You've got to do something about a.) that hiss in your recorded voice, and b.) that singing-hum in the background. They are annoying, hamper our attention, and leave an impression of incompetence in production on your part.
In school, I saw history folded into science and math classes as a waste of time. Now as a teacher, I appreciate how historic personalities can be used to connect new abstract knowledge to the students’ subject base knowledge. Thank you for the presentation.
Do your students agree with you, or do they see it as a waste of time as you once did? ;) Serious question - as an adult, I really enjoy understanding the historical and biographical stories of human discovery, and it definitely enriches my grasp of the subject - but then I have the luxury of not needing to learn a new topic from scratch, against a time limit.
@@Confuseddave Students thrive on human connections, so I exploit historical storytelling to engage them. Weirdly, admins don't want anything that strays from "teaching to the test." Florida Statewide Assessments make or break schools.
If you like the historical development of mathematics, you will appreciate the _awesome_ book by Edna Kramer, _The Nature and Growth of Modern Mathematics_
Fascinating. I studied applied math and theoretical physics at the university of St. Andrews, Scotland (1976-80) and I taught math in a South African high school (1999-2019). I used to 'entertain' learners with stories about complex numbers and their use in fluid mechanics, but I had never heard of this origin story. In fact, I cannot remember how complex numbers were introduced to us all those years ago - I think we were just shown their use rather than their origin...
Damn. This makes you see how math was literally telling you that i was a thing but we humans refused to believe it back then. It's kinda like Math has a life of its own. Math can never be wrong but our understanding of it might be flawed. Math is a beautiful subject that speaks to you. I loved it. Thanks
This is a very interesting video. I think , however, that you could improve it substantially and really drive home the point of the need for complex numbers in the last segment where x is the sum of the cube root of 2+11i and the cube root of 2-11i. What is happening is that the formula requires complex numbers, but they conspire to produce a real value because it is a sum of a complex number and its conjugate. This idea is profound and important, and comes up in a number of physics problems where the answer must be real, and what happens is a sum of a complex value and its conjugate "conspire" when added to produce such. It is similar to how the fibonacci sequence can be represented as a set of discrete difference equations whose solutions are sums of powers of weird values containing irrational roots but which, when summed, conspire to produce integer values.
In Europe, it was the rigorous treatment of negative numbers that enabled (and forced) the acceptance of complex numbers. Complex numbers are not so great a conceptual leap beyond negative numbers, in spite of the way they are obscured in contemporary education. Once the representation of the real numbers (or some dense subset) and a line was understood and internalized--that is to say, once real numbers were understood at all--complex numbers followed very quickly. For centuries, Europeans only considered polynomials over the positive numbers, and addition and subtraction were treated separately. For instance, there was not one general quadratic univariate polynomial equation ax²+bx+c=0, but four: ax²+bx+c=0, ax²+bx=c, ax²+c=bx, and ax²=bx+c. Even then, when Scipione del Ferro solved every special case of the general cubic equation, he did not admit negative solutions, yet he encountered negative discriminants. This was immensely frustrating and required a number of lengthy workarounds to avoid using complex numbers. It was just a few decades between this algebraic masterpiece, working effectively in the positive algebraic numbers of degree 4 or less, and Rafael Bombelli's non-rigorous but fairly modern treatment of the complex numbers. Of course, a proper formalism even of the real numbers remained centuries away, but that is related to a conceptual difficulty of the continuum, not of the imaginary axis. It is important to remember that the geometric intuitions we take for granted did not yet exist for real numbers. There was no concept of the real line or the Cartesian plane. Descartes's intuition caught fire for a reason: algebra was desperately in need of visualization. So this idea of plotting complex numbers on a plane was inconceivable to an establishment that had not yet thought of plotting real numbers on a line. In truth, the algebraic understanding came first, in terms of the rules that must necessarily apply to abstract quantities, and the geometric understanding came later, in terms of the rectangular and polar representations of complex numbers.
A mistake @ 1:14 - the 2nd term in the Cardano-Tartaglia formula has the operation -q/2 - sqrt(…) applied twice instead of once. The two cube roots should look identical except for the sign of the square root inside. Thank you for your work!
Amazingly concise, yet beautifully conveyed video. Thank you for summarising this fascinating intersection of mathematics, history, and human discovery.
WOW I thought I was watching a youtube video from someone with 500k+ subs, you produce the same or even better content than them! :D Checkpoint: 2.1k subs - 35,531 views
" possible and necessary for mathematics ". More precisely stated : use of ' I = sqrt(-1) ' can produce results which are demonstrably correct [ as in the example of the intersection of a line and the simple cubic curve ], and it also points a way to derive two solutions to every quadratic. In mathematics we can always find examples of pragmatic use of a concept on the one hand, and rigorous reasoning on the other. One example is G.H.Hardy's work in putting Newton's calculus on a firm basis based on clear axioms. The invention of the Argand method is certainly pragmatic and useful in arriving at conclusions which are demonstrably correct. This should be good enough for most purposes, but will be offensive to those who believe that a priori reasoning based on rigorously defined axioms, which in turn are akin to laws of the universe - handed down from on high, pristine and immutable. Kant asks the question did mankind bring mathematics to the world, or does it exist in some absolute sense. In posing his famous set of problems David Hilbert states " wir muessen wissen, wir werden wissen " - a statement which was refuted by Kurt Goedel, so that we cannot always " know " even though there is no evidence to the contrary. A further example of the utility of complex numbers, as if it were needed, is the its use in taking higher powers of (a+I*b) to evaluate tan(2*x), tan(3*x), .... where tan(x)=b/a. This is much simpler that embarking on a proof based on Euclidean geometry. The proof of the pudding is in the eating.
Phenomenal video. Complex numbers always looked arbitrary to to me. Was also told the old false origin story about them. This video solves both issues, satisfyingly too.
I saw complex numbers for the first time and just accepted it, because it worked. It made all the more sense when capacitive and inductive reactance are complex values.
Galois did it more elegantly: Clearly x²+1 is irreducible over ℝ. So when a new element i that is a root of x²+1 is added, then ℂ = ℝ(i) is a larger field that extends ℝ.
A fascinating thing about the history of math and science is that it shows how many dead ends were taken first to get where we are today, so that the development of these subjects isn't anywhere nearly as clean-cut as you might think, and for all we know, we're still heading towards lots of dead ends! I for one suspect string theory is one of them!
It is not really a comment on your comment, but it fits here kind of anyway: complex numbers is not so called because it is “difficult”. It is so called because it is “consisting of different and connected parts”.
