14 ms·

How imaginary numbers were invented [video]

187 points by peter_d_sherman 5y ago

nabla9 5y ago
That was one of the most interesting 20 min history lessons I have ever watched.
- VHRanger 5y ago
  Veritasium always makes great stuff
- satori99 5y ago
  I wish I was taught this history when I was taught the math. My schooling provided next to no historical context when learning mathematics. My math teachers just taught the math, and my history teachers were probably largely unaware of math history.
timonoko 5y ago
Ouch. There is some logic and sanity in e^ix.
- coremoff 5y ago
  you might like this: https://www.youtube.com/watch?v=mvmuCPvRoWQ https://www.youtube.com/watch?v=mvmuCPvRoWQ
- mode80 5y ago
  you would like this: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...
motohagiography 5y ago
If we still had math duels, my life would have taken a very different trajectory. We need these.
adamnemecek 5y ago
I think that i is not interesting because i^2 = -1 but because i * conjugate(i) = 1.
- jonsen 5y ago
  It's the same thing i * i = -1 <=> -i * i = 1
  - adamnemecek 5y ago
    I know but the semantics are different.
    - jonsen 5y ago
      In what way does multiplying an equation with -1 change its semantics?
      - adamnemecek 5y ago
        It obviates the unit circle and more importantly, for split complex numbers, you’ll have j*conjugate(j) = -1 and the -1 obviates the connection between split-complex numbers and reflection.
VHRanger 5y ago
I'd say mathematical concepts are "discovered" rather than "invented". Even something like complex numbers.
- mxwsn 5y ago
  I wonder if one can place mathematical concepts on a spectrum from discovery to invention? To me, the pythagorean theorem feels much more like a discovery of a "hidden" eternal truth that was once beyond our grasp. But to me, complex numbers seem more like a notational choice, more akin to an invention. To put it another way, I would expect aliens to have the pythagorean theorem, but perhaps different notation for complex numbers, like representing them with matrices and computing over them with linear algebra. Such a representation would reduce the distinction of the "imaginary axis" which is really not that special. I may be naive; I haven't worked or thought with complex numbers since completing my basic education.
  - gizmo686 5y ago
    Definitions are invented, theorems are discovered. The Pythagorean theorem is a discovery about Euclidean geometry. But in order for that discovery to be meaningful, one must first invent Euclidean geometry, or at least something sufficiently similar to Euclidean geometry.
- count 5y ago
  Pure math concepts like complex numbers are not naturally existing, just waiting for us to find them, they're human-defined tools to describe things. Like new words, they're invented. They wouldn't be there without us, as they are, for the most part, artifacts of our cognition.
  - pohl 5y ago
    If that were the case, would you expect another civilization on some rock in different galaxy to arrive at entirely different concepts, or ones isomorphic to our own?
mrtedbear 5y ago
Veritasium is fantastic, I can't think of many current or past TV shows that are of a similar quality to his videos. A while back he did a video of his own startup story, which was very cool: https://www.youtube.com/watch?v=S1tFT4smd6E https://www.youtube.com/watch?v=S1tFT4smd6E
travisgriggs 5y ago
I love Veritasium. Derek has done a number of videos that are awesome like this. The videos are great and educational, but he also manages to subtly weave in an almost spiritual subtext to them as well. For example, in this one, you learn about imaginary numbers in some cool ways, history, how awesome e^ix is, etc, but also this yen-yang balance between embracing not only what is real but taking a chance on what is imaginary. Some of my other favorites: - https://m.youtube.com/watch?v=rhgwIhB58PA https://m.youtube.com/watch?v=rhgwIhB58PA - https://m.youtube.com/watch?v=HeQX2HjkcNo https://m.youtube.com/watch?v=HeQX2HjkcNo - https://m.youtube.com/watch?v=OxGsU8oIWjY https://m.youtube.com/watch?v=OxGsU8oIWjY - https://m.youtube.com/watch?v=3LopI4YeC4I https://m.youtube.com/watch?v=3LopI4YeC4I - https://m.youtube.com/watch?v=pTn6Ewhb27k https://m.youtube.com/watch?v=pTn6Ewhb27k I think he’s going to lose the $10,000 bet on the car though.
- kzrdude 5y ago
  I didn't click through all (mobile, too slow) so don't know if it's in there, but one favourite i have is the one where he has an ice cube, a metal block and a plastic block and ask people on the street which block is colder. It teaches a lot about what temperature vs feels cold vs heat is.
- lxe 5y ago
  Some of popular educational/informational channels and podcasts make this patronizing assumption about the audience, watering everything down to make it "accessible". Veritasium doesn't do this, yet maintains huge popularity.
- agumonkey 5y ago
  That's my favorite video of him. This one ties history, education, simple geometry, advanced abstractions in an entertaining script. Pretty high bar.
- jasonwatkinspdx 5y ago
  > I think he’s going to lose the $10,000 bet on the car though. Are you referring to the wind powered propeller car bet or is there a new one?
cool_dude85 5y ago
