm Leibnitz. Monads are units of force which are dual -- monads are dual. Energy = force * distance. If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual. Apples fall to the ground because they are conserving duality. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. "Always two there are" -- Yoda.
@@nathanlevesque7812 Asinine! Duality allows you to create new laws of physics:- Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non teleological physics (entropy). Making predictions to track targets and goals (objectives) is a syntropic process -- teleological. Complex numbers are actually dual numbers -- the complex plane. Poles (eigenvalues) are dual to zeros -- optimized control theory. Duality is everywhere if you look for it, male is dual to female.
I thought I was just going to browse the video but here am i going through it all and even rewinding. Thanks it was very engaging and brilliantly undertaken.
I’m 31 years old and remember learning about imaginary numbers but never taught why they exist, or what examples there were in nature. I absolutely loved this video and thank you for making it. It reminds me that things you thought were useless info in school have an immense impact on the collective knowledge of humanity.
This is a common problem in school, especially math. You are not taught why, just how to apply it (and sometimes you don't even get taught when to apply it).
I lost interest in mathematics after never being told why we use such things as imaginary numbers and their applications. After years of doing math problems you get to a point of questioning why you are doing all of this with no context of application. I think this is a real problem with how math is taught, the student is never explained the relevance of what they are actually doing in terms of the real world. It is sad that math is taught like a menial task to be performed until the correct solution is found yet you don't understand why you are doing it.
@@Leaptab The problem is, the field of mathematics in the pure sense indeed is never concerned with the application of math itself. You will notice to in this video, that the "invention" of imaginary number theory far predate the application of the imaginary number itself in real life. If you find math to be menial task just because you don't know the application in the real world, chances are that will be the same feeling you'll get if you ever dig deeper into the field of math in pure math research.
@@Leaptab You're conflating the application with the subject. What you're saying is "Why do I learn a physics concept instead of an engineering concept?" and it really misses the point.
I agree; however, I feel that you can never teach others: you can only motivate others to dive deeper into the material themselves, and this is a great video that fires up people’s interest to do exactly that.
Nahh i assure you, dumb and lazy students will always be like that. They would dismiss the historical story and ignore the illustration of solving an equation using geometry.
@@vandalm9311 it's about those who are really curious to know about the core concepts and essance of science. Here neither the dumb nor the intelligent has access to this quality of education.
Ah yes, just spend hundreds of hours on a video for a 1 hour lesson. I'm sure that will be highly efficient. Tell me you've never taught before without telling me.
This video was incredible, I cannot put into words the fantastic journey I experienced in these last few minutes, thinking about the realities of mathematicians, how problems that have been considered to be impossible for thousands of years are solved, and how we naturalize the legacy of these incredible minds. Thanks my friend
Math is riveting if you don't know math. If you actually learned at least high school math and some history behind it (like how was calculus developed), you'd know that it is riveting.
( colé izzy o/ ) the thing that i take from this is just how insane being a mathematician this day and age problaby is when you're needed to know all prior knowledge at the same time that you need to challenge it to find new questions and keep progressing mathematics towards the future.
Yeah, I was going to say, what’s sad 😞 is that you didn’t find it riveting to begin with as it literally is the language we use to describe the reality of our universe. But to each his own and I don’t blame you.
Instead of letter grades A through D, 8th graders should get a grade placement based on which century of Italian mathematics they most closely align with.
If you think 8th graders are learning about imaginary numbers, solving cubic equations, or quadratics for that matter you either don't remember primary school or were an exceptionally gifted child. My guess is the concept of a variables is introduced in 7 or 8th grade, probably putting 8th grades some where in the dark ages. Probably where they belong from what I've seen , haha.
@@901blitz yeah variables are taught in the middle of primary school and in 8th you learn algebra 1 which definitely has quadratics, imaginary numbers, etc. im in 12th grade right now taking calculus using the prior knowledge from middle school with graphs and algebra needed for complicated derivatives
It’s always funny that the answer to why something is called what it is called, is often right in front of you, but (in the case of completing the square) due to how math is taught in school it often comes off as nonsense.
This is my favorite math history episode ever! I love the "cubic battle" and the invention of "imaginary numbers" so much, thanks for making it even more interesting with your narrative.
Most of your work is educational yet highly entertaining but this particular video deserves an award. One of my favorite channels on the platform. Proud to have subscribed to it over 10 years ago.
Complex numbers are dual to real numbers. Perpendicularity or orthogonality = DUALITY! Column vectors are dual to row vectors -- group theory. Electro is dual to magnetic -- Maxwell's equations. The electric field is perpendicular (dual) to the magnetic field -- probability waves. Positive charge is dual to negative charge -- electric fields. North poles are dual to south poles -- magnetic fields. Electro-magnetic energy or photons are dual. Points are dual to lines -- the principle of duality in geometry. Group theory:- the image is a copy, equivalent or dual to the factor or quotient group. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference). Homo is dual to hetero, same is dual to different. Injective is dual to surjective synthesizes bijective or isomorphism. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger. Duality (energy) creates reality. Action is dual to reaction -- Sir Isaac Newton (the duality of force). Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force. Monads are units of force -- Gottfried Wilhelm Leibnitz. Monads are units of force which are dual -- monads are dual. Energy = force * distance. If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual. Apples fall to the ground because they are conserving duality. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. "Always two there are" -- Yoda.
When things were at their very worst: 2 Suns, Cross in the sky, 2 comets will collide = don`t be afraid - repent, accept Lord`s Hand of Mercy. Scientists will say it was a global illusion. Beware - Jesus will never walk in flesh again. After WW3 - rise of the “ man of peace“ from the East = Antichrist - the most powerful, popular, charismatic and influential leader of all time. Many miracles will be attributed to him. He will imitate Jesus in every conceivable way. Don`t trust „pope“ Francis = the False Prophet - will seem to rise from the dead - will unite all Christian Churches and all Religions as one. One World Religion = the seat of the Antichrist. Benedict XVI is the last true pope - will be accused of a crime of which he is totally innocent. "Arab uprising will spark global unrest - Italy will trigger fall out" "The time for the schism in the Church is almost here and you must get prepared now." The Book of Truth.
Seriously, as an Iraqi student, I have been studying complex numbers for a month and a half, and after all of this, I do not know why “i” is written with “y” this intuitive piece of information, loves god what teaching have I been learning throughout this period? Really, thank you, god bless you
I’m a physics major. I’ve always had trouble understanding complex numbers and why they exist in equations. It’s like my professors were just handing out the equations like the Schrodinger equation without really explaining what they mean. As I went on throughout college I gathered an understanding, but this video gave me that “aha!” moment. Thank you Veritasium, Your videos are something special and I appreciate every single one that gives me more insight on how the universe works.
yes he is a good "teacher", knows how to explain and visualize things. In math one would learn that "things" exist as parts of other things, real numbers are a subset of complex numbers. And then there's quaternions ... and one would think, does it ever stop... and yes it does, but that means walking into set theory and such, everything is kinda 'connected'. I worked a lot with physicists, they always came to me "to pick my brain".
I majored in math and I always got that impression from the courses taught by physics profs. They seem to think of math as a set of tools that "just work". If you ever want to unravel those tools and figure out why they work, the field of math you are looking for is called "analysis" (probably "complex analysis" and "real analysis" in most universities). Those were always my favorite courses because they helped so much to explain all of those weird formulas from calculus and differential equations.
This comment got a lot longer than I initially intended, but covers a number of things about constructions of the complex numbers and how else they can be thought of: As a pure mathematics student, I like to think of imaginary numbers as a construction. They are not “numbers” in the same sense as real numbers, but they can be paired with real numbers to produce a helpful construction which allows people to manipulate things in ways that may not initially seem possible. In the context of ring theory, we consider general systems of numbers with addition and multiplication. You can add and multiply polynomials with coefficients from any given ring as one might expect. And it turns out that a construction basically the same as imaginary numbers appears when you “quotient” the polynomial ring of the real numbers by “the ideal generated by” X^2 + 1. If you’re not familiar with this language, that basically means that if you take this ring, but now consider that whenever X^2 + 1 appears (or any of its multiples), it is now considered to be 0. You can see that this can produce the complex numbers intuitively since we are essentially just treating X as a number which squares to -1. Which is exactly equivalent to this idea behind the complex numbers: allowing the number i to exist and square to -1. All the properties of rotation then naturally appear through all the classical studies of complex numbers. However, a possibly interesting idea from this is that if instead someone had decided that they wanted a *different* cube root of 1, let’s call it j. Then j^3 = 1, and (-j + 1)^3 = -1 + 3j^2 - 3j + 1 = 3j^2 - 3j. But noting that j^2 = 1/j (and allowing some algebraic manipulation with some extra assumptions), we can find that (-j + 1)^3 = 1 as well (which matches what you’d get if you treated j as either of the complex cube roots of unity). I think we can all be pleased that no one did find this, because working with this number is a lot more tedious than working with complex numbers as we know them, but it does work. From the ring theoretic perspective, for this construction we wouldn’t set X^3 + 1 = 0, but instead we’d use X^2 - X + 1 = 0 which is (X^3 + 1)/(X - 1). This is basically so that we don’t have worries about X actually being 1. Obviously this comment misses out on a lot of rigour, but is intended to provide the general intuition behind the fields mentioned and give an alternative perspective of complex numbers.
Complex numbers are pretty cool once you get really into the theory on it. Like it’s applications and stuff is cool but things like reimanns hypothesis or schrodingers equation are very interesting
The phrase "completing the square" makes much more sense now. Holy crap my mind is blown. I really wish math was taught like this. I thought I hated math but I'm finding that isn't actually the case when I learn through mediums such as TH-cam. Does anyone have any suggestions for other videos that combine math and history like this one?
@@fatitankeris6327 History has many fields. History of England, History of the 16th Century, History of Agriculture and yes, History of Science. Did you say GOAT? Shemakhinskaya Bayaderka Festival / Yana Kremneva / 201. Science GOAT. James Burke Connections, Ep. 4 "Faith in Numbers". From 1978, pre-internet. I actually used Hollerith cards (punch cards) to write my first computer programs in Algol inn 1973.
HS math teacher here: thanks for showing the weird and cool the history of the equations, and visually describing how they all relate back to basic geometric shapes (even when they then veer off into the imaginary land). Definitely borrowing this for class.
Complex numbers are dual to real numbers. Perpendicularity or orthogonality = DUALITY! Column vectors are dual to row vectors -- group theory. Electro is dual to magnetic -- Maxwell's equations. The electric field is perpendicular (dual) to the magnetic field -- probability waves. Positive charge is dual to negative charge -- electric fields. North poles are dual to south poles -- magnetic fields. Electro-magnetic energy or photons are dual. Points are dual to lines -- the principle of duality in geometry. Group theory:- the image is a copy, equivalent or dual to the factor or quotient group. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference). Homo is dual to hetero, same is dual to different. Injective is dual to surjective synthesizes bijective or isomorphism. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger. Duality (energy) creates reality. Action is dual to reaction -- Sir Isaac Newton (the duality of force). Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force. Monads are units of force -- Gottfried Wilhelm Leibnitz. Monads are units of force which are dual -- monads are dual. Energy = force * distance. If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual. Apples fall to the ground because they are conserving duality. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. "Always two there are" -- Yoda.
I wish I had this, I failed math one, then it built on this until geometry. I failed all three classes, it might not have been as bad if I understood what I was looking at, and this did that for me, at least better than I had it before. Hopefully your students get that out of it too
@@hyperduality2838 The first statement is completely false. The real numbers are contained in the complex numbers. You're thinking of imaginary numbers. Complex numbers and imaginary numbers are different things.
As a student I used to have a mindset that Math cannot be imagined and thus it is useless at higher levels compared to physics and chemistry which have practical use. After watching this video, i was enlightened!
I love how this video displays how difficult advancing mathematics can actually be, and the sheer imagination required to conceptualize another dimension and it’s properties when you contemporary mathematics has no answers for what’s going on and will treat the discovery as fiction. Some people think math is all logic and that the ancients were fools for not knowing what HS freshmen know today. They fail to understand what the process of advancing a field of knowledge is actually like.
Breakthroughs can take an unreasonable amount of time. But once it occurs, it can be passed on. I completely agree with you and this is why we have to honor the greats. They made it easier to push the limits just a tad bit further. It just takes that one eureka moment.
Anyone who thinks that way is not that intelligent. Why would someone think that people before us, with less access to knowledge, had less problem solving skills? It would be like claiming the person who figured out that rubbing two sticks together to make fire, was stupid. Sure we all know that now, but i dont think there is anyone in the comment section, that could figure this out, if they were raised by wolves. Knowledge is nurture not nature.
Math is all logic. Logic is what tells you that something is a mathematical statement or just conjecture. Also, it's not so much sheer imagination to come to work with imaginary numbers. Imaginary numbers and complex numbers are fundamentally the number system of algebra, and inherently come out of polynomial solutions. All that was needed was a pen, paper and the ability to not reject something simply because it doesn't immediately make sense.