@@roygalaasen Yes, and the "imaginary" part is just as "good" as the real part, so that is also a misleading term. A better word might be composite or compound numbers or similar 😉
Hey just a small detail, in the end there x is just 4, and you can see that since (2+i)^3=2+11i so you could have shown this since that's the crazy thing, because you end up using i to arrive at the number 4! Nice video, I've learned about it long ago but didnt remember but this is the real thing, it isn't only the need to expand the range of possible roots, but that the complex numbers already were in the real roots. Edit: no pun intended
What a staggering result! I wish I had seen this simple explanation of the necessity of imaginary numbers years and years ago!! Why isn’t this part of every standard course in math??? Can you elaborate more on why exactly multiplication in the complex plane results in rotation? I’m sure it’s simple but Id really like to see a clear path from that to Eulers identity. Also, 3B1B’s popularized python animation library strikes again :)
Essentially, what it comes down to is that complex numbers are isomorphic to a subgroup of 2 by 2 real matrices. In fact, it would be far more natural for you to first discover the mathematics of the rotation group of 2 by 2 real matrices, and only afterwards extrapolate that these matrices can, in themselves, be treated as numbers that constitute a simple extension of the real numbers. For example, there is no real number r such that r^2 = -1. However, if I consider this equation for 2 by 2 real matrices instead of for real numbers, then there do exist real matrices R such that R^2 = -I. In fact, there are infinitely many such matrices, but one thing you will find is that the generating matrix i is a rotation matrix: namely, it is the matrix with first column (0, 1) and second column (-1, 0). This matrix satisfies the property that it rotates a real vector in 2-dimensional space by 90° counterclockwise. Here is an interesting fact: if you scale this matrix by a real number B, you still have a 90°, but now the length of vector being rotated scales according as well, with negative multipliers resulting in reflection. If to this scaled rotation, you add a scaling matrix, which is just a real number A multiplied by the 2 by 2 identity matrix, then you get a matrix with first column (A, B) and second column (-B, A), which is just A·I + B·i. If you studied linear algebra, then you probably know that every matrix of the above form is actually a scaled rotation matrix, and this can be proven rather easily using some geometry if you translate from numbers to geometrical space. In particular, A = s·cos(t), B = s·sin(t), where s is the scaling factor, and t is the rotation angle. To this, add the important fact that the matrices of this form actually form a multiplicative group if excluding the 0 matrix! In fact, not only is this true, but these matrices form a field. From the relationship between A, B and s, t as delineated above, you can further prove that the product of two scaled rotation matrices itself a scaled rotation matrix, where the scaling factors multiply, and the angles of rotation add. Finally, since these matrices form a field with a well-suited notion of absolute value, and since the scalimg matrices are themselves just equivalent to real numbers, you can very much just treat these matrices as a set of numbers of the form A + B·i, where A and B are real numbers, and i is just some new number introduced into the number system in question. We call these the complex numbers, but since they are just structurally equivalent to the matrices above, the rotation visualization built into the multiplication of these numbers has the same rotation properties as the matrices themselves do, and this is ultimately the answer to your question. So in summary, it all boils down to the way 2 by 2 real matrices work.
@@xaviconde √(-2)*√(-3)=√(2i²)*√(3i²) and then its either A: √(2i²)*√(3i²) = √(6i⁴)=√6 or B: √(2i²)*√(3i²) = i√2*i√3=i²*√6=-√6 UPD B is the case, cause √(-2) and √(-3) are both complex numbers with phase equal to pi/2, so if we multiply them we get number with phase equal to pi (real negative number)
@@kendakgifbancuher2047 I understand that square root function returns 2 results, positive and negative. Sqrt (4) is both 2 and -2. so sqrt(x) = -sqrt(x), since sqrt(x) means both the positive and the negative. However according to en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers there are some operations with negative numbers that are not defined, this could be one of the cases.
What is the difference between the binomial factorization, in the real numbers x and y, of any given rectangle and one whose sides happen to be equal? The way I learned it, you always can draw the figure in the first and third quadrant if it is positive, or the second and fourth if it is negative. As with any equation that doesn't pass the vertical line test, or for that matter to travel backwards in time and find the exact preconditions that were determinant for any given outcome, we can never be certain which factorization was the immediate predecessor of the value. But no need to introduce unnecessary "complexity".
@@angelmendez-rivera351 so what is taking the square root? It is a binomial factorization. When the factorization is integral (in the integers) then the square in question is no different than some other integer sided rectangle - say 21 = 3 x 7. 21 can also = -3×-7. No need for complex numbers explaining this. Where as 36 = 6 × 6, or -6 × -6. The square root of 36 is either 6 or-6. The square root of-36 is exactly the same with opposite signs on the factors. Taking the square root of 37 gives no meaningful factorization. But simply the geometric mean of the factors (1 and 37 in this case) Taking the square root of -37 gives the mean of either -1 × 37. Or 1 × -37. No need for i
@@davidshechtman4746 *so what is taking the square root?* The square root of a nonnegative real number r is defined as the unique nonnegative real number s that solves the equation s^2 = r, and we denote that with a radical symbol, or as sqrt(r). An exact formula is given by sqrt(0) = 0, sqrt(r) = exp[ln(r)/2] for r > 0. For complex numbers, the generalization of the formula is given by Sqrt(0) = 0, Sqrt(z) = exp([ln(|z|) + Arg(z)·i]/2) for C\{0}. In the complex case, this gives the unique complex ζ number in Union(C+, {0}) that solves the equation ζ^2 = z, where C+ := {z in C : Re(z) > 0, or Re(z) = 0 & Im(z) > 0}. Here, Sqrt is the analytic continuation of sqrt to C with a cut along Re(z) < 0. *It is a binomial factorization.* This is related to the above, but not actually equivalent. ζ^2 = z is equivalent to ζ^2 - z = 0, which is a polynomial equation of degree 2 with respect to ζ. Every polynomial of degree 2 with complex coefficients can be uniquely factored into two polynomials of degree 1 with complex coefficients. As such, ζ^2 - z = (ζ - α)·(ζ - β) is guaranteed. What you can prove is that β = -α, and that of α and -α, exactly one of the two values is an element of C+, unless α = 0. Whichever of α and -α is an element of C+ is called the square root of z, although some authors prefer to call it the principal square root of z, for emphasis. *When the factorization is integral (in the integers) then the square in question is no different than some other integer-sides rectangle - say 21 = 3 x 7.* Is that "x" symbol supposed to denote multiplication? If so, then I would recommend you that you actually use correct mathematical notation. Otherwise, your sentences are going to be nigh-impossible to understand. With that in mind, I do not see in what manner do you say that perfect square integers are no different than any other integer rectangles. Perfect square integers do indeed satisfy properties that other integer rectangles do not, so they are indeed different. For example, a polynomial x^2 - n is only factorizable with integer coefficients if n is a perfect square integer. *21 can also = -3x-7. No need for complex numbers explaining this.* Complex numbers are for taking the square root of negative numbers. Now, let me ask you a question: is 21 a negative integer? Answer: no, it is not. So your statement is nonsensical. *Where as 36 = 6x6 or -6x-6. The square root of 36 is either 6 or -6.* The square root of 36 is simply 6. -6 is -1 multiplied by the square root of 36. *The square root of -36 is exactly the same with opposite signs on the factors.* No. There is no integer or real number r such that r^2 = -36. Yes, you can factor -36 as 6·(-6), but this is irrelevant to the topic of the square root of -36, so your statement is still nonsensical. *Taking the square root of 37 gives no meaningful factorization.* Wrong. You can factor 37 as sqrt(37)·sqrt(37). *But simply the geometric mean of the factors (1 and 37 in this case) Taking the square root of -37 gives the mean of either -1x37. Or 1x-37. No need for i.* Yeah, this is completely incoherent, it is gibberish. I would think that this was written by a cat who arbitrarily jumped on the keyboard and randomly pressed keys. There, I gave you specific objections. Now, before you reply, I VERY STRONGLY suggest that you read a few textbooks on algebra. Consult your math teacher for some recommendations.
