Imaginary Numbers Are Real [Part 3: Cardan's Problem]
ฝัง
- เผยแพร่เมื่อ 19 มิ.ย. 2024
- More information and resources: www.welchlabs.com
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: www.welchlabs.com/resources. - วิทยาศาสตร์และเทคโนโลยี
![Imaginary Numbers Are Real [Part 4: Bombelli's Solution]](http://i.ytimg.com/vi/DThAoT3q2V4/mqdefault.jpg)
![Imaginary Numbers Are Real [Part 4: Bombelli's Solution]](/img/tr.png)
![Imaginary Numbers Are Real [Part 13: Riemann Surfaces]](http://i.ytimg.com/vi/4MmSZrAlqKc/mqdefault.jpg)
This is what I love about math. If you have a problem that you know is solvable but you can't solve it, you can say "let there be another type of number."
¡And what it's more exciting than that it's that it works!... but then it gets curious... ¿was that number inside your biological logic? ¿did you discover that number? ¿if you discovered that number... what sense made you capable of it? and after that it turns philosophical... and you don't want to get philosophical solving math problems... at least not pure philosophical hahahah
Dead Beetle
If only this always worked as nicely as it does in this case! Unfortunately, new sorts of numbers sometimes lead to a contradiction, and to repair this, so as to prevent a contradiction, the supposed numbers must lack some properties of normal numbers, which can make them not the solution you thought they would be.
Numbers don't actually exist. Which is why you can invent any number you want. As long as you follow your own internal logic, it's all gud.
Geek
Does that mean undefined stuff like infinity are numbers now? Time to math duel someone
This really shows how amazing math mathematicians really are. These guys took 12 or so episodes just to prove one point. I love you.
Words are just not enough to explain how awesome these series are. Only true mathematician can understand that how difficult it is to prepare such lectures.
I have been searching for such kind of study from many years.
I am truly great-full for this series.
beautiuflly explained videos... true masterpieces !!
I still remember when i thought
_____
\/ - 17
Was a joke in multiple choices
You deserve much more fame.
And a nail clipper
now he does
This is a really well produced series.
You have earned another subscriber :)
+avatar098 woohoo!
Welch Labs Another subscriber!
Ditto.
dito
And another.
Videos are long enough to get me hooked, short enough to feel not long enough. Very, very well done.
Really great series - the best that I have ever seen on lateral/negative numbers!
Thank you so much for this series. It's really great. specially with the accompanying pdf. just one small problem:
the picture used for Rafael Bombeli is actually François Viète. It appears in both the video and the pdf, so I though I would mention it.
Thanks again.
I’ve been here since part 1 of this mini series and I’m hooked! A new subscriber 😊 please keep up the good work 😁
you should do a couple of videos on calculus(limits and stuff)the Italians had their share of fighting over math so did Newton and Leibnitz!(kinda) plus your narration is awesome.
great video....
circuits brought me hear although i have completed differential equations and never have i heard such a brilliant explanation
Even though I don't yet fully understand the algebra you're working with in these videos, you present the story in such a thrilling way that I CAN'T WAIT to see the next video!
awesome video man, i'm glad i found this channel.
one minor thing, i noticed that the picture used for Rafael Bombelli is actually Francois Viete. i guess someone who uploaded it mixed the names up.
Yours videos are the best about imaginary numbers I've ever seen!! 🤩
Thank you very much to make this topic so interesting to listen and learn. 😄
This is a great series. You're are really good at this.
This is what should have been taught in "Introduction to Complex Numbers"
HISTORY of either Science and Maths plays a very crucial role in learning. What our textbooks have become is what i now reffer to as "L.A.M.E- Lost And Mug-up Era"
Love you brother........
I am just a Simple Sound Engineer and a Musician....
You teach here so Well and So Simply .. that i understand Everything....
And i feel like i am also a Mathematician .....
Thank you very Much.. I wish if you ware my math teacher in my High School.....
thank you... love you brother.. ♥
wow... what a ton of work and the the materials are fantastic.. God Bless you for taking some of my math fear away!!! I am wanting to study electronics and em theory and being comfortable with complex numbers should be so so helpful
these "complex numbers" series are awesome !! ... keep up man ! =)
Love this series! Amazing work
Your videos are incredibly creative and interesting to watch!