I’m a high school math teacher and I created a worksheet to go along with this video last year and showed it in my Algebra classes. I just got done showing it again this week while I was preparing my students for our unit on Complex Numbers and the imaginary number. I show a lot of videos from this channel in my class as they’re both educational and very fun. Thanks for the great content.❤
I'm a student who has a lot of trouble in school - You sound like a great teacher. Teachers who don't hesitate to relate to students with modern forms of education are ALWAYS the best. Learning in the modern age is *still* so underutilized in schools, American schools anyway. We have the world at our fingertips and we still use outdated textbooks from 30 years ago?Come on. Just saying, I think it's great Veritasium is being used as a source of education. Keep it up and good luck, Mr. Brown :)
@@GongGirl-ie5wy You are right that most people don't use higher math in everyday life. However being able to do it proves that you can think and tackle hard problems. For me, that has been incredibly valuable in my career because my employers seek me out and pay me not because I'm doing X hours per week, but that I'm differentiated by my ideas, thinking, and problem solving. That's allowed me to get really far ahead!
As someone who really struggled with math all throughout school, and still to this day struggles with it, I wish they would have taught this kind of stuff in school. I started watching these videos not too long ago and it's sparked a passion for the beauty that is mathematics, and inspired me to learn more about their fundamentals. Thank you to Veritasium, and thank you to all the teachers I had who wanted to teach me things like this, but were unable to because of strict, unchanging curriculum.
blows my mind how these guys figured out math. Studying math today it feels like everything is pretty much figured out. I guess you need to be at an incredible level to figure out what does not yet exist.
On the contrary Victor; you can quickly catch up as the hard work has been done leaving you with advancement. The only hiccup is when an axiom turns out to be false; which is only discovered when it is pushed to the limits.
Somewhere out there in this world. Someone again made a huge mathematical, or physics discovery, but didn't think much about it because they thought other physicist already considered it but didn't publish because it's wrong. If back in the past, people kept great discoveries secret, now it could be possible that someone out there made a discovery but isn't confident about their own ability to explain it. I mean, we already got the internet. If a person thought of something, they could search the internet for answers, or to confirm something. If it doesn't show up, it may be because it was a stupid idea that noone ever considered it, or it may have not existed considering them to be the first to come up with it. It is possible that the latter could happen. It's not such a bad situation though because some other people could just come up with the same idea, until one actually is confident enough to go public with it. Same with the story in this vid.
Related to complex numbers is a mathematical problem called the Reimann Hypothesis.... It's widely considered to be the hardest mathematical problem. It's one of the 7 millennium problems, of which only one has been solved to date. Solving any of these problems would be revolutionary and would win the solver a prize of a million dollars Some poeple like to joke saying "There are much easier ways to earn a million dolllars than to solve the reimann hypothesis" Very intresting stuff IMO
Like a lot of people on here, when I took advanced maths in school and was shown imaginary numbers, the course material made no attempt to describe the fundamentals of how imaginary numbers work, or why. This video in 15 mins made more sense than a year of schooling. Having this globally available on youtube is a gift to humanity
@@kienthanhle6230 I'm in my 3rd year of Electrical engineering. One of the fields that uses imaginary number the most. Now I actually understand what it means. I finally get why e^x and cos(x) and sin(x) are related.
@@kienthanhle6230 That's BC highschool only prepares us to take and pass tests. Fundamentals and true understandings are never required. BC schools only worry about average grades and test results for funding.
That would require _good_ math teachers, though. You only get those at the graduate-school level, because that's where they all end up. The teachers who end up teaching algebra, geometry, and trigonometry in grade-school are the people who just barely graduated with their math degrees -- they aren't qualified to teach anything harder than "x + 5 = 10" or "sohcahtoa". Whereas anything _simpler_ than algebra, geometry, and trigonometry doesn't require a math degree to teach it _at all,_ so there's nowhere lower for those bottom-tier math teachers to go. So _every single kid's_ first introduction to math that requires _actual thinking_ (instead of memorizing tables) is with a math teacher who, objectively, sucks at math.
@Hypnotize: Honestly most grade-school teachers are too burned out from grading homework until midnight and paying for classroom supplies with their own money, and aren't looking for even more ways to make their lives more complicated. If you have a good real-world example and explanation for a specific math concept, pass it along to your kid's math teacher. It will help them.
I mean... I've had math classes where we had to read about the history behind things like this, and I found it uttermost boring and useless. I've also had classes where we had to read some of the proof for the new concept/rule we were learning. This was sometimes boring, sometimes helpful. But, sadly, sometimes there isn't a way to showcase the proof, or at least not on a high school level, so you just have to accept it. Which sometimes suck, but you get used to it. XD (damn, I said sometimes a lot.)
Thanks…. I got stuck and quit math. I crushed geometry and sucked at algebra. In this video at 3:48 I felt my brain click, like a dislocated shoulder snapping back in place. Now, I’m pretty excited. Thanks again.
@@hckoenig no no no, I think you mean "Understand". Overestimate means praising the capability of a person or something that doesn't even have the ability to do that. Underestimate means doubting the capability of something or someone. Understand is knowing how something works.
@Pradeep Singh Dear Mr Singh, I kindly advise you to go and find the book "My Big Toe" by Thomas Campbell. It will explain a whole lot more as well to you. Have a nice day.
*VERITASIUM* is my inspiration!! My mom said that if I got 30k subscribers!! She definitely buy me a professional mic!! *begging you GUYS alot* literally begging.!.
Also isn't it surprising no nobel prize in mathematics and it is continuing, the members of nobel committee should announce that nobel prize should also be given to mathematicians for their work
@@b0mby1 Although your word made me feel insulted, guess it cant be helped. Well you see. First of all,I dont talk english. Second, I know what he talk about when he talk about that cubic stuff. Third, I understand all of this except that last part where he start do some reality bending edit, turning 2d into 3d before turning it into 2d again. The only thing I need to watch back is the part the -5 cube being introduce. That all.
@@ayuballena8217 I think so. He said something like im not good at math and something like that, which I agree. Im not math genius. However, his sentence feel like trying to say that im dumb, which is true, but I dont think he have good meaning behind it
The part where he explains how to solve the equation with literal visuals in my opinion should be taught in schools. It helps people grasp the concept much more easier.
I too think it should be shown, but only later on. I was shown the geometric version of the Pythagorean theorem and it didn't click until it was shown to me later on
While I agree that this video is very well done and engaging, the moment that your teachers would have made you start solving equations even after showing you this video, you would've get bored of maths again.
Meanwhile 10 million people die from preventable cancer every single year and not a peep from the press or any politician. Solve that unsolvable problem.
I just started my higher mathematics, got stuck in the Depressed Equations, and why to substitute x ---> x - b/3a , couldn't find any proper explanation , later suddenly unknowingly decided to watch this video , and without my known , I got everything I was looking for ! Man , this is how maths should've been taught , without just mugging up results , algorithms and solutions 🙌🏻 You're a genius Sir !
Everything that I learned in high school suddenly has an image for itself, algebra suddenly can be visualised , Complex numbers have suddenly so much meaning in my head. This video is just mind blowing!! Way to good!!
If you’re really curious about understanding the true realit of complex numbers, I highly recommend the “imaginary numbers are real” series by the channel Welch Labs. An amazing explanation of just exactly how complex numbers apply to the real world.
Imho, this is how mathematics should be taught. Rather than throwing the conclusion and have students arbitrarily plugging in numbers; an introduction with a few key points in its discovery would go a long way to understand how one stems from another and create better understanding and thus foundation to the subject.
Complex numbers are dual to real numbers. Perpendicularity or orthogonality = DUALITY! Column vectors are dual to row vectors -- group theory. Electro is dual to magnetic -- Maxwell's equations. The electric field is perpendicular (dual) to the magnetic field -- probability waves. Positive charge is dual to negative charge -- electric fields. North poles are dual to south poles -- magnetic fields. Electro-magnetic energy or photons are dual. Points are dual to lines -- the principle of duality in geometry. Group theory:- the image is a copy, equivalent or dual to the factor or quotient group. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference). Homo is dual to hetero, same is dual to different. Injective is dual to surjective synthesizes bijective or isomorphism. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger. Duality (energy) creates reality. Action is dual to reaction -- Sir Isaac Newton (the duality of force). Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force. Monads are units of force -- Gottfried Wilhelm Leibnitz. Monads are units of force which are dual -- monads are dual. Energy = force * distance. If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual. Apples fall to the ground because they are conserving duality. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. "Always two there are" -- Yoda.
and after that you have to plug the red wire into the socket to make sure the engine boots at launch. Wrap the green wire around it's coil that sits directly beside the A button. After you put the back shell on, place the battery in the slot. Screw the Vr26 Jeeper back up and press the reset button. If everything worked according to plan you're device should show a thumbs up sprite. Plug the HDMI port into a monitor and wait three seconds. If it boots up on TV your in the good side. If it doesn't boot in less then 5 seconds quickly unplug. This can severely damage your TV and possibly start a fire
That s because it is a poor statement to begin with.... it is the arrogance that blinds us.... in our quest for knowledge with each step forward, we stop to admire ourselves and claim that we now have acquired foundation for reality. Until we learn it is still out of our grasp.
we need to move on from old rules so we can seek further newer better ones. we need to move on from our prior definition and understanding of what reality is, to seek new better definitions of reality! in the end, it's OUR perception of reality.
@@francoisiannacci2615 I agree, there's always further to seek. standing on one discovery and thinking it's the end of the line, is such a toxic narcissistic thing to do.
this "lecture" is a real gem. I find it very, very inspirational for those students that area interested in knowing how the world works and how we found out about how to describe it. I hope to convince my children two watch this when the right time comes. Thanks a lot.
They didn’t because teachers didn’t know , they were just looking to get paid on the 15th and 30th of each month, rarely you will find a true math teacher.
@@aaronovski9949 math teacher here. Reality is, we often cannot go too far beyond the math curriculum given to us. I plan on showing this to my students little by little, because if we did this instead of class I'd be in trouble
Not sure why YT has recommended me this vid. I am too dumb to understand this shite. Just gonna casually scroll pass and listen to CardiB WAP and watch Logan Paul. Bye Ngl.
This is the first time I've seen an explanation of imaginary numbers that made sense to my brain. The first time I ever had someone say that mathematics was disconnected from reality, and how it moved away from geometry. This just flipped something in my brain. I wish I had this 20 years ago when I was dealing with calculus!
More like, it's not math that's disconnected from reality, but human beings. The "reality" we construct in our heads is more real to us than actual reality.
This REALLY feels like a lost episode of the new Cosmos, from the pacing, the language used, the explanation strategy, the animations, to the deep dive into the history, making it all concrete. This is a masterpiece, and it might be the best video you've ever made.
This seems really over the top, Cosmos has extremely high production values for an educational program, and is a lot more accessible to general audiences.
Just found this video yesterday. OMG what an amazing video, I have worked with 'i' a lot as an engineer, but was never taught where it came from. Thanks :)
As a grade 12 just finishing the last few of my finals this week... this video has blown my mind I NEVER understood why the method was called 'completing the square' until just now when you literally complete the square in front of me. If only all teachers and textbook writers were as dedicated as you.
The story of how cubic equations led to the invention of Imaginary Numbers was the favorite one that my school teacher has told us. This story was one of the reasons why I fell in love with Mathematics, and why I and my other classmates became engineers, physicists, scientists. But the storytelling quality and details of this video is on another level!
The history of math is always so fascinating. Lots of mathematical concepts seem so obvious now, but it’s almost impossible to get those connections without being taught about them. Amazing how so many brilliant people figured it all out over the centuries.
A man who can hold the attention of millions without using memes, gifs, music, effects or silly entertainment is why this channel is so great at has over 10M subs.
Seeing the effort that went into these innovations and discoveries in math makes me feel ashamed for our lack of appreciation for the material we are taught in school.
I'm an engineer, I use maths as a tool and just wow this video blew my mind away and gives a new context to everything that I do now. Suddenly I am less inclined to believe that math is a man-made invention and more an intrinsic property of the universe and reality itself.
Math is a general-purpose tool for **description**. We have specifically tooled and optimized it to be able to **describe** literally any sort of relation. The world happens to be relational, we should be very careful not to marvel too much that our tool, which works for all relations, real or imagined, has anything at all to do with how reality works. The map, as they say, is not the territory.
19:48 "So when you're multiplying by i, what you're really doing is rotating by ninety degrees on the complex plane." Oh my gosh that was brilliant. So very well done. I had to stop right there to leave this compliment on a great job!
You should look into the polar coordinate system in the complex number field. It basically uses a similar system. Instead of an X/Y coordinate system (where X is the real number and Y imaginary) it uses a magnitude and some rotation.