You have a number line. You decide to place an "origin" somewhere - where this line "ends" - and on the "other side" the same number line is reflected and you call it "negative numbers". This is all so arbitrary. Reflected around what axis precisely? Introducing this arbitrary "origin" is completely equivalent to introducing another orthogonal axis without which this weird "reflection" argument makes no sense. You create a 2D space with this move. In fact, we should be weirded out by negative numbers alone. How can you just make this move? Reflect a line in "another" direction? Which direction? How can you just do this? And what do you do with this remaining dimension you added? Complex number on the other hand just make perfect sense. I now challenge you to apply the "complex plane" to conceptual space: if at 1 is "existing" and at -1 is "non-existing", what does a 90degree CCW rotation away from 1 represent? If you'll applaud in silence I will go gently on you during my Nobel acceptance speech.
I know now why we never discussed origins of complex numbers. It just makes things more complicated. I don't understand how people can live with something so abstract. I guess that's how even the scientists before found this concept as psychologicaly disturbing.
@@MetaMaths actually yea, now I realize it. Back then I was the kind of student that reads a lot of math things but never dive deep. And I always believed in things like, 1/0 is infinity and quadratic equations always have roots, even tho we were working with real numbers.
A very surprising fact is that quantum mechanics, explaining everything that surrounds us, requires inevitably complex numbers. Not that it is a convenient tool to calculate predictions done by quantum mechanics, no, complex numbers are unavoidable.
I am a musician. However,I do not understand why so many otherwise good video producers choose to spoil their videos with musical distraction. It is truly annoying when great informative videos, such as this, are spoilt by unnecessary, irritating, distracting (usually bad) music. Although, I am a musician, I have studied many other subjects and have a University Diploma in Mathematics - specialising in Applied Mathematics. However, I have never attended a serious lecture where the lecturer deliberately decided to distract the concentration of pupils with music (good or bad) - unless the lecture was about music.
My doubts... 1. Proof by contradiction is used in maths so many times, so wen they encoubtered a negative root, y dint they thought that original equation was false. 2. If final result for x is a real number and intermediary is a complex, concluding that complex numbers must exist if final result is real, this idea if applied in other problems, fir example 4 divided by 2, multiplying numerator and denominator by 0, coz multiplying is possible, 4 by 0 into 0 by 2, 0 cancels out giving a real number, answer 2, then is division by zero possible? Or 0 by 0 exists? ... still cant comprehend the existence of complex numbers...
Yes, you get real numbers. If you didn't get real numbers, we wouldn't have complex numbers. The point is that, much like the quadratic formula, there is a cubic formula. For certain cubics, the cubic formula will always result in considering square roots of negative numbers. The difference is that, for quadratics, getting a square root of a negative can is equivalent to having no (real) roots. But for a cubic, you _always have at least one real root._ So cubics led mathematicians to the following situation: every cubic polynomial has at least one (real) root, but some cubic polynomials will always yield square roots of negative numbers when using the cubic formula. In other words, mathematicians encountered a situation in which the only systematic way to find real-valued solutions to real-valued problems was to work with square roots of negative numbers.
I am a musician. However,I do not understand why so many otherwise good video producers choose to spoil their videos with musical distraction. It is truly annoying when great informative videos, such as this, are spoilt by unnecessary, irritating, distracting (usually bad) music. Although, I am a musician, I have studied many other subjects and have a University Diploma in Mathematics - specialising in Applied Mathematics. However, I have never attended a serious lecture where the lecturer deliberately decided to distract the concentration of pupils with music (good or bad) - unless the lecture was about music.
@@MetaMaths First off, let me confirm how much I enjoyed learning from your excellent lecture - which is why I have subscribed to receive more. The point that I tried to make is that your informative, well narrated, good lecture on this important topic was, in my view, spoilt by a distracting, unnecessary background sound . If you intend to make a video about the relationship between music, physics and maths I am happy to oblige - otherwise my advice is to have no music at all in any video which has zero musical content in the actual lecture..
@David Brown So you’re saying that music shouldn’t be used in any videos that don’t explicitly have or reference musical content? Music evokes emotional responses which is why it can be used to great effect in movies, advertising, therapy, etc. All kinds of art are used in contexts that don’t directly or explicitly involve that art itself.
At school,mathwmatixs histiry should have been thought fron an apreciation fir the mathematians who spent decades in solving problems. It would give a cultural appreciation also.
I was expecting that you will show that the number is a real number and the value. Since you did not it not clear why we should accept complex numbers instead of discarding the formula. it seems that the formula derivation just assumed an inequality.
His point that it must be real since it is on the graph is sufficient but yeah he should've showed a little extra step since x=4 there. But I've wondered a bit and one can see that the two values are actually conjugate to each other, and since one has positive complex part and the other negative, the complex part is zero so it is only real (just workout (2+i)^3 and you'll see)
I believed in complex numbers because I trusted that my high school Algebra teacher said they were a Thing. Later on, as I was finishing up my college studies, I had the choice of either taking partial differential equations or a complex numbers course. By this time, the thought of which rabbit hole to go down led me to a decision not to go on to study mathematics in graduate school, but rather to tackle the Ultimate Complexity: so I went to Theological Seminary instead--but I still love these math videos: they're sort of like a college class reunion, only without all the cocktails. Keep calculating!
@@MetaMaths keep it up bro. Music choice have minor impact. Your editing skills are already great and you'll keep on improving at storytelling. Just one video at a time
It’s such a shame that the terms “imaginary” and “complex” are used because these terms have semantic associations which distort our perception of the properties of these numbers.
1:44 It's sort of true, if you accept that sqrt(6) has 2 values - one positive and one negative, as do sqrt(-2) and sqrt(-3) for that matter, thought those are positive and negative imaginary numbers instead of positive and negative real numbers.
You say it's true history of complex number yet you don't talk anything about the progress of complex numbers through 16th to 18th century. In grad classes and moocs, complex numbers are introduced with the example of Bombelli and his motivation for it. There is nothing new in this video.
Right, but this video does not seem to be aimed at graduate-level students. In my experience, at secondary school, "it appears in the quadratic formula" is indeed about as much as pupils are taught. Even at undergraduate level this history was not taught. Some lecturers did give historical context, but mostly from the later 19th and 20th centuries - by then it was assumed that the audience was familiar with complex numbers, and didn't need more motivation for studying them.
today: it took humanity centuries to realize that you can consider negative numbers... complex numbers frightened mathematicians throughout generations... few centuries later: imagine those ancient days when mathematicians were afraid of division by zero...
There is a branch in mathematics that allow division by zero. Basically you need to assume that positive infinity equals negative infinity before division by zero becomes logically possible. This works but by allowing division by zero you lost the concept of higher infinities. Mathematics is not a single thing it is a collection of systems that sometimes does not agree with each other. As long as you specify what are the axioms you are using and only use theorems provable with your list of axioms then that branch of mathematics is internally consistent.