I love these videos they are so professionally made
this is magnificent! I am finally realizing what this all meant I forcibly learnt back in high school! Thank you for making these great episodes!
You really sparked my interest in imaginary numbers. Awesome video !!
I love this series so much... I have never understood what "square roots of negatives" are until now
Your videos are so engaging and elegant, time for a binge sesh
Man, you really did an awesome job! I am thankful for your videos! Ah I am so happy that you shared your knowledge beautifully here. Again, thank you!
25 years ago I've lerned this in high school .... now I'm starting to understand. Tkanki you
This series is pure gold!!
you have done real great job man, please make more, I will share your vidoes with whomever I know. Mankind should know you
really brillant!!
when i was in high school i had good grades in maths, but never really understood complex numbers (although I could follow the rules and work out the answers)
you really got deep into the problem but present it in such an easy to understand, and interesting way.
Thanks Sir!
Wow, amazing explanation on the history of imaginary numbers. Fantastic videos!
Thanks! I finally understand imaginary numbers thanks to this series. Definitely subscribed. She don’t they put it this way in math books....?
Really love this series.
MAN YOUR VIDEOS ARE GREAT!
"If you are a number act like one" LOL. I really wish my math teachers had the sense of humor like this.
These are really interesting and entertaining
Very good video!! It's great to see how can we teach math with fun!!
The best explanation I have seen on imaginary numbers!
Those videos are incredibly great.
Those videos have an amazing quality of content
His series is amazing!
to the makers of these videos - great work all of you👍
like literally
I didn't knew these things about imaginary numbers
4:05 those numberphile videos in the recommended though...
great series
Hi Welch Labs,
The portrait you used for Bombelli is in fact french mathematician François Viète (it says so in the portrait!).
I'm not even a math major or anything but you make learning so fun I just wanna watch it
1:25 That's image of F. Vieté, not Bombelli.
Yes, negative and complex numbers perfectly makes sense in geometrical (positional) interpretation. Great videos!
Lo acabo de encontrar y son muy buenas estas video-explicaciones, ventajas del internet y del youtube. Espero que tenga mas videos.
your videos are simply awesome!!!!
Like so many of these comments, I love the videos and show them yearly to my algebra 2 students learning about square roots of negative numbers. I especially appreciate all of the embedded humor in these videos! Outstanding job in every respect! I tip my imaginary hat to you!!
Great videos, thank you!
You could have just showed some graphical representations with voice overs and be done with it. Instead, you chose to do all that work, taking much time to explain what you really want. Great material, great use of humour. Don't even for a second think that all of us watching did not notice your hard work. The tiny high schooler inside the 26 year old me thinks these series is pretty cool. Thanks mate!
Thanks for watching!
An amazing series, so helpful for a school project, thanks so much
Thanks for watching!
This videos are gold.
This is fascinating thank you.
From around 700 something views to 700,000 something views. Good work man 👍
Incredibly interesting video!
U deserve SOOOO much more attention! Subscribing!
Great knowledge!
the most important that I learned is by inderstanding history we can learn principals that can help us solve problems like Bombelli noticed that every time people extend the number system (ex:adding fraction) they solve problems.
My new favorite Channel
nice series of videos!
You deserve more subscribers and likes!!
Gotta see the next one!
First video and I'm already a fan :)
wonderful video
next time my math teacher says "This problem has no answers, its UNDEFINED", I'm giving my math teacher a lesson about this
Retro Gaming - Clash Of Clans sqrt 0
its reasonable for her to say this though if it is not in your syllabus, it will just bring more confusion, you will probably learn it sooner or later
root zero is zero tho :/ as it is the only number that works consistently whats truly undefined is 1/0
The only thing in math that is undefined is diving by zero, and derivatives at a peak
It actually is undefined. I can prove it using Infinite Series
So interesting. I definitely want to look into imaginary numbers again. So far I haven't really encountered them much in diffy eq or multivari calc though, odd.