Complex numbers are just great. You basically do a "Let's assume that i is a number, and that its square is -1" exercise. You get all the basic maths really fast. For example: Let's assume that (a + b i) and (c + d i) are complex numbers, and a, b, c, d are all real numbers. Then, (a + b i) + (c + d i) = (a + c) + (b i + d i) = ((a+c) + (b+d) i), (a + b i) - (c + d i) = (a - c) + (b i - d i) = ((a-c) + (b-d) i), (a + b i) (c + d i) = (a c) + (a d i) + (b c i) + (b d i²) = (a c - b d) + (b c + a d)i. Division is a bit more complicated, but you start with a twist: try multiplying (a + b i) by (a - b i): (a + b i) (a - b i) = a² - a b i + a b i - b² i² = a²+b², a real number. So, (a + b i) (a - b i) / (a²+b²) = 1, as long as the denominator isn't zero. In other words, the inverse of (a + b i) is (a - b i) / (a²+b²). (That's a division by a _real number,_ or a multiplication by 1/(a²+b²), so no problem there.) To get complex division, you merely have to multiply the inverse of the second term by the first.
i love how there’s this torch that’s passed along from generation to generation of geniuses to figure out crazy things and eventually someone in the future solves a problem from the work of all that came before them.
The history of all human achievement has been written this way. Unfortunately, the efforts of hundreds, thousands even, are often attributed to a single person.
If I were a HS Algebra teacher, when we hit imaginary numbers we would definitely be watching this video in class. It might be the best how and why video I've ever seen on this subject.
Brilliant research and presentation! I had no idea complex numbers arose from the solution to cubic equations. Things make so much more sense when you explain the path of their discovery.
Theres nothing "real" or "fundamental about complex numbers. As a matter of fact, for a long time they were met with skepticism from mathematicians because its nonsensical to try to give them any inherent meaning. As Gauss, personally responsible for making complex numbers acceptable, explicitly explains, complex numbers are basically nothing but planar vectors and they make two variable real calculations more compact
@@Someone-ig7we It is found in quadratics too. But quadratics can find solutions without the imaginary unit showing up, and for quadratics where there were no real solutions, mathematicians could just dismiss those as not being solvable. Further, and more importantly, deriving a general solution for quadratics does not utilize an imaginary number, but for cubic equations, imaginary units are a necessary intermediary step for deriving a general formula. So while imaginary numbers show up in some quadratics, they're a necessary feature for getting the general solution to cubics.
@@hamidrezamahmoudian2710 False. Complex numbers are the fundamental number system of algebra. To understand this process, you need to first look at how we get integers from real numbers and addition by way of looking at additive inverses. Then look at the way we get rational numbers by look at multiplicative inverses of integers. Then look at how we get algebraic numbers from those, and you'll come to see that in the end, complex numbers are where we end up from the natural operations and their inverses. Complex numbers can be represented through other means (but so can any number system), but complex numbers are both real and fundamental as much as any other number is. The fundamental theorem of algebra should be the first thing you look into. Yes, they are planar vectors, but they also possess a unique behavior under multiplication, and it's this behavior that shows they're more fundamental than the biased view of them you're presenting. In order to represent them as vectors, you need to introduce a special operation for multiplication that's derived from the behavior of how sqrt(-1) acts under our typical understanding of multiplication. This new vector multiplication, distinct from any other multiplicative operations on vectors, cannot be derived from pairs of real numbers alone. It can only be derived from observing how the behavior of sqrt(-1) behaves under our traditional concept of multiplication. Since you cannot derive the behavior of complex numbers simply from pairs of real numbers, it follows that they are not "basically nothing but planar vectors that make two variable real calculations more compact" because it dismisses the entire field of complex dynamics, which only follows from the how sqrt(-1) behaves under multiplication.
This sort of took me back to when I used to enjoy learning mathematics from a teacher who always prefaced each lesson with the story of the history of the origins of each topic.
engineer here. in my schooling, i was never shown that i correlation on e^iX. Imaginary numbers were so confusing to me, but i could plug and chug formulas for them. Now, 10 years later, seeing that description of how it combines sin and cos into 3 dimensions is simply astounding. I never understood the why of imaginary numbers so i couldn't visualize it properly in my brain.
@@Perceptious37 Same here and I had trouble with the wave equation too because it departed fully from anything I could use as an analogy in my mind. When a thing becomes just spade work then I lose all mental interest. I needed to actually grasp what it meant in my mind or chugging equations just made my brain hang up. The guy who taught Schrödinger was first generation German with the thick accent and, combined with all the strange Greek notation, I just stopped caring about it all.
This is a faultless presentation of one of the most inspiring naratives in history, maths and physics. Congratulations! You have set a new paradigm YT. Could you do the same for Dirac's equation?
@@WestExplainsBest I was just saying to someone that I sorely wish the history of all of this had been taught to me back when I was learning it. I went on to study math in college, but I still wish that someone earlier on had showed us the humanity in math, the bickering scientists and the disbelief/hope that a solution would ever exist. I think it would be awesome for you to show it
Dirac and Schrodinger were both genius. Although Dirac's equation gets the fame, Schrodinger was the one who built the mathematical framework for Quantum particles.
@@vikraal6974 s`right, but beauty of following through to Dirac is bringing matrices into the story and thus antimatter! How maths reveals the world. After that, string theory?
I am simply awestruck by this amazing trip through mathematical history that brings us all the way to the Euler's equation. The graphic explaining e^ix = cos x + i sin x is the best I have seen. Congratulations and thank-you, Veritasium! BTW, is that an iron ring on your finger?
Seeing the spiral of e^ix helped me understand Euler’s formula in a way that I’d never understood it before, despite having used it repeatedly in math, physics and engineering classes. A visual rather than algebraic analysis.
Ikr. It's a common Sci Fi trope but I love the thought of these guys throughout history being aliens or something changing our whole civilisation with what is high school maths to them 🤣
Perhaps the reason why mathematics developed so slowly before the Enlightenment era was due to this 'competitive' notion of mathematics, which encouraged people to keep their discoveries secret, to use as 'concealed weapons' for any adversary trying to challenge them on their position. I'm glad science has transitioned to a more collaborative process, it just makes more sense for overall progress.
back then, what prevented science from being collaborative was religion. now it's not cared for, but not cured and done for entirely. you have to have no humans left that insist on such primitive thoughts, + make every person interested in science and that's how you reach peak humanity. (i'm including both linguistic and computational aspects of science along with music, art or whatever comes to mind of course, they are all science.) i'm not saying people should deny the existence of an higher power, nobody can ever know that, agnosticism and deism exists for a reason. just not be stuck with all the books and the prophets and whatnot. that's how you get going, in terms of humanity.
I mean, maybe with some things it has gotten better, but this "I will keep this discovery, which would positively impact society at large if made public, a secret to myself for personal gain" is alive and well today, encouraged even, in the tech and pharmaceutical industries; and just like the video reminds us, it could take decades for a discovery made now to be made public, and really put to use in proyects that benefit is all
@@astridkjellberg This is nonsense. Explain in what waay religion "prevented science from being collaborative". Referring to religion as "primitive thoughts" says a lot about you friend. Making such wide statements regarding the history of science ( which I doubt you know anything about ) and regarding religion is pointless; if your thesis is that religion hinders scientific advancement you ought to provide evidence for it. The conflict thesis ( the enlightenment idea that religion and science are fundamentally incompatible ) has been rejected by most historians and scientist, because it's ridicuolous and outdated.
@@martiddy In a framework where God exists ( the Abrahamic God I mean ) then how could miracles contradict science? From that point of view God can just do anything without regard for any natural laws, I don't see any contradiction.
I never understood why "i" sat on a different plane until now. The fact that i * i = -1, and that it is a rotation in a different plane never occured to me before. So simple, yet obscure. It makes so much sense now.
“Only by giving up maths connection to reality could it guide us to a deeper truth about the way the universe works.” What a lovely statement to end this insightful video.
But imaginary numbers exist. They just don't follow human logic. We use imaginary numbers for real things all the time. You cannot be an electric engineer without understanding them.
@@Wylie288 But for almost 2 century from the "invention" of imaginary number to the finding of its use in wave equation, imaginary number is just a concept that is only of interest to "math geek". I could get with the statement above that only by giving math "real" connection to reality that it become of interest to the "mainstream", just like what Schroedinger does to imaginary number with wave equation.
@@martinsusanto510 Imaginary numbers are required for electrical engineering. They are part of the basis of your entire life. They are required knowledge for us to send these messages.
This is probably my favourite video on YT. It is the best maths one by a mile, I really, really enjoyed it - I learned so much. The enthusiasm of the presenter was tangible!
Quick reminder that the solution to fifth-degree equations was discovered by a political revolutionary who died in a possibly unrelated duel. The history of this part of mathematics has been almost excessively dramatic.
@@btf_flotsam478 I see you are talking about Evariste Galois and the Galois theory,but he did not find a solution to fifth degree equations,he just proved that they don't have a general solution
@@78anurag Abel also did it independently, but Galois did it a bit earlier and at a younger age. The only reason many people know about Abel and not Galois is because after Galois sent his work to his friend and died,his works were suppressed by other mathematicians(because they themselves did not understand it). Also Galois's work was much more detailed and generalized.
I went through school and university not really understanding the 'why' of imaginary numbers - just really using them as a 'magic spell' that happens to work. I wish I'd been able to see this explanation twenty five years ago, it's brilliant.
I am a big history buff and I always hated math in school.... Until I was introduced to the history of math. Now, it is just far more interesting and especially sensible for me than ever before.
@@Wasserkaktus My brother was telling me about a friend of his who was on a football scholarship and took a course on the history of mathematics. That got his interested in math and he eventually got a PhD in mathematics.
when you get to Engineering and start drawing Root Locus plots, you'll want to ignore these solutions (and any positive real values) when finding Breaking Points. I'm honestly trying to revise on TH-cam comments right now
Seeing the historical origin of mathematics, physics, and technology makes it understandable much better because then you can fathom why they need it in the first place. If you make something about Fourier like this video, I want to see it
you can checkout 3Blue1Brown youtube channel... He has made a few videos about Fourier, Complex numbers and many more topics with awesome animations and best explanations.
This is half a module on the History of Mathematics (trust me - I took such a module at university!) and about a quarter of a module in Quantum Physics. In 20 mins. That is quite the achievement - well done!
I wrote an essay on this topic in a History of Mathematics unit in 3rd year, about 20 years ago. Got up to a little of Galois theory. Good stuff! (Now I'm an algebraist, though a different kind.)
Meanwhile 10 million people die from preventable cancer every single year and not a peep from the press or any politician. Solve that unsolvable problem you ignorant sheeple!
@@ProfessorWumbology "Algebra" to a mathematician means something very different from what you learned algebra was in high school. Roughly speaking algebra in mathematics is the study of "algebraic structures" such as groups, rings, modules and fields. A ring roughly speaking is just a set R together with two operations + and × called plus and multiplication such that a×(b+c) = a×b + a×c. R could be the set of integers, rational numbers or the real numbers, R could also be the set of matrices with + and × matrix addition and multiplication. You can also consider the ring Z_2 for example which contains only 0 and 1 with + and × given as usual except where you define 1+1 = 0.
I wish schools taught the geometric way of solving quadratics and cubics first, then the algebraic way would’ve made much more sense for many students.
@@nathanlevesque7812 yeah that’s not it. You can have a sponsored video without having the whole video be the ad, so you avoid facts so your ad doesn’t fall short. It’s incredibly disingenuous, especially for a channel purporting to be about facts.
I was a chemistry major and tolerated math because I had to and I was fairly good at it. I might have been far more inspired and really good at math due to applied inspiration if I had seen this video 20 years ago. Every student taking algebra and calculus should have to watch this video at the beginning of those courses.
Maths is so incredibly exciting. The way you get taught several seemingly unrelated things which then end up seamlessly working together is absolutely brilliant.
Are you ready for some smartellic to give solutions in geometric format instead of algebraic format? Just curious. There's a lot of overlap between math and electronics. I sometimes gave math solutions to electronic circuits and circuit solutions to math problems. Surprisingly I got away with it IF my answer was correct. Operational amplifier circuits love differential equations. It's a shame that I don't remember either not having worked with them for decades. But they were beautiful when I was working with them.
I wish someone would have shown me the quadratic equation solved with shapes demo when I was in school. Literally have never seen that before and it makes total sense.... Despite the problems with negative area.
If they taught in school about the history of math and how we use it in real world, I'm sure most of people who "hate math" will see how magnificent it is.
@@yuseisatouissuffering they shouldn't be supposed to memorize the history . It should just be told to inform them on how maths is actually done, i.e. how new stuff is actually figured out. The focus on memorizing formulae to solve most math problems is what i think is stopping math education from being fun.
This video proves that wrong though. Maybe in the short term (like the student who tried stealing credit from his teacher) but the experienced mathmetician knew something was up. Sure enough just 500 years later we are here and we know the truth.
Visualizing "i" as describing a value that cancels itself out and seeing the CG of e^ix described as a spiraling 3-dimensional waveform with the X & Y functions 90° out of phase may have contributed more to my understanding of physics and mathematics than the entirety of my college calculus courses.