@@kazedcat Positive and negative infinities as limits for real number sequences, you mean? These weren't part of the real numbers anyway though. You can similarly extend complex numbers to obtain the Riemann sphere, but there is only one "number at infinity" indeed. But what do you mean by "losing the concept of higher infinities"? You mean the Aleph-naught and higher infinities from set theory? Those refer to cardinal numbers, but you dont get aleph-naught infinity by dividing by zero (division doesn't work like that for cardinals, if defined at all). Btw: there are other extensions with infinities, like the hyperreal numbers, but these still don't allow division by zero. Take home message: the concepts of introducing infinities and of allowing division by zero are not the same.
Dave Langers When i said you lost the concept of higher infinities what I mean is that to allow division by zero you need to define a singular point of infinity. This means you lost the concept of omega+1, omega+omega, omega*omega. Now it is fine to work with mathematical system without omega but there are branches of mathematics that need the concept of omega to prove things. So you have two incompatible branch of mathematics. Either you have omega but then you cannot have division by zero. If you want division by zero then you lost omega because now you only have one point at infinity. This is similar to Eucledian geometry and Non Euclidean geometry, Either you agree that the fifth pustulate is correct or you agree that the fifth pustulate is wrong,
An easy way to realize that sqrt(-1) exists is simply (sqrt(-1))^2 = -1 . Obviously, -1 exists. If sqrt(-1) didn't exist, then we have a case where non-existence leads to existence. Nonsense.
Sadly, your line of thought is flawed. If you start from a false premise you can prove anything. If you want to prove that sqrt(-1) exists you need to start from a true statement and arrive at the conclusion that sqrt(-1) exists, not the other way around. For example: Starting from 1=2, you can multiply both sides by 0 to have 0=0. Although the conclusion is true, the initial statement 1=2 is still false.
@@gotikona To generalize 1=2, take x=y. Multiply both sides by 0 always gives 0=0. This may be interpreted as nullifying a false statement. In other words, no matter the starting statement, we end up with nothing. Any starting statement is thrown out. The case of sqrt(-1) is different because we start with a precise statement. If we had the case of (any statement)^2 = -1, then this equation would mean no valid statement to start with. So, sqrt(-1) is valid and does exist.
most of math is congenial to logic and not inconsistent with reality in terms of counting apples and oranges......however the square root of negative one does not exist, which is why it is called imaginary, and it remains offputting and discordant to most human beings........the upshot is that root minus one, though 'useful' and 'it works,' still has something deeply fundamentally wrong about it ..... it remains in the companionship of things like infinity and dividing by zero and renormalization, which seem to apply and be workable only in the realm of the unseen and the unknowable such as the electron and the quark .... these electrons and quarks are imaginary concepts that are described with imaginary mathematics that amounts to modern alchemy/astrology....it 'works' after a fashion, to describe a perturbation in a field where both the field and the perturbation are forever beyond the examination by human sensory organs, and therefore the electron becomes an invisible workhorse described only by symbols which symbols themselves denote no actual sensory shape or form but only movements and energies......we end up describing an unseeable imaginary idea of what we conceive an electron to be by invoking imaginary non-logical disconserting mathematical ideas to give it a kind of mental 'reality' that we can manipulate mentally.................but is this actually 'knowledge?' ....is Michaelangelo's "David" sculpture actually a man?
I was taught ALL numbers are purely abstract concepts that are very usable and necessary in order to solve many types of problems, but, as said, they are ABSTRACT cpncepts. Therefore one cannot say that any number EXISTs. You cannot obsevre a number in any way, be it by seeing, hearing, feeling, smelling, or whatever. One cannot put a number on the table like you can with a coffee cup or so. The latter EXISTs. Numbers don't. But we need them to solve problems, no arguing about that. It is just the word EXIST that bothers me.
Complex is a bad traduction from the Germanic languages. The right semanticaly word is likely to be "Strange" or "awkward". In portuguese we call "número complexo", but a few authors are renaming to "número incómodo".
Hey, thanks for watching ! Please subscribe to help me create ground breaking content !
You've got to do something about a.) that hiss in your recorded voice, and b.) that singing-hum in the background.
They are annoying, hamper our attention, and leave an impression of incompetence in production on your part.
@@David_Lloyd-Jones Harsh, but true words.
Improvement comes with time and practice. Meanwhile, I've subscribed based on this content alone.
Would you please give the exact page number of the book 'visual complex analysis' where I can read it?
instablaster...
In school, I saw history folded into science and math classes as a waste of time. Now as a teacher, I appreciate how historic personalities can be used to connect new abstract knowledge to the students’ subject base knowledge. Thank you for the presentation.
thanks for watching !
Do your students agree with you, or do they see it as a waste of time as you once did? ;)
Serious question - as an adult, I really enjoy understanding the historical and biographical stories of human discovery, and it definitely enriches my grasp of the subject - but then I have the luxury of not needing to learn a new topic from scratch, against a time limit.
@@Confuseddave Students thrive on human connections, so I exploit historical storytelling to engage them. Weirdly, admins don't want anything that strays from "teaching to the test." Florida Statewide Assessments make or break schools.
If you like the historical development of mathematics, you will appreciate the _awesome_ book by Edna Kramer, _The Nature and Growth of Modern Mathematics_
Two things: I have gone back to some kind of classroom learning every year for over 60 years. And I'm quick to toss the crap.@@iamfighterman9646
Fascinating. I studied applied math and theoretical physics at the university of St. Andrews, Scotland (1976-80) and I taught math in a South African high school (1999-2019). I used to 'entertain' learners with stories about complex numbers and their use in fluid mechanics, but I had never heard of this origin story. In fact, I cannot remember how complex numbers were introduced to us all those years ago - I think we were just shown their use rather than their origin...
Damn. This makes you see how math was literally telling you that i was a thing but we humans refused to believe it back then. It's kinda like Math has a life of its own. Math can never be wrong but our understanding of it might be flawed. Math is a beautiful subject that speaks to you. I loved it. Thanks
Well, considering that there are people today who believe the Earth is flat, this is far from surprising.
Math ca never be wrong. This is so wrong. Physics is the proof of math else any math is just mental gym.
@@rs-tarxvfzYep! Logic is not right or wrong, it's simply either coherent or incoherent
You were always a thing, believe in yourself 💪🏻
There are mathematicians who actually don't consider complex numbers an useful concept
This is a very interesting video. I think , however, that you could improve it substantially and really drive home the point of the need for complex numbers in the last segment where x is the sum of the cube root of 2+11i and the cube root of 2-11i. What is happening is that the formula requires complex numbers, but they conspire to produce a real value because it is a sum of a complex number and its conjugate.
This idea is profound and important, and comes up in a number of physics problems where the answer must be real, and what happens is a sum of a complex value and its conjugate "conspire" when added to produce such.
It is similar to how the fibonacci sequence can be represented as a set of discrete difference equations whose solutions are sums of powers of weird values containing irrational roots but which, when summed, conspire to produce integer values.
In Europe, it was the rigorous treatment of negative numbers that enabled (and forced) the acceptance of complex numbers. Complex numbers are not so great a conceptual leap beyond negative numbers, in spite of the way they are obscured in contemporary education. Once the representation of the real numbers (or some dense subset) and a line was understood and internalized--that is to say, once real numbers were understood at all--complex numbers followed very quickly.