This is really cool approach to something being unsolvable
gem of a video!
excelent work
Nice one
Welch Labs, are you the russian mathematician I've seen in Numberphile's videos (IIRC) over the years? I think so.
If so, I've found the channel of my favourite math expert! Subbed AT ONCE.
Greetings from Italy. (=
Nice series
Best video on imaginary numbers
This channel definitely needs reputation. BTW, why are there dislikes to the video??
Amazingly well done presentation but why did you give Viete’s picture to Bombelli? Weirddd
You are a true math teacher mate
Great video! But the guy at the picture (0:27) is not Bombelli, that's François Viète
@cansomeonehelpmeout: Sure . . . but I make that mistake myself all the time.
True... and it is also written at the bottom of the picture
Julioprofe for the win. Nice video men. You won a new viewers.
OH my GOD. I can't believe this series is like some kind of HBO shows that keep me addicted!
Your writing is very neat
I was watching this video and just saw how old it was, 8 years like how did this pop up on my feed.
Your videos are awesome
Top marks!!
pleas keep making mathmatical videos like this. :D
best explanation ever !
Studying in korea, I didn't know anything about imaginary numbers except we call them i, and it's squar root of -1. watching your video, I feel relieved, due to your satisfying video full of inspiration and knowledge. Thank you very much.
thank you for this video and great work sir. i'm a physics undergrad from india and want to from know math theories more deeply , can you provide me names of
some good math history books or tell me how u researched on this topics.thank you sir
I like that office joke at 1:05
*Cough*
thank you sir....
real quality content
This is quality content
This video would have been great to have in high school.
0:25 Cardan's student, Rafael Bombelli made some incredible insights about what's really going on here. Let's remember why Cardan was stuck. The square roots of negative numbers ask us to find a number, that when multiplied by itself will yied a negative. Neither positive nor negative numbers will work. Bombelli's first big insight was simply to accept that if positive numbers won't work and negative numbers won't work, then maybe there's some other kind of number out there that will. Now if there is some other kind of number out there, a good follow-up question is, "What are we going to call it?" After all, we need to use it in our equations, Bombelli's approach was a very practical one. Rather than dream up a new name and symbol, Bombelli simply let the square roots of negatives be their own thing. In the past, mathematicians would have thrown in the towel here and declared the problem impossible, but Bombelli was able to press on simply by allowing the square roots of negatives to exist. 1:17 1:55 Is '√-1' a "real" thing? However, before we dismiss the square root of minus 1 as some abstraction invented to torture students, let's review what we've learned so far. ... 2:24 Let's make sure we're clear about what it means for the square root of negative 1 to be its own number. If our new number is truly a discovery and not an invention, it should behave like the other numbers we already know about. It should follow the established rules of algebra and arithmetic, and it turns out the square root of minus one does, for the most part. Just as we can split apart the root of the product of two positive numbers, we can also split apart the square roots of negatives. The square root of minus 25 splits into the square root of 25 times the square root negative 1. This process is important because it allows us to express the root of any negative using the square root of minus 1. The square root of minus 25 becomes 5√(-1). We can use this process to expand the root of any negative number, writing it as some number we already know about, thimes the square root of minus one. Let's quickly make sure that our new numbers follow the same algebra rules as our old numbers. In algebra problems with x, only like terms can be added and subtracted: 2x+3x = 5x, but 2+3x = 2+3x. Likewise, 2√(-1) + 3√(-1) is equal to 5√(-1), but 2+3√(-1) is just 2+3√(-1). Finally, unlike terms can be multiplied just as in algebra with x. 5 times x is just 5x, and 5 times √(-1) is just 5√(-1). Now, there are some cases where our new numbers behaves a little strangely, but these can often be avoided by first separating out the square root of minus one. √(-5) x √(-2) = √5 √2 (√1)^2 = -√10.
luv it. Just one question. If i (on the other side of the ocean) would like to print outyour ressources myself. Can I take an oath not to distribute it and get a digital version of your Workbook for a good price?
Always when i see an interesting way to explain something it makes me think about how schools are bad at teaching things.