That’s how I know you found the actual Easter egg in the video. It literally made me gasp and shout as soon as I saw why Euler formula used e. Integrating and differentiating e is always going to be the same and the way they reflect sin and cos and their interwind is simply mind blowing 19:57
The fallacy of your comment is that you assume that your "understanding" back then is equal to your understanding now, after accepting "not understanding" and dedicating your attention to your individual set-up. Being patient, curious and brave (having FAITH) - i.e. staying HUMBLE - are the only ingredients of LIFE. The best reference to LIFE being THE BIBLE!
Maths building out from geometry is amazing to me. I think people would understand it more if we taught it with geometric shapes alongside the formulaic advances.
This was an amazing video, Derek. I love how you’ve doubled down on TH-cam in recent years. I so hope early high-school students get shown this as a motivating factor for much of high-school math. Wow. I feel envious of them to have this at their age while I see it now at 36.
A bit of a rant and a thank you letter. People these days forgot (and even gasp in amazement, sometimes ridicule) when I say algebraic notations are new language and they tell a story. Much like learning a language, frequent use and familliarity, as well as occassional spritz of history and cultural knowledge, is essential for engineers to use equations beyond it's mundane descriptions. And then there're mathematicians, a cadre of people dedicated to studying the language and art itself. I think the educator stands in between, massaging the notations into words for the learners, opening new possibility in the minds and professions. And I think you do well as an educator. Thank you so much for this brilliant video. I hope the message lasts beyond your passing.
I didn’t understand a single thing and I enjoyed every minute of it. Jokes aside, it brought me back to my quantum chemistry lessons, many years ago! You have a gift to make mathematics interesting.
I find that knowing the history of things always makes them simpler. That's the beauty of time... If you don't find your answer in the present, you can look to the past, and if you can't find it in the past, you can build your future to find it...
@@derpatel9760 yeah agreed. we have to remember that most of these concepts were often discovered in a collaborative effort between multiple people across different generations, and if we don’t get it straight away that’s fine - the people who made them up didn’t too. at least that’s what helps me to not get my tiny brain overwhelmed by all this. also, nice pfp lol
As a JHS student, I used to think that Math should be taught in a way that would help us in reality. For example how to solve basic operations, basic algebra, etc. Then watching this video I've realized society would not progress if Math is tied to reality. We would be stuck with elementary/junior high school level! From now on I'm not gonna dread about math. And if i sometimes feel like giving up, i'll remind myself of the countless mathematicians who made it without having the knowledge we have in the 21st century. This made me love math.
@Pradeep Singh Three ways: 1. The solutions you get using your new method on the problems seem reasonable 2. Your peers agree with you 3. Your new method solves any relevant real-world examples. I went looking online for real-world examples of problems someone in the 1500s might have wanted to solve with the cubic. One was the volume of a container for water or grain. For example, if I change the length of this side of a box, how much less grain would it hold? The other is fishing. If I have to pay three men to fish in a body of water, but it slowly gets harder to fish the more they catch, and the more fish they bring back the less each fish sells for on the market...how long should I pay them to fish? How many should they get and when should they give up and return with what they have? All the other relevant real-world solutions with the cubic are from our time, like drag on an airplane, making computer animations look smooth, turning 3D scans into 3D models, displaying fonts, using encryption to protect a web page when it loads, etc. Hopefully this didn't post twice, I had responded already but my page had an error.
Everyone talking about the history and math, meanwhile I am amazed by all the visuals and illustrations to make this clear to non-mathematicians! The art style and illustrations was absolutely amazing and fit perfectly with the storytelling! Overall, an amazing video! Thanks TH-cam Algorithm! :D
I wholeheartedly believe that giving context to the history and slowly guiding students through the mindset of mathematicians is objectively better than spoon-feeding them equations.
exactly
Not just mathematics
My Algebra teacher in college used to tell us stories like that and I remember him telling us this one too. He later went on to become the minister of education in my country.
plus 1
We had this in every chapter's explanation in our books, a large one ~one page paragraph explaining the history or the person behind the concept, unfortunately it was never used
This video makes me want to do math. It’s inspiring in the best way
Totally agreed!
It hurts my brain
Math
@Faizan & Stuff wdym jk it’s not a joke he is a failure he didn’t know even in preschool at most
Legends know the original title was "This problem broke math(and led to quantum science)
As someone who's really bad with math, these visuals have helped me realize a lot of what I didn't understand with basic algebra and trig functions from school as a kid.
It's hard to teach math concepts in a memorable way. But that's what Derek does best!
Schools convert man to a learning engine so that he will grasp everything that makes humans worse than google . But human brain is much more than just a fact storage device.
me too! its actually simple to see that this way
@@chanderparkash4537 not a "learning engine" at all
@@austinhernandez2716 to some extent
"Only by giving up maths' connection to reality could it guide us to a deeper truth about how the universe works." Bravo!
A thoroughly professional presentation from algebraic dependence on visual geometry through Mediterranean ego vignettes segueing into physics, with remarkable insights along the way, culminating in the quote above.
All throughout grade school and college I struggled to understand the "why" portion of math beyond plug and chug. Usually professors couldn't give me an adequate explanation. Completing the square was one term that never really clicked for me. The first 3 minutes of this video are pure genius. So simple and understandable. This makes math so much more digestible.
@@bobmanbob341 meaning?
@@Siso_Mnguni bruh if u look around it’s full of absolutely random comments. I think there must be some kind of bot at work
Exactly the same for me. Incredible video.
3blue1brown did a video series on calculus in an equally visual way that helped me understand it better than any teacher in high school ever could. This has been the pattern of my learning after school in general tbh, the internet has been the most valuable resource in my learning journey.
If you'd like a book that focuses on visuals, I would recommend "Proof without words" - my professor recommended it to me recently & gotta say, it's a fun gift for those who like picture books.
History of mathematics should be taught as early as in middle school, and this video tells exactly the reason why it would immensely help students appreciate what they are taught.
History of everything should be taught.. otherwise the new student must do what literary fiction does.. cause suspension of disbelief. In other words.. believe in magic..
@@Sierrahtl This video is about mathematics, hence the my comment.
My 8 year old (still 3 years away from middle school) understood *just* enough of this video that I'd have to agree.
It should, but it isn't and it won't. Schools are more concerned about your kid's dress code violations and football than to teach them anything useful. And if they are close to flunking, there's always the Army to set them straight ( in all manners).
Borrrrriiiinnnnnggggggg. History is boring to teens, they do not care about the past, more about the future, or even more the present, for most anyway.
For the entirety of my higher education, I've been told to "complete the square," but 6 teachers and 4 professors have never explained this further than restating the equation. In one extremely brief visual and explanation, you've managed to answer a question I'd long since forgotten. I don't know how to describe my astonishment, nor my gratitude for your content.
No kidding! I recall that math was usually taught by coaches reading from a book. I had the same reaction as you to the "complete the square" part of the video. I could have been so much smarter had I only had math teachers with this gentleman's style...
Mathematics needs a James Burke. This video is totally on that path... talking about secrets, jobs, politics, challenges, motivation, compromises and re-derivation of formulas.
This comment right there.
@@allmotion_video_channel5434 whether would it make you smarter depends on what aspect are you talking about. if we are only talking about taking test and exam, explaining the equation to you wouldn't make you smarter in doing in math classes. It mostly depends on how much practise questions you've done. The same philosophy can also apply to college.
Surely, however, explaining the meaning behind mathematical equations and practices would help you have a better understanding of the world. Though it have little use in real life since mathematics are mainly used in real world as a tool to solve, not a tool to understand.
@@jimmyli319 interesting outlook on the subject you have. I am a flight instructor and have seen how different people learn in different ways. For me, I like to understand the underlying theory. That helps me judge whether the results of a “solution”makes sense. I also have beginning engineers that I work with that will just plug numbers in to a program like “MATLAB” but do not understand the fundamentals well enough to judge whether the “answer” is reasonable enough to be correct.
But, in your context, it is valid that one does not need to know how an internal combustion engine works just to drive a car…
I was one of the worst math graduates in my highschool class but recently I had a spark of love for maths and reteach myself everything. This video is nothing short of amazing. Its just mindblowing!!
Next brilliant video: "Your Daily Equation #2: Time Dilation" by Brian Greene. Only pythagoras and basic algebra needed. But, for bonus, you can try to find the unit circle that links time dilation with speed
Imagine minding your own business as a mathematician and suddenly someone challenges you to MATH DUEL, that can make you lose your job. Man, the older times were really intense for mathematicians.
😂😂😂
Blue Eyed W- erm, sorry wrong series 😂😂😂😂👏
Intense for everyone tbh
Imagine minding your own business as the Burger King and someone knees you in the stomach.
Its time to DUEEEEEEELLLLL
This REALLY brought an eye opener to my "how the heck did they figure this out" during math classes. Awesome explanation. Thanks
well said!
Man, change "depressed quadratic" to an obscure magic spell and you literally get a fantasy duel story, complete with a sage old mentor, an underdog protagonist, an enchantment and a boastful proud villain wtf
It is magic to people of the era. Guys with beards fight in a duel to the death. They use secret formulas as spells.
Frfr
Stories of wizards and stuff are probably inspired directly by mathematicians.
So of all those T-shirts comparing some profession to wizardry (you see it for engineers or IT a lot), the most legitimate claim is for the mathematician imo.
no idea how on the same page we are, I was coming up with a whole revolutionary way to look at insanity based magic systems too
All without eating a single pizza. Incredible.
Superb in everyway. This is how mathematics should be taught. You deserve a prestigious award.
I love it when complex equations come down to something elementary like 2+2=4
I don't know. It usually means I have chosen the wrong approach and lost the x.
same
Sums up my higher school years with math
That is what its supposed to be in the first place we are doing a top down calculation to simplify it to our understanding. The complex structure is still the answer regardless of whether it is solved (simplified) or not so we are not trying to solve it but make it understandable for ourselves, which means making it elementary
@Pradeep Singh I think your a key is dying
A History, Math and Science smoothie blended to perfection. Well done 👏
Would it be too random to declare my intend to recommend
my fellow science-youtuber-fans some... well... more science-youtuber?
I mean, in my mind, it just makes sense, but many call me B0t, so... your choice...
Complex numbers are dual to real numbers.
Perpendicularity or orthogonality = DUALITY!
Column vectors are dual to row vectors -- group theory.
Electro is dual to magnetic -- Maxwell's equations.
The electric field is perpendicular (dual) to the magnetic field -- probability waves.
Positive charge is dual to negative charge -- electric fields.
North poles are dual to south poles -- magnetic fields.
Electro-magnetic energy or photons are dual.
Points are dual to lines -- the principle of duality in geometry.
Group theory:- the image is a copy, equivalent or dual to the factor or quotient group.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Homo is dual to hetero, same is dual to different.
Injective is dual to surjective synthesizes bijective or isomorphism.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger.
Duality (energy) creates reality.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force.
Monads are units of force -- Gottfried Wilhelm Leibnitz.
Monads are units of force which are dual -- monads are dual.
Energy = force * distance.
If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
"Always two there are" -- Yoda.
Yummy
@@hyperduality2838 stroke
@@nathanlevesque7812 Asinine!
Duality allows you to create new laws of physics:-
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Making predictions to track targets and goals (objectives) is a syntropic process -- teleological.
Complex numbers are actually dual numbers -- the complex plane.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
Duality is everywhere if you look for it, male is dual to female.
"I did not deem him capable of finding such a rule on his own." Savage 😂
I know, right?
Tartalia was a beast. He had no chill. Ended the guy's career in 2 HOURS for something that was supposed to take 1.5 months.
Savage i tell you! Savage!
lol
though anyone can reach the level of a genius, it' only a matter of decades
I thought I was just going to browse the video but here am i going through it all and even rewinding. Thanks it was very engaging and brilliantly undertaken.
I rewinded too. It was a great video.
I’m 31 years old and remember learning about imaginary numbers but never taught why they exist, or what examples there were in nature. I absolutely loved this video and thank you for making it. It reminds me that things you thought were useless info in school have an immense impact on the collective knowledge of humanity.
This is a common problem in school, especially math. You are not taught why, just how to apply it (and sometimes you don't even get taught when to apply it).
I lost interest in mathematics after never being told why we use such things as imaginary numbers and their applications. After years of doing math problems you get to a point of questioning why you are doing all of this with no context of application. I think this is a real problem with how math is taught, the student is never explained the relevance of what they are actually doing in terms of the real world. It is sad that math is taught like a menial task to be performed until the correct solution is found yet you don't understand why you are doing it.
@@Leaptab The problem is, the field of mathematics in the pure sense indeed is never concerned with the application of math itself. You will notice to in this video, that the "invention" of imaginary number theory far predate the application of the imaginary number itself in real life. If you find math to be menial task just because you don't know the application in the real world, chances are that will be the same feeling you'll get if you ever dig deeper into the field of math in pure math research.
@@Leaptab You're conflating the application with the subject. What you're saying is "Why do I learn a physics concept instead of an engineering concept?" and it really misses the point.
Schools need to do better about teaching the history of this stuff
This is exactly how you teach people the aesthetics of something.
The beauty, the thing which motivates people.
I agree; however, I feel that you can never teach others: you can only motivate others to dive deeper into the material themselves, and this is a great video that fires up people’s interest to do exactly that.
