Imaginary Numbers Are Real [Part 5: Numbers are Two Dimensional]
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- เผยแพร่เมื่อ 20 ก.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
Explosion elements by BlinkFarm, available for free at / blinkfarm
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: www.welchlabs.c....
![Imaginary Numbers Are Real [Part 6: The Complex Plane]](http://i.ytimg.com/vi/z5IG_6_zPDo/mqdefault.jpg)
![Imaginary Numbers Are Real [Part 6: The Complex Plane]](/img/tr.png)
![Imaginary Numbers Are Real [Part 13: Riemann Surfaces]](http://i.ytimg.com/vi/4MmSZrAlqKc/mqdefault.jpg)
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That rotating by 90° thing blew my freaking mind!
It is very interesting. There is a simple way to describe these rotations using other complex numbers called the roots of unity. If you take a trigonometry course, this may pop up.
Rynabunny, yeah(!)...the same here! (Even if, watching this series, I had _somehow_ flashed it in my mind.)
In just about 4 mins you have given me something really unique Welch Labs...
Having seen complex and imaginary numbers my entire career as ugly, hacky tools that were not but a necessary evil... til now, where it is like a complete 180, i could not imagine a more elegant and complete description of mathematical operations, and their relationship to their geometric representation, completely tieing a bow between the two. And now it honestly feels like that extra dimension has always been there, and I just was not smart enough to understand that until now. Imaginary/complex numbers just finally give us such a simple and elegant access to that dimension, completing the overall space. Clearly, brilliant minds realized long before, making so many areas of math so much easier to solve with that more complete view of the problem, as well as having such any easy to visualize view what is happening.
As *Rynabunny* so perfectly summed up, you blew my mind.
Going back to my undergraduate degree in Advanced Mathematics at Waterloo in 1989, not a single prof, textbook or any colleague or source of knowledge since has ever explained that concept, and it is so simple, yet so clearly 'YES', that is the missing piece...
This definitely should be taught in schools, and University as well, it took 4 mins, had I seen this earlier in my studies, it would have saved me so much trouble, and it simply would have made this particular subject so much more interesting....
Same here
@@Arrogan28 what a time to be alive, right? :)
Oh my god...... I'm training to be a maths teacher and at no point during my school education or my university degree did anyone use geometry to explain why you need imaginary numbers. It makes perfect sense! No doubt I'm going to be using that technique to teach my students for years to come so hopefully they understand it way faster than I did! Thank you so much :D
LukeRM If you ever figure this out, let me know: if numbers can be two dimensional, can they also be three dimensional? As in another axis of numbers, perpendicular to the real and imaginary ones?
Conner Baird It turns out that we can’t have 3-dimensional numbers if we want to keep the commutative and associative properties. See this: www.quora.com/Is-there-a-third-dimension-of-numbers
*However* we can have 4-dimensional numbers, designated with the letter ℍ and called the quaternions. These are in the form of a + bi + cj +dk. Unlike going from the Real numbers (ℝ) to the complex numbers (ℂ), there isn’t any way to go from ℂ to ℍ. (Another way of saying this is that ℂ is algebraically closed). You can’t get to the quaternions by doing the √i or any other way.
And we can even go further to higher dimensions, but only by powers of 2.
8-dimensional numbers are the octonions represented by 𝕆
16-dimensional numbers are called the sedonions and are represented by 𝕊
However, each time we go to a higher-dimensional number set we lose properties and eventually the numbers become very abstract. This is one of the reasons why we don’t have 32- or 64-dimensional numbers defined currently.
But going back to your original question, while we can’t have 3-dimensional numbers, we can use a projection of the quaternions that ends up as 3D space. This is like how a shadow of a 3D object makes a 2d shadow. Imagine the shadow of a 4D object being a 3D shadow and this may be a little more clear.
For more information on the quaternions I would watch the Numberphile video on it
Conner Baird Let me know if you have any more questions
id adds reality to the imaginary,genius as far as teaching and explaining
@joshua james because it is defined that way, and no, imaginary numbers (lateral numbers), are not negative numbers.
Can't believe complex numbers aren't taught this way in school. This is amazing.
you have no idea how much you helped me.
woohoo!
You are stupid af cuz iam not supposed to learn this shit but i still understand it
Hassan Melhem bravo we have new einstein here guys!!!