For centuries, Europeans only considered polynomials over the positive numbers, and addition and subtraction were treated separately. For instance, there was not one general quadratic univariate polynomial equation ax²+bx+c=0, but four: ax²+bx+c=0, ax²+bx=c, ax²+c=bx, and ax²=bx+c. Even then, when Scipione del Ferro
solved every special case of the general cubic equation, he did not admit negative solutions, yet he encountered negative discriminants. This was immensely frustrating and required a number of lengthy workarounds to avoid using complex numbers. It was just a few decades between this algebraic masterpiece, working effectively in the positive algebraic numbers of degree 4 or less, and Rafael Bombelli's non-rigorous but fairly modern treatment of the complex numbers. Of course, a proper formalism even of the real numbers remained centuries away, but that is related to a conceptual difficulty of the continuum, not of the imaginary axis.
It is important to remember that the geometric intuitions we take for granted did not yet exist for real numbers. There was no concept of the real line or the Cartesian plane. Descartes's intuition caught fire for a reason: algebra was desperately in need of visualization. So this idea of plotting complex numbers on a plane was inconceivable to an establishment that had not yet thought of plotting real numbers on a line. In truth, the algebraic understanding came first, in terms of the rules that must necessarily apply to abstract quantities, and the geometric understanding came later, in terms of the rectangular and polar representations of complex numbers.
Well said!! It always boggles my mind when I remember how much of math was done without the cartesian plane, which we take for granted
You are correct.
good work with manim, keep it up :)
Cool to see more and more youtubers utilising Grants animation library!
A mistake @ 1:14 - the 2nd term in the Cardano-Tartaglia formula has the operation -q/2 - sqrt(…) applied twice instead of once. The two cube roots should look identical except for the sign of the square root inside. Thank you for your work!
I realized that too. Wasn't sure at first if I remembered to formula right, but then I was glad to read your comment. ;)
Correct
Amazingly concise, yet beautifully conveyed video. Thank you for summarising this fascinating intersection of mathematics, history, and human discovery.
"Judging Mathematics by its pragmatic value is like judging symphony by the weight of its score."
-Alexander Bogomolny
01:12 The right cubic root has a small error to correct.
I have been looking for this for so long. Great work.
1:28. François Viète called. He wants his portrait back.
WOW I thought I was watching a youtube video from someone with 500k+ subs, you produce the same or even better content than them! :D
Checkpoint: 2.1k subs - 35,531 views
Wow, thanks ! Let' s return to this comment in a year and see where the channel ends up !
@@MetaMaths Exactly! I forgot to checkpoint the time "2 weeks ago of 22/03/2021"
@@MetaMaths Maybe you should like more comments. There's a rumor, that _the alorithm_ really likes that ;-)
in the first instance, view " i " as an arithmetical operator. it tell's you to rotate by one right angle.
Very interesting and compelling. I don't think i have ever seen complex numbers motivated in this way. Thanks!
" possible and necessary for mathematics ". More precisely stated : use of ' I = sqrt(-1) ' can produce results which are demonstrably correct [ as in the example of the intersection of a line and the simple cubic curve ], and it also points a way to derive two solutions to every quadratic. In mathematics we can always find examples of pragmatic use of a concept on the one hand, and rigorous reasoning on the other. One example is G.H.Hardy's work in putting Newton's calculus on a firm basis based on clear axioms.
The invention of the Argand method is certainly pragmatic and useful in arriving at conclusions which are demonstrably correct.
This should be good enough for most purposes, but will be offensive to those who believe that a priori reasoning based on rigorously defined axioms, which in turn are akin to laws of the universe - handed down from on high, pristine and immutable. Kant asks the question did mankind bring mathematics to the world, or does it exist in some absolute sense. In posing his famous set of problems David Hilbert states " wir muessen wissen, wir werden wissen " - a statement which was refuted by Kurt Goedel, so that we cannot always " know " even though there is no evidence to the contrary.
A further example of the utility of complex numbers, as if it were needed, is the its use in taking higher powers of (a+I*b) to evaluate tan(2*x), tan(3*x), .... where tan(x)=b/a. This is much simpler that embarking on a proof based on Euclidean geometry. The proof of the pudding is in the eating.
Great👍! This has been the best explanation about how the complex numbers have come into being.
Excellent and very simply explained, the best explanation I've found so far :)
Phenomenal video. Complex numbers always looked arbitrary to to me. Was also told the old false origin story about them. This video solves both issues, satisfyingly too.
I saw complex numbers for the first time and just accepted it, because it worked. It made all the more sense when capacitive and inductive reactance are complex values.
Galois did it more elegantly: Clearly x²+1 is irreducible over ℝ. So when a new element i that is a root of x²+1 is added, then ℂ = ℝ(i) is a larger field that extends ℝ.
A fascinating thing about the history of math and science is that it shows how many dead ends were taken first to get where we are today, so that the development of these subjects isn't anywhere nearly as clean-cut as you might think, and for all we know, we're still heading towards lots of dead ends! I for one suspect string theory is one of them!
Also the music is great. I don't see where the problem is ha. Keep it up man. You earned a sub
Great explainer video, very informative and a good perspective to take especially with visual aids. Thank you for your presentation!
Thank you for watching !
I love your work. It is so well done!
Great video! I needed it so much during college times..😂
Wow, much complex, such numbers!
It is not really a comment on your comment, but it fits here kind of anyway:
complex numbers is not so called because it is “difficult”. It is so called because it is “consisting of different and connected parts”.
@@roygalaasen I know. I wanted to leave a comment for the algorithm to promote this video and this is the first thing that came to my mind.
honesty
@@roygalaasen Yes, and the "imaginary" part is just as "good" as the real part, so that is also a misleading term.
A better word might be composite or compound numbers or similar 😉
@@Bjowolf2 yes, exactly!
A new maths youtuber? Nice!
Hey just a small detail, in the end there x is just 4, and you can see that since (2+i)^3=2+11i so you could have shown this since that's the crazy thing, because you end up using i to arrive at the number 4!
Nice video, I've learned about it long ago but didnt remember but this is the real thing, it isn't only the need to expand the range of possible roots, but that the complex numbers already were in the real roots.
Edit: no pun intended
thanks !
could have shown.
@@azzteke oh boi, you're right, English is tricky , not my language, thanks anyway
Just discovered your channel. Keep up the good work.
Thanks !
What a staggering result! I wish I had seen this simple explanation of the necessity of imaginary numbers years and years ago!! Why isn’t this part of every standard course in math???
Can you elaborate more on why exactly multiplication in the complex plane results in rotation? I’m sure it’s simple but Id really like to see a clear path from that to Eulers identity.
Also, 3B1B’s popularized python animation library strikes again :)
Essentially, what it comes down to is that complex numbers are isomorphic to a subgroup of 2 by 2 real matrices. In fact, it would be far more natural for you to first discover the mathematics of the rotation group of 2 by 2 real matrices, and only afterwards extrapolate that these matrices can, in themselves, be treated as numbers that constitute a simple extension of the real numbers.
For example, there is no real number r such that r^2 = -1. However, if I consider this equation for 2 by 2 real matrices instead of for real numbers, then there do exist real matrices R such that R^2 = -I. In fact, there are infinitely many such matrices, but one thing you will find is that the generating matrix i is a rotation matrix: namely, it is the matrix with first column (0, 1) and second column (-1, 0). This matrix satisfies the property that it rotates a real vector in 2-dimensional space by 90° counterclockwise.