@@MrFrazerz Yeah, especially with pure math being mostly proofs a lot of them non-constructive, this medium would mostly be unavailable.
This is how students should be taught in schools and colleges. You are a perfect teacher.
Nahh i assure you, dumb and lazy students will always be like that. They would dismiss the historical story and ignore the illustration of solving an equation using geometry.
@@vandalm9311 it's about those who are really curious to know about the core concepts and essance of science. Here neither the dumb nor the intelligent has access to this quality of education.
He probably makes 10X a teacher's salary as a TH-cam celebrity.
Exactly! Motivate the students with the lore rather than just throwing numbers at them and expecting well performance.
Ah yes, just spend hundreds of hours on a video for a 1 hour lesson. I'm sure that will be highly efficient. Tell me you've never taught before without telling me.
This video was incredible, I cannot put into words the fantastic journey I experienced in these last few minutes, thinking about the realities of mathematicians, how problems that have been considered to be impossible for thousands of years are solved, and how we naturalize the legacy of these incredible minds. Thanks my friend
Dude makes math sound absolutely riveting... Incredible
Math is riveting if you don't know math. If you actually learned at least high school math and some history behind it (like how was calculus developed), you'd know that it is riveting.
I dont understand the Mathias he is doing but im interested
( colé izzy o/ ) the thing that i take from this is just how insane being a mathematician this day and age problaby is when you're needed to know all prior knowledge at the same time that you need to challenge it to find new questions and keep progressing mathematics towards the future.
Good job on getting 600 k subs
Yeah, I was going to say, what’s sad 😞 is that you didn’t find it riveting to begin with as it literally is the language we use to describe the reality of our universe. But to each his own and I don’t blame you.
Instead of letter grades A through D, 8th graders should get a grade placement based on which century of Italian mathematics they most closely align with.
Guys, we can reform standardized testing now! We found the perfect scoring system!
Congratulations! You have scored "Roman Republic" in Math!
If you think 8th graders are learning about imaginary numbers, solving cubic equations, or quadratics for that matter you either don't remember primary school or were an exceptionally gifted child. My guess is the concept of a variables is introduced in 7 or 8th grade, probably putting 8th grades some where in the dark ages. Probably where they belong from what I've seen , haha.
@@901blitz Variables are taught in 5th grade my guy. We live in a very different world than the one we grew up in.
@@901blitz yeah variables are taught in the middle of primary school and in 8th you learn algebra 1 which definitely has quadratics, imaginary numbers, etc. im in 12th grade right now taking calculus using the prior knowledge from middle school with graphs and algebra needed for complicated derivatives
I can't believe that now, a decade after struggling to understand it, I finally know what "completing the square" means.
congrats!
Not just you. Even some teachers don't know what it means they just memorize the process.
I don't think the teachers knew either, they were just cutting off the tip of the ham because everybody else had done it before them
my thoughts exactly. I always wondered why it was called that
It’s always funny that the answer to why something is called what it is called, is often right in front of you, but (in the case of completing the square) due to how math is taught in school it often comes off as nonsense.
This is my favorite math history episode ever! I love the "cubic battle" and the invention of "imaginary numbers" so much, thanks for making it even more interesting with your narrative.
If you take the time to understand the *why* in maths, it can take you a long way
This comment will blow up soon
Maybe idk
Generations, Prolly
This comment will indeed blow up, now we shall wait…
Math is the only subject which seeks absolute truth. The WHY will indeed take most amount of time here
Most of your work is educational yet highly entertaining but this particular video deserves an award. One of my favorite channels on the platform. Proud to have subscribed to it over 10 years ago.
Complex numbers are dual to real numbers.
Perpendicularity or orthogonality = DUALITY!
Column vectors are dual to row vectors -- group theory.
Electro is dual to magnetic -- Maxwell's equations.
The electric field is perpendicular (dual) to the magnetic field -- probability waves.
Positive charge is dual to negative charge -- electric fields.
North poles are dual to south poles -- magnetic fields.
Electro-magnetic energy or photons are dual.
Points are dual to lines -- the principle of duality in geometry.
Group theory:- the image is a copy, equivalent or dual to the factor or quotient group.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Homo is dual to hetero, same is dual to different.
Injective is dual to surjective synthesizes bijective or isomorphism.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger.
Duality (energy) creates reality.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force.
Monads are units of force -- Gottfried Wilhelm Leibnitz.
Monads are units of force which are dual -- monads are dual.
Energy = force * distance.
If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
"Always two there are" -- Yoda.
@@hyperduality2838 brain worms
When things were at their very worst:
2 Suns, Cross in the sky, 2 comets will collide = don`t be afraid - repent, accept Lord`s Hand of Mercy.
Scientists will say it was a global illusion.
Beware - Jesus will never walk in flesh again.
After WW3 - rise of the “ man of peace“ from the East = Antichrist - the most powerful, popular, charismatic and influential leader of all time. Many miracles will be attributed to him. He will imitate Jesus in every conceivable way.
Don`t trust „pope“ Francis = the False Prophet
- will seem to rise from the dead
- will unite all Christian Churches and all Religions as one.
One World Religion = the seat of the Antichrist.
Benedict XVI is the last true pope - will be accused of a crime of which he is totally innocent.
"Arab uprising will spark global unrest - Italy will trigger fall out"
"The time for the schism in the Church is almost here and you must get prepared now."
The Book of Truth.
Agreed. Veritasium is always a must watch. Every video is entertaining and educational, the best of what TH-cam is about.
Totally award worthy content
It's just wonderful to see how he is explaining math, physics and chemistry with such ease
I have to rewind the video a couple of time to get it
@@tonyng3285 at least you are taking an effort and finally coming to the understanding. :)
SUCH EASE????
Pretty easy for me, though I lack high knowledge I get the spirit.
Absolutely. But he should definitely get some more sleep. ;-)
Seriously, as an Iraqi student, I have been studying complex numbers for a month and a half, and after all of this, I do not know why “i” is written with “y” this intuitive piece of information, loves god what teaching have I been learning throughout this period? Really, thank you, god bless you
I’m a physics major. I’ve always had trouble understanding complex numbers and why they exist in equations. It’s like my professors were just handing out the equations like the Schrodinger equation without really explaining what they mean. As I went on throughout college I gathered an understanding, but this video gave me that “aha!” moment. Thank you Veritasium, Your videos are something special and I appreciate every single one that gives me more insight on how the universe works.
yes he is a good "teacher", knows how to explain and visualize things. In math one would learn that "things" exist as parts of other things, real numbers are a subset of complex numbers. And then there's quaternions ... and one would think, does it ever stop... and yes it does, but that means walking into set theory and such, everything is kinda 'connected'. I worked a lot with physicists, they always came to me "to pick my brain".
I majored in math and I always got that impression from the courses taught by physics profs. They seem to think of math as a set of tools that "just work". If you ever want to unravel those tools and figure out why they work, the field of math you are looking for is called "analysis" (probably "complex analysis" and "real analysis" in most universities). Those were always my favorite courses because they helped so much to explain all of those weird formulas from calculus and differential equations.
"i" mean imaginery ≈ imatter ≈ dark matter ??
Math came from nature phenomenon and "i" explainted it.
This comment got a lot longer than I initially intended, but covers a number of things about constructions of the complex numbers and how else they can be thought of:
As a pure mathematics student, I like to think of imaginary numbers as a construction. They are not “numbers” in the same sense as real numbers, but they can be paired with real numbers to produce a helpful construction which allows people to manipulate things in ways that may not initially seem possible. In the context of ring theory, we consider general systems of numbers with addition and multiplication. You can add and multiply polynomials with coefficients from any given ring as one might expect. And it turns out that a construction basically the same as imaginary numbers appears when you “quotient” the polynomial ring of the real numbers by “the ideal generated by” X^2 + 1. If you’re not familiar with this language, that basically means that if you take this ring, but now consider that whenever X^2 + 1 appears (or any of its multiples), it is now considered to be 0. You can see that this can produce the complex numbers intuitively since we are essentially just treating X as a number which squares to -1. Which is exactly equivalent to this idea behind the complex numbers: allowing the number i to exist and square to -1. All the properties of rotation then naturally appear through all the classical studies of complex numbers.
However, a possibly interesting idea from this is that if instead someone had decided that they wanted a *different* cube root of 1, let’s call it j. Then j^3 = 1, and (-j + 1)^3 = -1 + 3j^2 - 3j + 1 = 3j^2 - 3j. But noting that j^2 = 1/j (and allowing some algebraic manipulation with some extra assumptions), we can find that (-j + 1)^3 = 1 as well (which matches what you’d get if you treated j as either of the complex cube roots of unity). I think we can all be pleased that no one did find this, because working with this number is a lot more tedious than working with complex numbers as we know them, but it does work.
From the ring theoretic perspective, for this construction we wouldn’t set X^3 + 1 = 0, but instead we’d use X^2 - X + 1 = 0 which is (X^3 + 1)/(X - 1). This is basically so that we don’t have worries about X actually being 1. Obviously this comment misses out on a lot of rigour, but is intended to provide the general intuition behind the fields mentioned and give an alternative perspective of complex numbers.
Complex numbers are pretty cool once you get really into the theory on it. Like it’s applications and stuff is cool but things like reimanns hypothesis or schrodingers equation are very interesting
The phrase "completing the square" makes much more sense now. Holy crap my mind is blown. I really wish math was taught like this. I thought I hated math but I'm finding that isn't actually the case when I learn through mediums such as TH-cam. Does anyone have any suggestions for other videos that combine math and history like this one?
Try searching for "History of Science". A guy in my college dormitory actually goat a degree in this field.
3blue2brown
@@cachecollin6984 4blue1brown?
@@camgere goat a degree?
@@fatitankeris6327 History has many fields. History of England, History of the 16th Century, History of Agriculture and yes, History of Science. Did you say GOAT? Shemakhinskaya Bayaderka Festival / Yana Kremneva / 201. Science GOAT. James Burke Connections, Ep. 4 "Faith in Numbers". From 1978, pre-internet. I actually used Hollerith cards (punch cards) to write my first computer programs in Algol inn 1973.
HS math teacher here: thanks for showing the weird and cool the history of the equations, and visually describing how they all relate back to basic geometric shapes (even when they then veer off into the imaginary land). Definitely borrowing this for class.
Showing the geometric interpretation of completing the square is a must. Please teach this to your students.
Complex numbers are dual to real numbers.
Perpendicularity or orthogonality = DUALITY!
Column vectors are dual to row vectors -- group theory.
Electro is dual to magnetic -- Maxwell's equations.
The electric field is perpendicular (dual) to the magnetic field -- probability waves.
Positive charge is dual to negative charge -- electric fields.
North poles are dual to south poles -- magnetic fields.
Electro-magnetic energy or photons are dual.
Points are dual to lines -- the principle of duality in geometry.
Group theory:- the image is a copy, equivalent or dual to the factor or quotient group.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Homo is dual to hetero, same is dual to different.
Injective is dual to surjective synthesizes bijective or isomorphism.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger.
Duality (energy) creates reality.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force.
Monads are units of force -- Gottfried Wilhelm Leibnitz.
Monads are units of force which are dual -- monads are dual.
Energy = force * distance.
If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
"Always two there are" -- Yoda.
I wish I had this, I failed math one, then it built on this until geometry. I failed all three classes, it might not have been as bad if I understood what I was looking at, and this did that for me, at least better than I had it before. Hopefully your students get that out of it too
@@hyperduality2838 drugs
@@hyperduality2838 The first statement is completely false. The real numbers are contained in the complex numbers. You're thinking of imaginary numbers. Complex numbers and imaginary numbers are different things.
As a student I used to have a mindset that Math cannot be imagined and thus it is useless at higher levels compared to physics and chemistry which have practical use. After watching this video, i was enlightened!
Just had my mind blown learning that "complete the square" is literal.
Even after learning it in high school, it still sometimes blow my mind with how much sense it makes
I wish geometry was focused on more in schooling.
@@Trowa71 I hope schools now a days show these videos. I didn't pay attention at all in school, but now, I find myself deeply enthralled by it.
Lol I was thinking the same thing.
@@tyzxcj34 You commented this while I was watching the video.
I love how this video displays how difficult advancing mathematics can actually be, and the sheer imagination required to conceptualize another dimension and it’s properties when you contemporary mathematics has no answers for what’s going on and will treat the discovery as fiction. Some people think math is all logic and that the ancients were fools for not knowing what HS freshmen know today. They fail to understand what the process of advancing a field of knowledge is actually like.
Breakthroughs can take an unreasonable amount of time. But once it occurs, it can be passed on. I completely agree with you and this is why we have to honor the greats. They made it easier to push the limits just a tad bit further. It just takes that one eureka moment.
Crazy to think it takes one hell of an imagination to see and understand reality... lol
Anyone who thinks that way is not that intelligent. Why would someone think that people before us, with less access to knowledge, had less problem solving skills? It would be like claiming the person who figured out that rubbing two sticks together to make fire, was stupid. Sure we all know that now, but i dont think there is anyone in the comment section, that could figure this out, if they were raised by wolves. Knowledge is nurture not nature.