@@kenjimelhem6218 How does that make him stupid
@@strebicux6174 guys I forced my 13 years old brother to watch this video in exchange for letting him play a bit more of fortnite and he's the one who wrote the comment not me lol
OMG - I did a degree in Physics, and most grad work in opto-eletronics were "i" was used extensively........ and know 20 years later I finally feel I understand i for the first time. Wow, amazing, lost for words. You'll are amazing.
Me too
David Whyte wtf, im studying a degree in phisics and they do teach us this, and more in analisis. Its consider a fundamental subject, and you need it or you cant take any 4 couse subjets
How is ‘i’ used in physics?
@@markiyanhapyak349 exactly. How?
Tell us, tell us(!)!
I wish my high school maths teacher was as good as you. I'm a engineering grad student and use complex number all the time. Know the formulas and equations but I always think of imaginary numbers as a hack for computing. After watching this its becoming more meaningful to me. Thank you
so is there any end to the mind blowness of maths?
Nope. If you ever get a PhD and advance to the research frontier in some specialty of math, that just prepares you to add a little more to the mind blowness.
I respond “No.”. I'm not the only one.
Tal Roitberg, what is really “PhD”?
@@markiyanhapyak349 Piled Higher & Deeper
Mathew, than “Higher” must be like this: “higher”... .
I wish I could megalike videos. This is amazing.
Thank you!
No, thank *you*!
you can share all of his videos on all your social networks and to your friends on messages
This series is just amazing. Never understood how imaginary numbers are perpendicular to real numbers, and you just explained it so easily. Keep going with the awesome work!!
This is me, I was asking why the heck i is perpendicular to 1. And here I got the answer...
Thankyou! Why couldn't my £40 textbook just explain it as rotating through a second dimension?! Bloody waste of trees lol.
Ha-ha(←really quick), [a medium-long _sized_ pause here] bad searches… .
Ya actually I hate textbooks
That's why we use cosine and sine function
I literally cried at the end of the video as I got emotional over how beautiful the explanation is and how useful they really are
my understanding of imaginary numbers have increased so much from this XD thank you
As a visual learner, this video series has done more to explain imaginary numbers than any of my AP classes ever did.
I am trying to learn audio signal processing and wanted to revisit some algebra and complex numbers. Bumped into this video, and realised how easy they can be if studied in the correct way. This video is just genius how it's explained. It's like now I will never forget what complex numbers anymore :D
only 700 views? shame on your people, this guy is the only person who describes imaginary numbers in an understandable way, that was a question which baffled my mind for years and I got the answer here
+roozbeh halvaei dude, everyone reading your message is by definition one of the people who *has* viewed this video. of course you know this
It's up to is raise that number.
I know this is an old comment but it should make u happy tht now there is over 800k views. Lesson in life is patience.
I don't have any math that says so, although I feel like it should exist by now, but if numbers are two dimensional, could they be three?
+coltonHD Great question! This was an open question in mathematics (I believe in the early 19th century) - I can't recall the complete argument at the moment - but it was proven that 2 dimensions are it - we don't need more. Later in the series we'll talk about mathematical closure - which will realate to this. Thanks for watching!
+coltonHD The "complex" numbers are two-dimensional, and the Fundamental Theorem of Algebra says that's all you'll ever need when adding, multiplying, and raising to powers (just like negative numbers are all you need when working with addition). There are higher-dimensional numbers, though. The quaternions are four-dimensional, the octonions are eight-dimensional, and the sedenions are sixteen-dimensional! Surprisingly (to many mathematicians!), it's impossible to make three-dimensional numbers. You simply can't make a three-dimensional system behave like numbers should behave.
+chunkyq Do you know where I can go to find more about why three dimensional numbers don't work?,
I don't know of anything for a non-mathematician, but you can try these:
en.wikipedia.org/wiki/Hypercomplex_number
www.quora.com/Complex-Numbers/Do-higher-dimensional-numbers-exist-Or-do-they-stop-with-2-dimensions-complex-numbers
math.stackexchange.com/questions/751106/complex-number-with-3-dimensions
+chunkyq And as I know after octonions (or sedenions) we are not able to do math any more: addition and multiplication are breaking down and those higher dimensional numbers have no sense as an algebraic system.
"Numbers are 2 dimensional" Mind blown! and I know is just the tip of the iceberg.
Can't believe how interesting this simple problem can get... Literally getting more and more excited after watching each part....
Everything ties in beautifully now. I have honestly learned more in 15 minutes than an entire year's worth on the subject in school could have offered. Thank you!
Wow I never though about negative multiplication like that 2:20
Maybe because it's to overly complicated and would waste too much paper drawing each time you came across -a*b, a*-b, and -a*-b .