Here is an interesting fact: if you scale this matrix by a real number B, you still have a 90°, but now the length of vector being rotated scales according as well, with negative multipliers resulting in reflection. If to this scaled rotation, you add a scaling matrix, which is just a real number A multiplied by the 2 by 2 identity matrix, then you get a matrix with first column (A, B) and second column (-B, A), which is just A·I + B·i.
If you studied linear algebra, then you probably know that every matrix of the above form is actually a scaled rotation matrix, and this can be proven rather easily using some geometry if you translate from numbers to geometrical space. In particular, A = s·cos(t), B = s·sin(t), where s is the scaling factor, and t is the rotation angle. To this, add the important fact that the matrices of this form actually form a multiplicative group if excluding the 0 matrix! In fact, not only is this true, but these matrices form a field.
From the relationship between A, B and s, t as delineated above, you can further prove that the product of two scaled rotation matrices itself a scaled rotation matrix, where the scaling factors multiply, and the angles of rotation add. Finally, since these matrices form a field with a well-suited notion of absolute value, and since the scalimg matrices are themselves just equivalent to real numbers, you can very much just treat these matrices as a set of numbers of the form A + B·i, where A and B are real numbers, and i is just some new number introduced into the number system in question. We call these the complex numbers, but since they are just structurally equivalent to the matrices above, the rotation visualization built into the multiplication of these numbers has the same rotation properties as the matrices themselves do, and this is ultimately the answer to your question.
So in summary, it all boils down to the way 2 by 2 real matrices work.
Hey there! I wanted to ask you, whether I could use ur video in one of my assignments. If yes, what licence does this Video have? CC BY?
You can cite the book which I mention in the video description.
Nice use of Manim, but please upload in higher resolution to make the video more crisp
1:48
Shouldn't sqrt(-2)sqrt(-3) = -sqrt(6) =/= sqrt(6) ?
Yup. Euler was wrong.
I'm sure it's a typo
A^n * B^n = (A * B) ^ n. Hence (( - 2) ^1/2) * ((-3)^1/2) = (-2*-3)^1/2=6^1/2
@@xaviconde √(-2)*√(-3)=√(2i²)*√(3i²)
and then its either
A: √(2i²)*√(3i²) = √(6i⁴)=√6
or
B: √(2i²)*√(3i²) = i√2*i√3=i²*√6=-√6
UPD B is the case, cause √(-2) and √(-3) are both complex numbers with phase equal to pi/2, so if we multiply them we get number with phase equal to pi (real negative number)
@@kendakgifbancuher2047 I understand that square root function returns 2 results, positive and negative. Sqrt (4) is both 2 and -2. so sqrt(x) = -sqrt(x), since sqrt(x) means both the positive and the negative. However according to en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers there are some operations with negative numbers that are not defined, this could be one of the cases.
What app do you use? Thinking of doing similar ones
Manim and final cut
What is the difference between the binomial factorization, in the real numbers x and y, of any given rectangle and one whose sides happen to be equal? The way I learned it, you always can draw the figure in the first and third quadrant if it is positive, or the second and fourth if it is negative. As with any equation that doesn't pass the vertical line test, or for that matter to travel backwards in time and find the exact preconditions that were determinant for any given outcome, we can never be certain which factorization was the immediate predecessor of the value. But no need to introduce unnecessary "complexity".
What are you even talking about?
@@angelmendez-rivera351 so what is taking the square root? It is a binomial factorization. When the factorization is integral (in the integers) then the square in question is no different than some other integer sided rectangle - say 21 = 3 x 7. 21 can also = -3×-7. No need for complex numbers explaining this. Where as 36 = 6 × 6, or -6 × -6. The square root of 36 is either 6 or-6. The square root of-36 is exactly the same with opposite signs on the factors. Taking the square root of 37 gives no meaningful factorization. But simply the geometric mean of the factors (1 and 37 in this case) Taking the square root of -37 gives the mean of either -1 × 37. Or 1 × -37. No need for i
@@davidshechtman4746 Nothing of what you said there is coherent or intelligible.
@@angelmendez-rivera351 pretty sure it is. Maybe you have an objection that is specific
@@davidshechtman4746 *so what is taking the square root?*
The square root of a nonnegative real number r is defined as the unique nonnegative real number s that solves the equation s^2 = r, and we denote that with a radical symbol, or as sqrt(r). An exact formula is given by sqrt(0) = 0, sqrt(r) = exp[ln(r)/2] for r > 0. For complex numbers, the generalization of the formula is given by Sqrt(0) = 0, Sqrt(z) = exp([ln(|z|) + Arg(z)·i]/2) for C\{0}. In the complex case, this gives the unique complex ζ number in Union(C+, {0}) that solves the equation ζ^2 = z, where C+ := {z in C : Re(z) > 0, or Re(z) = 0 & Im(z) > 0}. Here, Sqrt is the analytic continuation of sqrt to C with a cut along Re(z) < 0.
*It is a binomial factorization.*
This is related to the above, but not actually equivalent. ζ^2 = z is equivalent to ζ^2 - z = 0, which is a polynomial equation of degree 2 with respect to ζ. Every polynomial of degree 2 with complex coefficients can be uniquely factored into two polynomials of degree 1 with complex coefficients. As such, ζ^2 - z = (ζ - α)·(ζ - β) is guaranteed. What you can prove is that β = -α, and that of α and -α, exactly one of the two values is an element of C+, unless α = 0. Whichever of α and -α is an element of C+ is called the square root of z, although some authors prefer to call it the principal square root of z, for emphasis.
*When the factorization is integral (in the integers) then the square in question is no different than some other integer-sides rectangle - say 21 = 3 x 7.*
Is that "x" symbol supposed to denote multiplication? If so, then I would recommend you that you actually use correct mathematical notation. Otherwise, your sentences are going to be nigh-impossible to understand. With that in mind, I do not see in what manner do you say that perfect square integers are no different than any other integer rectangles. Perfect square integers do indeed satisfy properties that other integer rectangles do not, so they are indeed different. For example, a polynomial x^2 - n is only factorizable with integer coefficients if n is a perfect square integer.
*21 can also = -3x-7. No need for complex numbers explaining this.*
Complex numbers are for taking the square root of negative numbers. Now, let me ask you a question: is 21 a negative integer? Answer: no, it is not. So your statement is nonsensical.
*Where as 36 = 6x6 or -6x-6. The square root of 36 is either 6 or -6.*
The square root of 36 is simply 6. -6 is -1 multiplied by the square root of 36.
*The square root of -36 is exactly the same with opposite signs on the factors.*
No. There is no integer or real number r such that r^2 = -36. Yes, you can factor -36 as 6·(-6), but this is irrelevant to the topic of the square root of -36, so your statement is still nonsensical.
*Taking the square root of 37 gives no meaningful factorization.*
Wrong. You can factor 37 as sqrt(37)·sqrt(37).
*But simply the geometric mean of the factors (1 and 37 in this case) Taking the square root of -37 gives the mean of either -1x37. Or 1x-37. No need for i.*
Yeah, this is completely incoherent, it is gibberish. I would think that this was written by a cat who arbitrarily jumped on the keyboard and randomly pressed keys.