Math is all logic. Logic is what tells you that something is a mathematical statement or just conjecture. Also, it's not so much sheer imagination to come to work with imaginary numbers. Imaginary numbers and complex numbers are fundamentally the number system of algebra, and inherently come out of polynomial solutions. All that was needed was a pen, paper and the ability to not reject something simply because it doesn't immediately make sense.
I’m a high school math teacher and I created a worksheet to go along with this video last year and showed it in my Algebra classes. I just got done showing it again this week while I was preparing my students for our unit on Complex Numbers and the imaginary number. I show a lot of videos from this channel in my class as they’re both educational and very fun.
Thanks for the great content.❤
I'm a student who has a lot of trouble in school - You sound like a great teacher. Teachers who don't hesitate to relate to students with modern forms of education are ALWAYS the best. Learning in the modern age is *still* so underutilized in schools, American schools anyway. We have the world at our fingertips and we still use outdated textbooks from 30 years ago?Come on. Just saying, I think it's great Veritasium is being used as a source of education. Keep it up and good luck, Mr. Brown :)
You should put your worksheet on TpT!
I don't get how people use math in everyday life. I don't see people use it. I am sorry. I don't want to offend you. I hope I didn't.
@@GongGirl-ie5wy You are right that most people don't use higher math in everyday life. However being able to do it proves that you can think and tackle hard problems. For me, that has been incredibly valuable in my career because my employers seek me out and pay me not because I'm doing X hours per week, but that I'm differentiated by my ideas, thinking, and problem solving. That's allowed me to get really far ahead!
Read your bible! (KJV, preferably) ♥
As someone who really struggled with math all throughout school, and still to this day struggles with it, I wish they would have taught this kind of stuff in school. I started watching these videos not too long ago and it's sparked a passion for the beauty that is mathematics, and inspired me to learn more about their fundamentals. Thank you to Veritasium, and thank you to all the teachers I had who wanted to teach me things like this, but were unable to because of strict, unchanging curriculum.
blows my mind how these guys figured out math. Studying math today it feels like everything is pretty much figured out. I guess you need to be at an incredible level to figure out what does not yet exist.
I'm sure they thought the same way back then hahaha
On the contrary Victor; you can quickly catch up as the hard work has been done leaving you with advancement.
The only hiccup is when an axiom turns out to be false; which is only discovered when it is pushed to the limits.
Yeah I can't imagine that level of comprehension to the subject.
Somewhere out there in this world.
Someone again made a huge mathematical, or physics discovery, but didn't think much about it because they thought other physicist already considered it but didn't publish because it's wrong.
If back in the past, people kept great discoveries secret, now it could be possible that someone out there made a discovery but isn't confident about their own ability to explain it.
I mean, we already got the internet.
If a person thought of something, they could search the internet for answers, or to confirm something. If it doesn't show up, it may be because it was a stupid idea that noone ever considered it, or it may have not existed considering them to be the first to come up with it.
It is possible that the latter could happen. It's not such a bad situation though because some other people could just come up with the same idea, until one actually is confident enough to go public with it. Same with the story in this vid.
Related to complex numbers is a mathematical problem called the Reimann Hypothesis....
It's widely considered to be the hardest mathematical problem.
It's one of the 7 millennium problems, of which only one has been solved to date.
Solving any of these problems would be revolutionary and would win the solver a prize of a million dollars
Some poeple like to joke saying "There are much easier ways to earn a million dolllars than to solve the reimann hypothesis"
Very intresting stuff IMO
Like a lot of people on here, when I took advanced maths in school and was shown imaginary numbers, the course material made no attempt to describe the fundamentals of how imaginary numbers work, or why. This video in 15 mins made more sense than a year of schooling. Having this globally available on youtube is a gift to humanity
actually I feel like this video is worth more than 3 years of high school
@@kienthanhle6230 I'm in my 3rd year of Electrical engineering. One of the fields that uses imaginary number the most. Now I actually understand what it means. I finally get why e^x and cos(x) and sin(x) are related.
@@kienthanhle6230 That's BC highschool only prepares us to take and pass tests. Fundamentals and true understandings are never required. BC schools only worry about average grades and test results for funding.
@@kienthanhle6230 Wow you actually said something this cringy
@@BygoneT I don't know how good your teacher at high school is, but mine is pretty bad.
I swear, if more of my math classes were like this and explained the "why" behind the concepts, the content would've been much easier to grasp.
That would require _good_ math teachers, though. You only get those at the graduate-school level, because that's where they all end up. The teachers who end up teaching algebra, geometry, and trigonometry in grade-school are the people who just barely graduated with their math degrees -- they aren't qualified to teach anything harder than "x + 5 = 10" or "sohcahtoa". Whereas anything _simpler_ than algebra, geometry, and trigonometry doesn't require a math degree to teach it _at all,_ so there's nowhere lower for those bottom-tier math teachers to go. So _every single kid's_ first introduction to math that requires _actual thinking_ (instead of memorizing tables) is with a math teacher who, objectively, sucks at math.
If so then we wouldn't have enough teachers
@Hypnotize: Honestly most grade-school teachers are too burned out from grading homework until midnight and paying for classroom supplies with their own money, and aren't looking for even more ways to make their lives more complicated.
If you have a good real-world example and explanation for a specific math concept, pass it along to your kid's math teacher. It will help them.
not "easier" " interesting" is the correct word
I mean... I've had math classes where we had to read about the history behind things like this, and I found it uttermost boring and useless. I've also had classes where we had to read some of the proof for the new concept/rule we were learning. This was sometimes boring, sometimes helpful. But, sadly, sometimes there isn't a way to showcase the proof, or at least not on a high school level, so you just have to accept it. Which sometimes suck, but you get used to it. XD
(damn, I said sometimes a lot.)
Thanks…. I got stuck and quit math. I crushed geometry and sucked at algebra. In this video at 3:48 I felt my brain click, like a dislocated shoulder snapping back in place. Now, I’m pretty excited. Thanks again.
One can't possibly overestimate the amount of work that has gone into producing this amazing video.
@Melon Husk Yes. Fixed.
@@hckoenig no no no, I think you mean "Understand". Overestimate means praising the capability of a person or something that doesn't even have the ability to do that. Underestimate means doubting the capability of something or someone. Understand is knowing how something works.
@Pradeep Singh Dear Mr Singh, I kindly advise you to go and find the book "My Big Toe" by Thomas Campbell. It will explain a whole lot more as well to you.
Have a nice day.
It seens you underestimated my power!
8574694746473853 universes and a singular rubber ducky probably went into the production
One can't possibly overestimate the amount of work that has gone into producing the subject mathematics
This was a fascinating insight into the origins of the mathematics that's so familiar. Wonderful. Thanks Derek!
Cool
Hey guys this is Derek from more plates more dates
Yeah, video was really interesting
*VERITASIUM* is my inspiration!! My mom said that if I got 30k subscribers!! She definitely buy me a professional mic!! *begging you GUYS alot* literally begging.!.
Also isn't it surprising no nobel prize in mathematics and it is continuing, the members of nobel committee should announce that nobel prize should also be given to mathematicians for their work
It's shocking how thoroughly you managed to deceive me into thinking I almost understood this topic. You, sir, are phenomenal!
lmaoo same
He was like "even kid can do this" and pull into abyss call math
@@b0mby1 Although your word made me feel insulted, guess it cant be helped. Well you see. First of all,I dont talk english. Second, I know what he talk about when he talk about that cubic stuff. Third, I understand all of this except that last part where he start do some reality bending edit, turning 2d into 3d before turning it into 2d again. The only thing I need to watch back is the part the -5 cube being introduce. That all.
@@unknownman5090did the comment get deleted?
@@ayuballena8217 I think so. He said something like im not good at math and something like that, which I agree. Im not math genius. However, his sentence feel like trying to say that im dumb, which is true, but I dont think he have good meaning behind it
Please separate these videos from others and make a separate playlist for mathematics. It is fun understanding maths in this fashion ❤.
The part where he explains how to solve the equation with literal visuals in my opinion should be taught in schools. It helps people grasp the concept much more easier.
I too think it should be shown, but only later on. I was shown the geometric version of the Pythagorean theorem and it didn't click until it was shown to me later on
I mean, the whole idea of it makes negative numbers literally impossible, so that might not be the best way to teach math.
2d variant is taught in schools. It is called the Pythagorean theorem. And all sqare equations could be transformed into a perferct square.
Well its quite similar to algebra however alot more understandable.But anyways there is no easy way to show maths.
Much easier*. No need to add “more”.
Math teachers, please, please, show this kind of stuff during class. It would've changed my life.
While I agree that this video is very well done and engaging, the moment that your teachers would have made you start solving equations even after showing you this video, you would've get bored of maths again.
@@pitthepig good point.
I did and they said.... it was boring.
@@kevinbugusky7416 Kids don't want to learn or think anymore.
Wait, they don't teach that everywhere?
Was expecting cool math, didn’t expect the crazy history story, but it was my favorite part:D
Meanwhile 10 million people die from preventable cancer every single year and not a peep from the press or any politician. Solve that unsolvable problem.
@@TheFirstBubbaBong population control. They don’t want the USA to turn into India.
@@TheFirstBubbaBong Not the place to discuss that, you will attract extremely biased opinions
i loved the animations with it too!
@@TheFirstBubbaBong What have you done in that regard?
I just started my higher mathematics, got stuck in the Depressed Equations, and why to substitute x ---> x - b/3a , couldn't find any proper explanation , later suddenly unknowingly decided to watch this video , and without my known , I got everything I was looking for !
Man , this is how maths should've been taught , without just mugging up results , algorithms and solutions 🙌🏻
You're a genius Sir !
Everything that I learned in high school suddenly has an image for itself, algebra suddenly can be visualised , Complex numbers have suddenly so much meaning in my head. This video is just mind blowing!! Way to good!!
If you’re really curious about understanding the true realit of complex numbers, I highly recommend the “imaginary numbers are real” series by the channel Welch Labs.
An amazing explanation of just exactly how complex numbers apply to the real world.
@@connorcoultas9629 Hey Connor, thanks for the suggestion. I'll be sure to check it out! Would love to know more!!
Imho, this is how mathematics should be taught. Rather than throwing the conclusion and have students arbitrarily plugging in numbers; an introduction with a few key points in its discovery would go a long way to understand how one stems from another and create better understanding and thus foundation to the subject.
Complex numbers are dual to real numbers.
Perpendicularity or orthogonality = DUALITY!
Column vectors are dual to row vectors -- group theory.
Electro is dual to magnetic -- Maxwell's equations.
The electric field is perpendicular (dual) to the magnetic field -- probability waves.
Positive charge is dual to negative charge -- electric fields.
North poles are dual to south poles -- magnetic fields.
Electro-magnetic energy or photons are dual.
Points are dual to lines -- the principle of duality in geometry.
Group theory:- the image is a copy, equivalent or dual to the factor or quotient group.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Homo is dual to hetero, same is dual to different.
Injective is dual to surjective synthesizes bijective or isomorphism.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger.
Duality (energy) creates reality.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force.
Monads are units of force -- Gottfried Wilhelm Leibnitz.
Monads are units of force which are dual -- monads are dual.
Energy = force * distance.
If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
"Always two there are" -- Yoda.
@@hyperduality2838 lol this troll again
This level of animation deserves appreciation.
True
Is this manim like @3blue1brown?
and after that you have to plug the red wire into the socket to make sure the engine boots at launch. Wrap the green wire around it's coil that sits directly beside the A button. After you put the back shell on, place the battery in the slot. Screw the Vr26 Jeeper back up and press the reset button. If everything worked according to plan you're device should show a thumbs up sprite. Plug the HDMI port into a monitor and wait three seconds. If it boots up on TV your in the good side. If it doesn't boot in less then 5 seconds quickly unplug. This can severely damage your TV and possibly start a fire
@@pattyryopotybuttongamer3063 why… are you trying to teach us how to hotwire a car?
"Only by abandoning math’s connection to reality could we discover reality’s true nature." I cannot shake these words from my head.
That s because it is a poor statement to begin with.... it is the arrogance that blinds us.... in our quest for knowledge with each step forward, we stop to admire ourselves and claim that we now have acquired foundation for reality. Until we learn it is still out of our grasp.
@@francoisiannacci2615 go back to your fairy tale about a ghost that literally claims to be the greatest thing in the universe
@@Flaystray go back to your fairy tale world with triangle and infinite number?
we need to move on from old rules so we can seek further newer better ones.
we need to move on from our prior definition and understanding of what reality is, to seek new better definitions of reality!
in the end, it's OUR perception of reality.
@@francoisiannacci2615 I agree, there's always further to seek. standing on one discovery and thinking it's the end of the line, is such a toxic narcissistic thing to do.
this "lecture" is a real gem. I find it very, very inspirational for those students that area interested in knowing how the world works and how we found out about how to describe it.
I hope to convince my children two watch this when the right time comes. Thanks a lot.