Also, on a slightly different note, he only showed that demonstration for exponets meaning multiplication which makes me wonder how you would draw qoutients, sums, and differences involving negatives?
*NoOo...‽*
i wish i had seen this video twenty five years ago, when i was in school and struggling with these concepts. thank you.
This series strikes the skeptical bee, with such simplicity and clarity, leaving no option but to yield to the propositions.
Great efforts. Thankyou
Haha, awesome. Thanks for watching!
Well, not *ALL* real numbers, strictly speaking, get bigger as they get higher powers. 0.5^2 = 0.25, and both 1^2 and 0^2 stay the same (1 and 0)
Good Call! I'll fix this in the workbook.
+Welch Labs Some times formula breaks when the varient x ∈ [0, 1], like x² ＞ x is be true only if x ∉ [0, 1]
+Michael Owen is true*, gosh bloody grammar
You just gave me a clue about how to differentiate between the real numbers line along with imaginary numbers line that they are perpendicular to each other and the input of a function graphed along with its perpendicular output line.
They hide in a perpendicular dimension is absolutely beautiful. Pure unadulterated genius!
I feel very sleepy now, but I could not stop myself from watching these series till part5. Fantastic
I go on, and on, and on( instead). 😏 😏·😁 😁·😏 😁 😊 😇 😎 😎
About 1:30 in, you wrote 2^1 = 1
So im not crazy
Ooooo, there's even a part six. Kickass!
This is an amazing series thank you so much
This is making a lot of sense. I have used imaginary numbers, but I never visualized them this way. I had no problem accepting negative numbers, this is just the next step
these videos are amazing
In college, I studied electrical engineering, signals, phasers, etc.. we _used_ complex numbers in this way all the time, but I can't remember ever learning this fundamental explanation of _how_ imaginary numbers represent rotations. This is brilliant!
2^1 is 2, not 1.
I'm lost of words to express, how easy he made the entire concept... Never seen a master like this!... I am a big fan of you, turned the bell icon on for all !
1:30 did you just say 2^1=1... jk youre awsome dude
He probably meant to say 2^0=1 then 2^1=2
The other videos are very good but this one was just amazing. Such a good way of explaining it! Cool stuff.
MIND=BLOWN
This type of video that teaches math through the historical perspective of the evolution of the ideas is so helpful to me.
What an awesome and cool explanation
Damn dude that moment when you explained multiplication as rotation on a number line was eye opening. it makes so much things clearer
Great!
So is there a third dimension?
Best complex number explanation hands down!!!
this was Great ! I came here because I have heard that imaginary numbers are real. However, now I think that even negative numbers are not real P: I started thinking of all things we took it for granted (why -3 * -3 = +3) and other stuff. you did really really great work. I hope your next video will answer my Qs
Thank you .. Really!
+أحمد عبدالرحمن العودة We use Negative numbers in reality, subtraction can be seen as the addition of negative numbers, which proves them pretty quickly to me.
This is the best explanation I have ever heard! Not just about this subject, but just the best explanation of any subject ever
3:31 - Mind = Blown
So awesome!
Woohoo!
Seriously, this series has reignited my interest in mathematics. It's had such a profound effect, thank you!
It's totally changed the way I think about mathematics.
@@insideoutface Dude it's just 6 fucking videos, If this little amount of info on complex numbers gets you so excited, a course on complex analysis would certainly make your heart stop beating.
I've worked a lot with complex numbers in the past, but this straight up blew my mind. "i" being not only halfway from -1 to 1, multiplicatively, but a physical rotation of the number is remarkable. I immediately thought, "What about 45 degrees?" and proved to myself that 1+i rotates by 45 degrees per multiplication, and I assume all other rotations work the same. That is simply amazing. Thank you for broadening my view of numbers!
Multiplying 1+i also stretches by sqrt(2) ;)
Is there 3 dimensional numbers ? 😮
What do you mean with 3 dimensional numbers? There are obviously 3 dimensional vectors (meaning vectors which can be written as a*i+b*j+c*k) which you can add and scale by real/complex numbers just fine, but don't have such nice properties when multiplied like the complex numbers or the quaternions
I truly appreciate you breaking down complex number as such. It is unbelievable.....I have never heard or read it explained in such an easy manner
You stretch out one video topic to how ever many there is in this series and waste so much time repeating what has been said in previous videos and the information has nothing to do with understanding complex numbers. I never leave negative comments but I feel like this is just and effort to get more ads per minimal content.
I particularly haven't seen much repetition. Considered the time of all videos so far (less than 25min), I'm amazed how much I've learned.