There, I gave you specific objections. Now, before you reply, I VERY STRONGLY suggest that you read a few textbooks on algebra. Consult your math teacher for some recommendations.
You have a number line. You decide to place an "origin" somewhere - where this line "ends" - and on the "other side" the same number line is reflected and you call it "negative numbers". This is all so arbitrary. Reflected around what axis precisely? Introducing this arbitrary "origin" is completely equivalent to introducing another orthogonal axis without which this weird "reflection" argument makes no sense. You create a 2D space with this move. In fact, we should be weirded out by negative numbers alone. How can you just make this move? Reflect a line in "another" direction? Which direction? How can you just do this? And what do you do with this remaining dimension you added? Complex number on the other hand just make perfect sense.
I now challenge you to apply the "complex plane" to conceptual space: if at 1 is "existing" and at -1 is "non-existing", what does a 90degree CCW rotation away from 1 represent?
If you'll applaud in silence I will go gently on you during my Nobel acceptance speech.
1:12 - MAJOR BUMMER!!
really nice 👏👏need more stuffs like this...
maybe for cross and dot products...
“Stuffs” isn’t a word. The plural of stuff is...stuff.
What's the name of soundtrack in the end? Something familiar but can't recall it
Nice history lesson! Thx!
The background sound gives it an incredible sensation of mystery. I love it!!! :D
It kept me on edge to the point I couldn't concentrate. I had to turn it off sadly.
Would you please give the exact page number of the book 'visual complex analysis' where I can read it?
Either in the introduction or in Chapter 1
@@MetaMaths I have found it. Great content
Beautiful ! Thanks !
Is there any word problem for imaginary numbers?
You go man! You are great!
I know now why we never discussed origins of complex numbers. It just makes things more complicated. I don't understand how people can live with something so abstract. I guess that's how even the scientists before found this concept as psychologicaly disturbing.
This video is too short!! Please is there another one that's longer?
Awesome work! What program do you use to make your videos?
final cut
Cool thanks, it looks really clean and professional! What do you use to make all the text/graphs/diagrams? Keep up the good work!
1:20 it was discovered by a great Indian and this rule is aka SRIDHARACHARYA RULE
wow what a great video! liked and subscribed!
why is 3px + 2q a straight line?
p and q are constants, so we essentially have y = kx + c
Good Job Professor
I actually never heard anyone telling me that complex numbers were discovered _to solve quadratic equations_ . But good video!
That means you did not have brain- washed teachers )
@@MetaMaths actually yea, now I realize it. Back then I was the kind of student that reads a lot of math things but never dive deep. And I always believed in things like, 1/0 is infinity and quadratic equations always have roots, even tho we were working with real numbers.
A very surprising fact is that quantum mechanics, explaining everything that surrounds us, requires inevitably complex numbers. Not that it is a convenient tool to calculate predictions done by quantum mechanics, no, complex numbers are unavoidable.
Indeed.
Great Video! The music was a bit much vor me though. I lost the focus from time to time because it distracted me
I am a musician. However,I do not understand why so many otherwise good video producers choose to spoil their videos with musical distraction.
It is truly annoying when great informative videos, such as this, are spoilt by unnecessary, irritating, distracting (usually bad) music. Although, I am a musician, I have studied many other subjects and have a University Diploma in Mathematics - specialising in Applied Mathematics. However, I have never attended a serious lecture where the lecturer deliberately decided to distract the concentration of pupils with music (good or bad) - unless the lecture was about music.
My doubts...
1. Proof by contradiction is used in maths so many times, so wen they encoubtered a negative root, y dint they thought that original equation was false.
2. If final result for x is a real number and intermediary is a complex, concluding that complex numbers must exist if final result is real, this idea if applied in other problems, fir example 4 divided by 2, multiplying numerator and denominator by 0, coz multiplying is possible, 4 by 0 into 0 by 2, 0 cancels out giving a real number, answer 2, then is division by zero possible? Or 0 by 0 exists? ... still cant comprehend the existence of complex numbers...
I love the history of ideas.
Brilliantly captivating! 😇
Very good video but you should have shown the simplified value of the root in the end, x = 4.
This is dope and mindblowing 😮😂
Nice video except the final music? noise? is WAY LOUDER than the rest of the video. So annoying! Did you listen to it?
dunno what's wrong with the music, i really enjoyed it :D
Simply awsm
5:18 that whole thing is actually just 4 tho, which is the solution to the problem.
Yes, you get real numbers. If you didn't get real numbers, we wouldn't have complex numbers.
The point is that, much like the quadratic formula, there is a cubic formula. For certain cubics, the cubic formula will always result in considering square roots of negative numbers. The difference is that, for quadratics, getting a square root of a negative can is equivalent to having no (real) roots. But for a cubic, you _always have at least one real root._
So cubics led mathematicians to the following situation: every cubic polynomial has at least one (real) root, but some cubic polynomials will always yield square roots of negative numbers when using the cubic formula. In other words, mathematicians encountered a situation in which the only systematic way to find real-valued solutions to real-valued problems was to work with square roots of negative numbers.
I am a musician. However,I do not understand why so many otherwise good video producers choose to spoil their videos with musical distraction.
It is truly annoying when great informative videos, such as this, are spoilt by unnecessary, irritating, distracting (usually bad) music. Although, I am a musician, I have studied many other subjects and have a University Diploma in Mathematics - specialising in Applied Mathematics. However, I have never attended a serious lecture where the lecturer deliberately decided to distract the concentration of pupils with music (good or bad) - unless the lecture was about music.
Could you help me choose the right music for my next videos ?
@@MetaMaths First off, let me confirm how much I enjoyed learning from your excellent lecture - which is why I have subscribed to receive more.
The point that I tried to make is that your informative, well narrated, good lecture on this important topic was, in my view, spoilt by a distracting, unnecessary background sound . If you intend to make a video about the relationship between music, physics and maths I am happy to oblige - otherwise my advice is to have no music at all in any video which has zero musical content in the actual lecture..
@David Brown So you’re saying that music shouldn’t be used in any videos that don’t explicitly have or reference musical content? Music evokes emotional responses which is why it can be used to great effect in movies, advertising, therapy, etc. All kinds of art are used in contexts that don’t directly or explicitly involve that art itself.
awesome video!
Wow!
Great video !! New sub
At school,mathwmatixs histiry should have been thought fron an apreciation fir the mathematians who spent decades in solving problems.
It would give a cultural appreciation also.
The video was great!! The only thing bothering was the ominous sound in background, it was like you were telling a ghost story lol
That was the coolest part
I was expecting that you will show that the number is a real number and the value. Since you did not it not clear why we should accept complex numbers instead of discarding the formula. it seems that the formula derivation just assumed an inequality.
His point that it must be real since it is on the graph is sufficient but yeah he should've showed a little extra step since x=4 there.
But I've wondered a bit and one can see that the two values are actually conjugate to each other, and since one has positive complex part and the other negative, the complex part is zero so it is only real (just workout (2+i)^3 and you'll see)
Bonelli’s example of the Cardono formula is amazing
manim is simply gorgeous
Will love to see a video on riemannian sphere with some "special" content ❤️
"special" like proving Poincare' s conjecture ?)