If math had been explained to me like this in school I would have actually remembered it
They didn’t because teachers didn’t know , they were just looking to get paid on the 15th and 30th of each month, rarely you will find a true math teacher.
@@aaronovski9949 math teacher here. Reality is, we often cannot go too far beyond the math curriculum given to us. I plan on showing this to my students little by little, because if we did this instead of class I'd be in trouble
Math is still math tho
Not sure why YT has recommended me this vid. I am too dumb to understand this shite. Just gonna casually scroll pass and listen to CardiB WAP and watch Logan Paul.
Bye Ngl.
@@MrUssy101 I SCROLLED THROUGHT IT FOR 15 TIMES IT CANT BE GONE
This is the first time I've seen an explanation of imaginary numbers that made sense to my brain. The first time I ever had someone say that mathematics was disconnected from reality, and how it moved away from geometry. This just flipped something in my brain. I wish I had this 20 years ago when I was dealing with calculus!
More like, it's not math that's disconnected from reality, but human beings. The "reality" we construct in our heads is more real to us than actual reality.
Noob
Vaguely...
@@aks1993kumar lol
Yg8h8g has x the cg
This REALLY feels like a lost episode of the new Cosmos, from the pacing, the language used, the explanation strategy, the animations, to the deep dive into the history, making it all concrete. This is a masterpiece, and it might be the best video you've ever made.
I got that same exact feeling. This is TV quality.
This seems really over the top, Cosmos has extremely high production values for an educational program, and is a lot more accessible to general audiences.
This is great. Cosmos is garbage. Don't compare the two.
@@aguywithanopinion8912 cosmos is really good and so is this video. Wtf are you on about?
@@SahilP2648 what is cosmos?
Just found this video yesterday.
OMG what an amazing video, I have worked with 'i' a lot as an engineer, but was never taught where it came from.
Thanks :)
As a grade 12 just finishing the last few of my finals this week... this video has blown my mind
I NEVER understood why the method was called 'completing the square' until just now when you literally complete the square in front of me.
If only all teachers and textbook writers were as dedicated as you.
And I'm a 55 year old with an honurs degree in astrophysics and PhD in physics ... and it blew my mind too! :-)
This comment thread warms my heart.
Same here !
@@duncpott what
Literally lmao. Things I had to memorize are now, after watching a short video, intuitive and easy. Magic 🙉
The story of how cubic equations led to the invention of Imaginary Numbers was the favorite one that my school teacher has told us. This story was one of the reasons why I fell in love with Mathematics, and why I and my other classmates became engineers, physicists, scientists. But the storytelling quality and details of this video is on another level!
Actually it's because you're Asian
The history of math is always so fascinating. Lots of mathematical concepts seem so obvious now, but it’s almost impossible to get those connections without being taught about them. Amazing how so many brilliant people figured it all out over the centuries.
@@xxpatrick204xx must be because i never heard of multiple classmates getting jobs like engineer, physicists and scientist
@@babaloons4887 with a passionate, dedicated teacher, it shouldn't be _that_ hard to imagine it.
@@BrendonGreenNZL its hard to imagine a teacher like that because u can barely find them
A man who can hold the attention of millions without using memes, gifs, music, effects or silly entertainment is why this channel is so great at has over 10M subs.
Seeing the effort that went into these innovations and discoveries in math makes me feel ashamed for our lack of appreciation for the material we are taught in school.
I'm an engineer, I use maths as a tool and just wow this video blew my mind away and gives a new context to everything that I do now. Suddenly I am less inclined to believe that math is a man-made invention and more an intrinsic property of the universe and reality itself.
I've always been in the "invented" boat and I think this video reinforces that school of thought.
Math is a general-purpose tool for **description**. We have specifically tooled and optimized it to be able to **describe** literally any sort of relation. The world happens to be relational, we should be very careful not to marvel too much that our tool, which works for all relations, real or imagined, has anything at all to do with how reality works. The map, as they say, is not the territory.
I mean we basically discovered fractals and then observed they were everywhere in nature like the way branches grow from trees or cauliflower
true. I just use equations and algorithms like a mechanic would a hammer. It's so fascinating to see all this in its historical origins.
Meanwhile atheists don't see proof of God.
19:48 "So when you're multiplying by i, what you're really doing is rotating by ninety degrees on the complex plane."
Oh my gosh that was brilliant. So very well done. I had to stop right there to leave this compliment on a great job!
mindblown
Same as multiplying by -1 but with intermediary steps.
Hell yes that's cool- teaching my students that this week...to see their faces light up made my school year.
You should look into the polar coordinate system in the complex number field. It basically uses a similar system. Instead of an X/Y coordinate system (where X is the real number and Y imaginary) it uses a magnitude and some rotation.
Complex numbers are just great. You basically do a "Let's assume that i is a number, and that its square is -1" exercise. You get all the basic maths really fast. For example:
Let's assume that (a + b i) and (c + d i) are complex numbers, and a, b, c, d are all real numbers. Then,
(a + b i) + (c + d i) = (a + c) + (b i + d i) = ((a+c) + (b+d) i),
(a + b i) - (c + d i) = (a - c) + (b i - d i) = ((a-c) + (b-d) i),
(a + b i) (c + d i) = (a c) + (a d i) + (b c i) + (b d i²) = (a c - b d) + (b c + a d)i.
Division is a bit more complicated, but you start with a twist: try multiplying (a + b i) by (a - b i):
(a + b i) (a - b i) = a² - a b i + a b i - b² i² = a²+b², a real number. So,
(a + b i) (a - b i) / (a²+b²) = 1, as long as the denominator isn't zero.
In other words, the inverse of (a + b i) is (a - b i) / (a²+b²). (That's a division by a _real number,_ or a multiplication by 1/(a²+b²), so no problem there.) To get complex division, you merely have to multiply the inverse of the second term by the first.
i love how there’s this torch that’s passed along from generation to generation of geniuses to figure out crazy things and eventually someone in the future solves a problem from the work of all that came before them.
@DON'T ok I won’t
The history of all human achievement has been written this way. Unfortunately, the efforts of hundreds, thousands even, are often attributed to a single person.
"I stand in the shoulders of giants"
Such is the power of written language.
If I were a HS Algebra teacher, when we hit imaginary numbers we would definitely be watching this video in class. It might be the best how and why video I've ever seen on this subject.
Brilliant research and presentation! I had no idea complex numbers arose from the solution to cubic equations. Things make so much more sense when you explain the path of their discovery.
Which most of things are explained that way in Wikipedia (history).. it just more convenient and easier to understand it from video tho😅
My math teacher told us it was for quadratics lol
Theres nothing "real" or "fundamental about complex numbers. As a matter of fact, for a long time they were met with skepticism from mathematicians because its nonsensical to try to give them any inherent meaning. As Gauss, personally responsible for making complex numbers acceptable, explicitly explains, complex numbers are basically nothing but planar vectors and they make two variable real calculations more compact
@@Someone-ig7we It is found in quadratics too. But quadratics can find solutions without the imaginary unit showing up, and for quadratics where there were no real solutions, mathematicians could just dismiss those as not being solvable. Further, and more importantly, deriving a general solution for quadratics does not utilize an imaginary number, but for cubic equations, imaginary units are a necessary intermediary step for deriving a general formula. So while imaginary numbers show up in some quadratics, they're a necessary feature for getting the general solution to cubics.
@@hamidrezamahmoudian2710 False. Complex numbers are the fundamental number system of algebra. To understand this process, you need to first look at how we get integers from real numbers and addition by way of looking at additive inverses. Then look at the way we get rational numbers by look at multiplicative inverses of integers. Then look at how we get algebraic numbers from those, and you'll come to see that in the end, complex numbers are where we end up from the natural operations and their inverses. Complex numbers can be represented through other means (but so can any number system), but complex numbers are both real and fundamental as much as any other number is. The fundamental theorem of algebra should be the first thing you look into.
Yes, they are planar vectors, but they also possess a unique behavior under multiplication, and it's this behavior that shows they're more fundamental than the biased view of them you're presenting. In order to represent them as vectors, you need to introduce a special operation for multiplication that's derived from the behavior of how sqrt(-1) acts under our typical understanding of multiplication. This new vector multiplication, distinct from any other multiplicative operations on vectors, cannot be derived from pairs of real numbers alone. It can only be derived from observing how the behavior of sqrt(-1) behaves under our traditional concept of multiplication. Since you cannot derive the behavior of complex numbers simply from pairs of real numbers, it follows that they are not "basically nothing but planar vectors that make two variable real calculations more compact" because it dismisses the entire field of complex dynamics, which only follows from the how sqrt(-1) behaves under multiplication.
This sort of took me back to when I used to enjoy learning mathematics from a teacher who always prefaced each lesson with the story of the history of the origins of each topic.
Same here
engineer here. in my schooling, i was never shown that i correlation on e^iX. Imaginary numbers were so confusing to me, but i could plug and chug formulas for them. Now, 10 years later, seeing that description of how it combines sin and cos into 3 dimensions is simply astounding. I never understood the why of imaginary numbers so i couldn't visualize it properly in my brain.
@@Perceptious37 Same here and I had trouble with the wave equation too because it departed fully from anything I could use as an analogy in my mind. When a thing becomes just spade work then I lose all mental interest. I needed to actually grasp what it meant in my mind or chugging equations just made my brain hang up.
The guy who taught Schrödinger was first generation German with the thick accent and, combined with all the strange Greek notation, I just stopped caring about it all.
that's exactly how it should be taught
too bad most math teachers suck
This is a faultless presentation of one of the most inspiring naratives in history, maths and physics. Congratulations! You have set a new paradigm YT. Could you do the same for Dirac's equation?
I am seriously contemplating showing this video to my Alg 2 class. Visual demonstration of completing the square and math history? Too good!
@@WestExplainsBest I was just saying to someone that I sorely wish the history of all of this had been taught to me back when I was learning it. I went on to study math in college, but I still wish that someone earlier on had showed us the humanity in math, the bickering scientists and the disbelief/hope that a solution would ever exist.
I think it would be awesome for you to show it
Dirac pls
Dirac and Schrodinger were both genius. Although Dirac's equation gets the fame, Schrodinger was the one who built the mathematical framework for Quantum particles.
@@vikraal6974 s`right, but beauty of following through to Dirac is bringing matrices into the story and thus antimatter! How maths reveals the world. After that, string theory?
I am simply awestruck by this amazing trip through mathematical history that brings us all the way to the Euler's equation. The graphic explaining e^ix = cos x + i sin x is the best I have seen. Congratulations and thank-you, Veritasium! BTW, is that an iron ring on your finger?
One of your best ever videos. Wonderful stuff.
You do some wonderful stuff yourself.
your comment is going to blow up
By going into interesting history he made advanced math engaging. It’s an incredibly smart method executed well.
Im no surgeon, but I did help create life once......Does that make me God, or dumb for not wearing a condom.
Hey my other favorite TH-camr!
Seeing the spiral of e^ix helped me understand Euler’s formula in a way that I’d never understood it before, despite having used it repeatedly in math, physics and engineering classes. A visual rather than algebraic analysis.
graphical over symbolic
Both are important
The explanation and animation was great indeed! It made me immediately understand why applying it for the wave function obviously makes sense.
It's actually a helix, rather than a spiral.
bro imagine going back in time with like HS calculus on your belt you would be an oracle
Ikr. It's a common Sci Fi trope but I love the thought of these guys throughout history being aliens or something changing our whole civilisation with what is high school maths to them 🤣
As soon as they say "now proof it" you are done
Yea and they're gonna ruin your dreams with a simple: "And why is that?"🙂
@@agam9085 "Let me explain the next few hundred years of math to you"
or burned at the stake...
Thanks! Finally beginning to understanding Imaginary numbers relavance to quantum physics
Perhaps the reason why mathematics developed so slowly before the Enlightenment era was due to this 'competitive' notion of mathematics, which encouraged people to keep their discoveries secret, to use as 'concealed weapons' for any adversary trying to challenge them on their position. I'm glad science has transitioned to a more collaborative process, it just makes more sense for overall progress.
back then, what prevented science from being collaborative was religion. now it's not cared for, but not cured and done for entirely. you have to have no humans left that insist on such primitive thoughts, + make every person interested in science and that's how you reach peak humanity. (i'm including both linguistic and computational aspects of science along with music, art or whatever comes to mind of course, they are all science.) i'm not saying people should deny the existence of an higher power, nobody can ever know that, agnosticism and deism exists for a reason. just not be stuck with all the books and the prophets and whatnot. that's how you get going, in terms of humanity.
I mean, maybe with some things it has gotten better, but this "I will keep this discovery, which would positively impact society at large if made public, a secret to myself for personal gain" is alive and well today, encouraged even, in the tech and pharmaceutical industries; and just like the video reminds us, it could take decades for a discovery made now to be made public, and really put to use in proyects that benefit is all
@@astridkjellberg This is nonsense. Explain in what waay religion "prevented science from being collaborative". Referring to religion as "primitive thoughts" says a lot about you friend. Making such wide statements regarding the history of science ( which I doubt you know anything about ) and regarding religion is pointless; if your thesis is that religion hinders scientific advancement you ought to provide evidence for it.