Well, I'll say this, I learned more than I expected to and probably more mathematical history that I'll ever remember. But I guess he's just further explaining why imaginary numbers are real.
This actually makes sense. Thank you for making math enjoyable. It helps me when I understand the history of the problems involved and I understand how the logic developed.
the use of geometry to explain how i behaves is fantastic.
incredible video series!
I literally had a Eureka moment with every video in this series. You are awesome.
woohoo!
ive never had a math lecture this easy to understand
seriously, thanks
I’ve seen “rotating imaginary (lateral) numbers by 90” in a fun book about math when I was like 10, obviously understood nothing and thought “cool, they have a carousel!” (because the numbers in that book actually had a carousel :)). So seeing this years later when I know just enough math to actually understand it and it making so much sense is so amazing!!! Actually blew my mind!
Also, agree, lateral numbers are such a cool name! The numbers in the book were so upset at being called imaginary, it stuck with me😂
This 4:37 minute video answered questions i had to years. To give an example, i always wondered why by multiplying 2 negative numbers we get a positive. The arrow demonstration almost made me freak out. And the rotation answered things I learned in collage in calculus and shit. This series is more valuable than gold
I enjoyed the background on how historical figures came across imaginary numbers, I'd be curious to hear how someone first made the leap to modern geometric understanding.
Dude i'm seriously saying our teachers never explained these approaches to us you are actually a good teacher . also try to make videos on pre-calculus and calculus. it would we regarded as heavens itself
I never completely understood these during high school, now that I saw this it makes perfect sense to me!
Sir, that explanation is perfect, that’s the way it should be introduced in math classes!!!
I like how this is an excellent example of the power of generalization. Great job by highlighting it geometrically 🎉
I always looked at them as if they were a secondary, temporary graph (creating a supplementary origin) to conveniently move around problems or create new ones. This was another interesting perspective that's much easier to put into an explanation.
Delightful presentation. I have employed imaginary numbers in my work routinely for decades, but never had so much insight into them until watching your series.
this particular video should have 7.7billion views.
I'm actually so deeply interested in learning in this even if in school this isn't even taught yet. It's just kind of enthralling
Oh my god, the representation at 2:40 is so mind opening
I love the small things that come up in the right bottom (funny stuff, counting from 90, and so on) and kept subtle, so only those you want to notice, do
OMG it was that simple?! I'm 28 and just now (12 years later) I could understand were those "imaginary" numbers fit! Thank you for such a comprehensive explanation!
Woohoo! Thanks for watching!
THIS is the video that makes all the difference. I've grokked the algebra of complex numbers for decades, but - probably because I was introduce to R^2 before I was introduced to C - never grasped the geometrical interpretation of complex numbers. Yes, I've seen it used countless times, but it has always been presented as a convenience, rather than something rooted in the nature of multiplication. Excellent presentation!
Wow!!!! I taught tutored algebra for years and never looked at imaginary numbers as if they were real. Ive NEVER learned this story and gives me such a deep sense of joy to learn this!! Like why were we not taught in that way. This makes sooooo much more sense. Its like I was walking a tight rope and just finally stepped on to a plane!!! WOW!!!
I haven't learnt this in school. but after watching your videos, I feel like knowing about it. thanks
Never thought I would be checking math videos one after another when I slack on TH-cam.
The simplicity with which you explain it so clearly makes me wish I had chosen a different career. Can't think of any words more powerful than that.
This piques my curiosity in quite the surprising way. We learned these in algebra but we only skimmed it. But it definitely looked interesting to use, and i was a little disappointed it wasn’t used more over X
The countdown in the lower right corner to blowing up our minds was spectacular
This channel deserves a lot more attention!
Beautiful series.. if only this was available in high school
the geometric representation is insanely clever! it makes so much sense, so easily. like everyone else has said, it blows my mind!
"Numbers are two dimensional!" Can't wait for the day when we discover even more dimensions. This is a great stimulus for learning--and the perfect response to when some did doesn't want to learn algeabra--"Do you want to keep thinking like cavemen? You're only thinking in one dimension!"