@@MetaMaths will love it..
freakin beautiful!!!
THANK YOU 'I HOPEFULLY CAN ABSORB THIS REVELLATION SOON'
What a history!
Really entertaining video!
"It's Math, Jim, but not as we know it"
It is amazing to see how nature has a wider understanding of beauty for things we initially find ugly.
I believed in complex numbers because I trusted that my high school Algebra teacher said they were a Thing. Later on, as I was finishing up my college studies, I had the choice of either taking partial differential equations or a complex numbers course. By this time, the thought of which rabbit hole to go down led me to a decision not to go on to study mathematics in graduate school, but rather to tackle the Ultimate Complexity: so I went to Theological Seminary instead--but I still love these math videos: they're sort of like a college class reunion, only without all the cocktails. Keep calculating!
Dude just explained a complicated topic with the most discomforting bgm
apologies, my music tastes are a failure
@@MetaMaths keep it up bro. Music choice have minor impact. Your editing skills are already great and you'll keep on improving at storytelling. Just one video at a time
It’s such a shame that the terms “imaginary” and “complex” are used because these terms have semantic associations which distort our perception of the properties of these numbers.
1:44 It's sort of true, if you accept that sqrt(6) has 2 values - one positive and one negative, as do sqrt(-2) and sqrt(-3) for that matter, thought those are positive and negative imaginary numbers instead of positive and negative real numbers.
WOW mind blown
so in essence we can prove it exists but we cant observe it on its own
All mathematics can be considered imaginary.
You say it's true history of complex number yet you don't talk anything about the progress of complex numbers through 16th to 18th century. In grad classes and moocs, complex numbers are introduced with the example of Bombelli and his motivation for it. There is nothing new in this video.
Right, but this video does not seem to be aimed at graduate-level students. In my experience, at secondary school, "it appears in the quadratic formula" is indeed about as much as pupils are taught. Even at undergraduate level this history was not taught. Some lecturers did give historical context, but mostly from the later 19th and 20th centuries - by then it was assumed that the audience was familiar with complex numbers, and didn't need more motivation for studying them.
today: it took humanity centuries to realize that you can consider negative numbers... complex numbers frightened mathematicians throughout generations...
few centuries later: imagine those ancient days when mathematicians were afraid of division by zero...
Well they arent affraid of it they just know its nosens by itself and ask way more calcul in function
@@tritojean7549 Nonsense? Only in some algebras.
en.m.wikipedia.org/wiki/Wheel_theory
There is a branch in mathematics that allow division by zero. Basically you need to assume that positive infinity equals negative infinity before division by zero becomes logically possible. This works but by allowing division by zero you lost the concept of higher infinities. Mathematics is not a single thing it is a collection of systems that sometimes does not agree with each other. As long as you specify what are the axioms you are using and only use theorems provable with your list of axioms then that branch of mathematics is internally consistent.
@@kazedcat Positive and negative infinities as limits for real number sequences, you mean? These weren't part of the real numbers anyway though. You can similarly extend complex numbers to obtain the Riemann sphere, but there is only one "number at infinity" indeed.
But what do you mean by "losing the concept of higher infinities"? You mean the Aleph-naught and higher infinities from set theory? Those refer to cardinal numbers, but you dont get aleph-naught infinity by dividing by zero (division doesn't work like that for cardinals, if defined at all).
Btw: there are other extensions with infinities, like the hyperreal numbers, but these still don't allow division by zero.
Take home message: the concepts of introducing infinities and of allowing division by zero are not the same.
Dave Langers When i said you lost the concept of higher infinities what I mean is that to allow division by zero you need to define a singular point of infinity. This means you lost the concept of omega+1, omega+omega, omega*omega. Now it is fine to work with mathematical system without omega but there are branches of mathematics that need the concept of omega to prove things. So you have two incompatible branch of mathematics. Either you have omega but then you cannot have division by zero. If you want division by zero then you lost omega because now you only have one point at infinity. This is similar to Eucledian geometry and Non Euclidean geometry, Either you agree that the fifth pustulate is correct or you agree that the fifth pustulate is wrong,
An easy way to realize that sqrt(-1) exists is simply (sqrt(-1))^2 = -1 . Obviously, -1 exists. If sqrt(-1) didn't exist, then we have a case where non-existence leads to existence. Nonsense.
Sadly, your line of thought is flawed. If you start from a false premise you can prove anything. If you want to prove that sqrt(-1) exists you need to start from a true statement and arrive at the conclusion that sqrt(-1) exists, not the other way around.
For example:
Starting from 1=2, you can multiply both sides by 0 to have 0=0.
Although the conclusion is true, the initial statement 1=2 is still false.
@@gotikona Good point! How else can sqrt(-1) be proven to exist?
@@gotikona To generalize 1=2, take x=y. Multiply both sides by 0 always gives 0=0. This may be interpreted as nullifying a false statement. In other words, no matter the starting statement, we end up with nothing. Any starting statement is thrown out. The case of sqrt(-1) is different because we start with a precise statement. If we had the case of (any statement)^2 = -1, then this equation would mean no valid statement to start with. So, sqrt(-1) is valid and does exist.
Great video!
only suggestion is changing the background music but that's an opinionated take.
Keep up the good work!
Complex numbers can be easily visualised at least. Now quaternions can't.
most of math is congenial to logic and not inconsistent with reality in terms of counting apples and oranges......however the square root of negative one does not exist, which is why it is called imaginary, and it remains offputting and discordant to most human beings........the upshot is that root minus one, though 'useful' and 'it works,' still has something deeply fundamentally wrong about it ..... it remains in the companionship of things like infinity and dividing by zero and renormalization, which seem to apply and be workable only in the realm of the unseen and the unknowable such as the electron and the quark .... these electrons and quarks are imaginary concepts that are described with imaginary mathematics that amounts to modern alchemy/astrology....it 'works' after a fashion, to describe a perturbation in a field where both the field and the perturbation are forever beyond the examination by human sensory organs, and therefore the electron becomes an invisible workhorse described only by symbols which symbols themselves denote no actual sensory shape or form but only movements and energies......we end up describing an unseeable imaginary idea of what we conceive an electron to be by invoking imaginary non-logical disconserting mathematical ideas to give it a kind of mental 'reality' that we can manipulate mentally.................but is this actually 'knowledge?' ....is Michaelangelo's "David" sculpture actually a man?
Nice explanation but that background drone feels really unsettling. The video would be better without it.
Sorry, need someone to educate me in music
I was taught ALL numbers are purely abstract concepts that are very usable and necessary in order to solve many types of problems, but, as said, they are ABSTRACT cpncepts. Therefore one cannot say that any number EXISTs. You cannot obsevre a number in any way, be it by seeing, hearing, feeling, smelling, or whatever. One cannot put a number on the table like you can with a coffee cup or so. The latter EXISTs. Numbers don't. But we need them to solve problems, no arguing about that. It is just the word EXIST that bothers me.
interesting
The name"complex" should be changed to "compound"
Complex is a bad traduction from the Germanic languages. The right semanticaly word is likely to be "Strange" or "awkward". In portuguese we call "número complexo", but a few authors are renaming to "número incómodo".