The conflict thesis ( the enlightenment idea that religion and science are fundamentally incompatible ) has been rejected by most historians and scientist, because it's ridicuolous and outdated.
@@marcovalentini863 Miracles contradict science though
@@martiddy In a framework where God exists ( the Abrahamic God I mean ) then how could miracles contradict science? From that point of view God can just do anything without regard for any natural laws, I don't see any contradiction.
I never understood why "i" sat on a different plane until now. The fact that i * i = -1, and that it is a rotation in a different plane never occured to me before. So simple, yet obscure. It makes so much sense now.
Its so simple once you understand it but so weird when you don't
awesome
Check out the videos by 3blue1brown. Best math videos ever, and a friend of Derek.
yeah the rotation explanation really was so insightful
this is why mathematics should only be taught by competent people, otherwise they just make a mess and you end up learning formulas to solve problems
“Only by giving up maths connection to reality could it guide us to a deeper truth about the way the universe works.”
What a lovely statement to end this insightful video.
Seconded
Kinda changed my view on maths
But imaginary numbers exist.
They just don't follow human logic. We use imaginary numbers for real things all the time. You cannot be an electric engineer without understanding them.
@@Wylie288 But for almost 2 century from the "invention" of imaginary number to the finding of its use in wave equation, imaginary number is just a concept that is only of interest to "math geek". I could get with the statement above that only by giving math "real" connection to reality that it become of interest to the "mainstream", just like what Schroedinger does to imaginary number with wave equation.
@@martinsusanto510 Imaginary numbers are required for electrical engineering. They are part of the basis of your entire life. They are required knowledge for us to send these messages.
This is probably my favourite video on YT. It is the best maths one by a mile, I really, really enjoyed it - I learned so much. The enthusiasm of the presenter was tangible!
One of your best Derek… Really made me feel grateful for the pre invented math we use and take for granted today…
His math videos are a league above the rest of his imho, and the rest are still great
That story about Ferro, Fior, Tartaglia, and Cardano could be a movie.
Quick reminder that the solution to fifth-degree equations was discovered by a political revolutionary who died in a possibly unrelated duel. The history of this part of mathematics has been almost excessively dramatic.
Kelsey Oakes's Aunt stopped living (LMAO 😂) because I upload bangers!
..,...
@@btf_flotsam478 I see you are talking about Evariste Galois and the Galois theory,but he did not find a solution to fifth degree equations,he just proved that they don't have a general solution
@@indianalphazero I think that was Abel
@@78anurag Abel also did it independently, but Galois did it a bit earlier and at a younger age. The only reason many people know about Abel and not Galois is because after Galois sent his work to his friend and died,his works were suppressed by other mathematicians(because they themselves did not understand it). Also Galois's work was much more detailed and generalized.
Just take to a second to appreciate the quality of this video ! All the characters, the math animations,… it's just mind blowing
Totally!! Loved it!!
This is a captivating story and should be required viewing for all math students
I went through school and university not really understanding the 'why' of imaginary numbers - just really using them as a 'magic spell' that happens to work. I wish I'd been able to see this explanation twenty five years ago, it's brilliant.
Facts
I am a big history buff and I always hated math in school.... Until I was introduced to the history of math. Now, it is just far more interesting and especially sensible for me than ever before.
@@Wasserkaktus My brother was telling me about a friend of his who was on a football scholarship and took a course on the history of mathematics. That got his interested in math and he eventually got a PhD in mathematics.
when you get to Engineering and start drawing Root Locus plots, you'll want to ignore these solutions (and any positive real values) when finding Breaking Points. I'm honestly trying to revise on TH-cam comments right now
Financial studies?^^
Seeing the historical origin of mathematics, physics, and technology makes it understandable much better because then you can fathom why they need it in the first place. If you make something about Fourier like this video, I want to see it
I literally came here to say exactly what you just did. All of it. Fourier FTW
you can checkout 3Blue1Brown youtube channel... He has made a few videos about Fourier, Complex numbers and many more topics with awesome animations and best explanations.
Yeah this is the best way to understand science and u will be able to create new things
Makes me want to re-watch James Burke's Connections.
This is half a module on the History of Mathematics (trust me - I took such a module at university!) and about a quarter of a module in Quantum Physics. In 20 mins. That is quite the achievement - well done!
I wrote an essay on this topic in a History of Mathematics unit in 3rd year, about 20 years ago. Got up to a little of Galois theory. Good stuff! (Now I'm an algebraist, though a different kind.)
A quarter of a module in quantum physics? Yeah nah
@@BozoTheBear So you specialize in Algebra. What does your job specifically entail?
Meanwhile 10 million people die from preventable cancer every single year and not a peep from the press or any politician. Solve that unsolvable problem you ignorant sheeple!
@@ProfessorWumbology "Algebra" to a mathematician means something very different from what you learned algebra was in high school. Roughly speaking algebra in mathematics is the study of "algebraic structures" such as groups, rings, modules and fields.
A ring roughly speaking is just a set R together with two operations + and × called plus and multiplication such that a×(b+c) = a×b + a×c.
R could be the set of integers, rational numbers or the real numbers, R could also be the set of matrices with + and × matrix addition and multiplication. You can also consider the ring Z_2 for example which contains only 0 and 1 with + and × given as usual except where you define 1+1 = 0.
Such a great work you have done! What a video!!! No words can describe the excellence.
I was totally blown away when I learned all of this. You're incredible putting this all together and deliver it to other people!
I wish schools taught the geometric way of solving quadratics and cubics first, then the algebraic way would’ve made much more sense for many students.
You're country !
Definitely
true
In the Montessori Curriculum, they do teach like this
Agree! I learnt more watching this video than on my 4 unit maths class!!
This man doesn't miss
Huh
Unless he's doing a sponsored ad lmao
@@leftythat5833 Oh no a sponsor segment in a video. How dare he exist in this market dominated world.
@@nathanlevesque7812 yeah that’s not it. You can have a sponsored video without having the whole video be the ad, so you avoid facts so your ad doesn’t fall short. It’s incredibly disingenuous, especially for a channel purporting to be about facts.
this man doesn't piss
I was a chemistry major and tolerated math because I had to and I was fairly good at it. I might have been far more inspired and really good at math due to applied inspiration if I had seen this video 20 years ago. Every student taking algebra and calculus should have to watch this video at the beginning of those courses.
Maths is so incredibly exciting. The way you get taught several seemingly unrelated things which then end up seamlessly working together is absolutely brilliant.
truly
One more like to become 600😱
@@DanielBaDanielBassey cool
As a High School Math teacher, I'm going to show this video to all of my students! From Algebra 2 to AP Calculus. This is amazing!
Are you ready for some smartellic to give solutions in geometric format instead of algebraic format? Just curious. There's a lot of overlap between math and electronics. I sometimes gave math solutions to electronic circuits and circuit solutions to math problems. Surprisingly I got away with it IF my answer was correct. Operational amplifier circuits love differential equations. It's a shame that I don't remember either not having worked with them for decades. But they were beautiful when I was working with them.
I wish someone would have shown me the quadratic equation solved with shapes demo when I was in school. Literally have never seen that before and it makes total sense.... Despite the problems with negative area.
If they taught in school about the history of math and how we use it in real world, I'm sure most of people who "hate math" will see how magnificent it is.
Facts
Chances are they might hate maths even more because now they have to learn about history of maths lmao
Teaching at school will never teach you mathematics, it is more the influence of parents who have been engaged in your development since the cradle.
na they wil still hate math because they hate doing maths, having a interesting backstory doesn't make it suddenly fun to do.
@@yuseisatouissuffering they shouldn't be supposed to memorize the history . It should just be told to inform them on how maths is actually done, i.e. how new stuff is actually figured out. The focus on memorizing formulae to solve most math problems is what i think is stopping math education from being fun.
As a mathematician I never cease to find joyous awe in the fact the complex exponentials have the wave behavior of triginometry.
“In science the credit goes to the man who convinces the world, not to the man to whom the idea first occurs.” - Sir William Osler
like in all domains, especially politics :)
Gregor Mendel: am I a joke to you?
Unfortunately…
This video proves that wrong though. Maybe in the short term (like the student who tried stealing credit from his teacher) but the experienced mathmetician knew something was up. Sure enough just 500 years later we are here and we know the truth.
i think that also applies in business.. and most things.
Visualizing "i" as describing a value that cancels itself out and seeing the CG of e^ix described as a spiraling 3-dimensional waveform with the X & Y functions 90° out of phase may have contributed more to my understanding of physics and mathematics than the entirety of my college calculus courses.
That’s how I know you found the actual Easter egg in the video. It literally made me gasp and shout as soon as I saw why Euler formula used e. Integrating and differentiating e is always going to be the same and the way they reflect sin and cos and their interwind is simply mind blowing 19:57
The fallacy of your comment is that you assume that your "understanding" back then is equal to your understanding now, after accepting "not understanding" and dedicating your attention to your individual set-up. Being patient, curious and brave (having FAITH) - i.e. staying HUMBLE - are the only ingredients of LIFE.
The best reference to LIFE being THE BIBLE!
@@ACuriousChildthe bible ain't all that
What is this place
Maths building out from geometry is amazing to me. I think people would understand it more if we taught it with geometric shapes alongside the formulaic advances.
This was an amazing video, Derek. I love how you’ve doubled down on TH-cam in recent years. I so hope early high-school students get shown this as a motivating factor for much of high-school math. Wow. I feel envious of them to have this at their age while I see it now at 36.
Yeah, rather frustrating we didn't have access to great teachers on youtube!
@@GoldenEDM_2018 And you don't need to remortgage your house to pay for it.
I am a highschooler rn and *YES* envy me!
Haha :P
@@YourLocalCafe Well you have to go to school every day and do exams ;).
I was thinking the same....and imagining how awesome it would have been if this video existing when I was in high school!
A bit of a rant and a thank you letter.
People these days forgot (and even gasp in amazement, sometimes ridicule) when I say algebraic notations are new language and they tell a story.
Much like learning a language, frequent use and familliarity, as well as occassional spritz of history and cultural knowledge, is essential for engineers to use equations beyond it's mundane descriptions.
And then there're mathematicians, a cadre of people dedicated to studying the language and art itself. I think the educator stands in between, massaging the notations into words for the learners, opening new possibility in the minds and professions.
And I think you do well as an educator. Thank you so much for this brilliant video. I hope the message lasts beyond your passing.
I didn’t understand a single thing and I enjoyed every minute of it.
Jokes aside, it brought me back to my quantum chemistry lessons, many years ago! You have a gift to make mathematics interesting.
😄😂
I find that knowing the history of things always makes them simpler. That's the beauty of time... If you don't find your answer in the present, you can look to the past, and if you can't find it in the past, you can build your future to find it...
@@derpatel9760 yeah agreed. we have to remember that most of these concepts were often discovered in a collaborative effort between multiple people across different generations, and if we don’t get it straight away that’s fine - the people who made them up didn’t too. at least that’s what helps me to not get my tiny brain overwhelmed by all this.
also, nice pfp lol
@@tecc9999 thanks lol
As a JHS student, I used to think that Math should be taught in a way that would help us in reality. For example how to solve basic operations, basic algebra, etc. Then watching this video I've realized society would not progress if Math is tied to reality. We would be stuck with elementary/junior high school level! From now on I'm not gonna dread about math. And if i sometimes feel like giving up, i'll remind myself of the countless mathematicians who made it without having the knowledge we have in the 21st century. This made me love math.
And made me love you too.
@Pradeep Singh Three ways:
1. The solutions you get using your new method on the problems seem reasonable
2. Your peers agree with you
3. Your new method solves any relevant real-world examples.
I went looking online for real-world examples of problems someone in the 1500s might have wanted to solve with the cubic. One was the volume of a container for water or grain. For example, if I change the length of this side of a box, how much less grain would it hold? The other is fishing. If I have to pay three men to fish in a body of water, but it slowly gets harder to fish the more they catch, and the more fish they bring back the less each fish sells for on the market...how long should I pay them to fish? How many should they get and when should they give up and return with what they have?
All the other relevant real-world solutions with the cubic are from our time, like drag on an airplane, making computer animations look smooth, turning 3D scans into 3D models, displaying fonts, using encryption to protect a web page when it loads, etc.
Hopefully this didn't post twice, I had responded already but my page had an error.
Me too my friend, me too
do you want a math duel? I can send you my homework and you send me back your homework.
@@래모루래모로 simp spotted
Everyone talking about the history and math, meanwhile I am amazed by all the visuals and illustrations to make this clear to non-mathematicians! The art style and illustrations was absolutely amazing and fit perfectly with the storytelling! Overall, an amazing video! Thanks TH-cam Algorithm! :D
Yessss, the 3D visual of the sine and cosine waves blew my mind. Never thought the z axis was a factor with those
You all should watch more of Grant's work. 3Blue1Brown.
i am 12 and i totally, 100% understand the stuff y'all talkin' about. :)
i know this is 2 years old but like this is probably the best video on youtube ive watched this tens of times over these 2 years