You've done in 20 mins what my 4 years of high school math could not
this is so beautiful and elegant
I have a degree in maths and physics, didn't learn any new maths but the ideas and history, Certainly! Awesome vids
As soon as you showed the repeating pattern of 4. I starting imagining in what way they could be linked together. First I visualized the symbols clumped together like kids sitting in a circle. I then imagined a plane splitting them in half. This would group (i -1 | -i 1) OR (-1 -i | i 1 ). I wasn't sure what to make out of that. So instead I visualized the numbers "across" the circle (their opposite, consequently the inverse) as linked by by a line. Viewed from above I immediately saw a quadrant, like in a graph. 2 dimensions of numbers. And that's when it clicked, multiplying by an imaginary number would rotate us 90° in that quadrant, basically flipping between imaginary and real. Just like multiplying by a negative flips the number between positive and negative.
And then when you showed the 90° demonstration. My mind was blown that I just understood that a few seconds earlier, by myself. The slow lead-up and historical explanation from your previous videos was great and I'm sure helped put me in the right mindset to understand the concept. Thank you.
Bing watching this series and enjoying more than any netflix series....just loving it
watched this video 20 years late, but now concept is clear :D
Who needs Netflix when there's this banger on TH-cam
Wow! just WOW! the rotating thing is extremely illustrative, simple and truly elegant! I've just discovered your channel and I've been watching the series and feeling like I've found a gem! Thank you!
You have answered a question I have had for as long as i've known imaginary numbers. A long ass time. Thank you
Maths is really wonderful.
That 90 degree concept really wonderful.
Very informative. A vastly superior way of teaching complex numbers. Makes total sense. Wish I’d been taught in this manner.
Glad it was helpful!
I haven't studied imaginary numbers yet, and I don't know why I am watching this video, but it's very clear and interesting. Thanks!
Got a new subscriber 💙
Geometrifying Trigonometry expands this view to more deeper permutation spaces
Geesh why didn’t they teach it to us this way in high school? I teach geometric planes in 8th grade science because of Astronomy and all the concepts associated with orbits and gravity. We graph in two dimensions and I explain the z axis and though imaginary numbers aren’t at all brought up in any concepts, it’s still great to see it explained as a perpendicular plane that extends above and below the x axis. Thanks!!
Funny thing is multiplication on the imaginary plane always corresponds to some rotation.
x+iy is the general expression for any number. x is the real part of the number and iy is the imaginary part. You can use that arrow notation to represent any number using a flat 2d surface. If you do so, you would kinda end up something like vectors. That notation is called the Argand notation and the plane which we plot the numbers on is called the Argand plane. If y=0 and x≠0 then your number is on the real axis and your number is on the imaginary axis for vice versa. If neither is 0 then you number is on the plane somewhere, not touching any of the aforementioned axis.
With Argand notation you can use summation in a very straightforward manner, you can just use vectors sums. But multiplication is a bit more interesting. Suppose you have two numbers n1 = x1+iy1 and n2 = x2+iy2 (It looks disasterous but im writing this on my phone so this is the best i can do). Summation is easy:
n3=n1+n2=(x1+x2) + i(y1+y2)
For multiplication, we could simply write them in non vectoral forms and get the answer
n3=(x1+iy1)*(x2+iy2)
=x1*x2 + i(x1*y2+x2+y1) - y1*y2
But in Argand's notation the resulting vector's length would be calculated by multiplying the lengths of the multiplicand vectors (in this case |n1| and |n2|). But dont forget that vectors' orientations also matter so we need to find it too. To do that we need the orientations of the each multiplicand vector. Say, the first vector n1 makes the angle θ1 with the real axis and the second vector makes the angle θ2. Then, the resulting vector n3 would make the angle θ3=θ1+θ2 with the real axis! I know it looks hard to understand but lets try the example in the video.
For n1=i and n2=i we have
x1 = 0
y1 = 1
x2 = 0
y2 = 1
The length of n1
n1 = √[ (x1)² + (y1)² ] = 1 and same goes for n2. So the length of n3 is
|n3| = 1*1= 1
The number n1 makes π/2 radians of angle with the real axis and so does n2. Then n3 must be a vector of length 1 and should make π radians of angle with the real axis. Then, the answer is -1.
Excellent explanation of Imaginary Numbers , series is great
Thanks for making this video series. I remember learning about i in high school but never fully understanding it. I still am having difficulty, but I'm approaching understanding thanks to these videos!
Thanks for watching!
90 degree rotation blew my mind...Super bro
I was lucky enough to meet this man and attend his lecture.
TH-cam should give us the option to give 2 thumbs up. Thank you for the series.
OMG These videos answer all my questions with regard to imaginary numbers for years! Thanks a lot!!!
Wish I would'a been taught his 50 years ago. Great explanation!
The easy way I understand this is that to conceptualize area as we do 1D numbers (e.g. negative area), you need 2D numbers.