I've never liked the framing of "imaginary" numbers as "not reflecting reality" or somehow being less "real" or something. It's just an ordered pair of numbers with a convenient extension of multiplication that preserves nice properties of the reals. I think it's because of the name. If you called them double numbers or paired numbers nobody would say that.
- amai 5y ago
  The problem is not the term "imaginary", the problem is the term "number. Unlike any other numbers imaginary numbers cannot be ordered by size. So in that sense they are not numbers at all. In fact they represent rotations in two dimensions. They should be better called rotation operators instead of imaginary numbers.
- mathnmusic 5y ago
  Gauss complained about the name too. He suggested using "Direct, inverse and lateral numbers" instead of "positive, negative and imaginary numbers". Imaginary numbers are not pairs. Complex numbers are. Continuing Gauss, I'd rename "complex numbers" as "planar numbers".
  - Natsu 5y ago
    I kinda think of them as 'orthogonal' numbers.
    - carlmr 5y ago
      That I think is the best alternative name I've heard yet. Since you describe two orthogonal dimensions. Also vector numbers could be somewhat of a useful name, since they behave like a 2D vector (or even higher dimensional vectors for e.g. quaternions)
      - ogogmad 5y ago
        "Vector numbers" for something purely 2D seems dodgy to me. There are lots of unital algebras, even in 2 and 4 dimensions, and none of them should hog the name "vector numbers" as they all have equal entitlement to the name.
ralusek 5y ago
> Imaginary numbers exist on a dimension perpendicular to the real number line. Correct me if I'm wrong, but I don't think that imaginary numbers exist in a dimension with any sort of fixed spatial relation to real numbers. Isn't the fact that we chose to graph the imaginary axis perpendicular to the real axis on the complex plane simply arbitrary? Or maybe better described as "a functional way to visually plot the negative root component of a number." My understanding is that visualizing the imaginary portion of complex numbers on a spatial plane with perpendicular axes also just happened to get a convenient 90 degree rotation by multiplying by i, and rotations like this just happen to show up all over nature. For example, the wave function they mention in the video, where they say it has an "imaginary component," seems like complete bullshit to me. You could write a program with real numbers in order to graph that wave...in fact I guarantee that's how a technical artist visualized it. In fact, it's the same with the Mandelbrot set, which supposedly "exists on the complex plane." No...it doesn't. It just happens that if you describe the arbitrary rotation operation that is performed in order to define whether or not a coordinate falls within the Mandelbrot set or not, the rotation from the way we've chosen to graph the complex plane means the equation can be written very concisely like this.
- AnotherGoodName 5y ago
  Every independent variable should be graphed perpendicular. X and Y when graphed are perpendicular. If they are not graphed perpendicular it indicates that a change in X position must change Y (or vice versa) whereas being perpendicular makes it clear that they are independent and you can change one without any thought that you're changing the value of the other. Complex numbers have 2 independent variables. The real and the imaginary parts. You can change one part without being forced to change the other. So you graph them perpendicular. It's not really a convention, it's a rule that each independent variable is perpendicular (or else they aren't independent!). Complex numbers have 2 independent variables.
  - ralusek 5y ago
    Well yes, if we want to plot any sort of independent values, we have 3 spatial axes we are familiar with and will use them to graph accordingly. But I'm not going to argue that "housing prices therefore exist on an axes perpendicular to square footage." I just plotted the relationship that way.
pphysch 5y ago
The innovation of "imaginary" numbers is that we can concretely reason about multiple dimensions in a unified manner (unlike classical geometry, which relies on geometric primitives). `i` is not some magic, "imaginary" value, it is an invented syntax that means something like "rotate by tau/4 radians" or "orthogonal to the default vector". It turns out that physical problems are much more accurately modeled in 2 dimensions than in 1 dimension, and even more so in higher dimensions. > Freeman Dyson: “Schrödinger put the square root of minus one into the equation, and suddenly it made sense … the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers.” This quote is emblematic of the mysticism that some mathematicians and academics cannot resist using to advance their careers as public intellectuals. Reality is certainly not based solely in "real numbers" (one dimension), but nor is it based solely in "complex numbers" (two dimensions). The idea that mathematics is "the language of the universe" that can be precisely "discovered" by brilliant minds is a ridiculous notion that only serves the status of the mathematical elite. Mathematics is fundamentally about designing models and abstractions that help us reason about real phenomena with minimal cognitive resources. Everyone does it, and anyone can do it. Disclaimer: I have a degree in mathematics. "All models are wrong, but some are useful" [1] [1] - https://en.wikipedia.org/wiki/All_models_are_wrong https://en.wikipedia.org/wiki/All_models_are_wrong
- BeetleB 5y ago
  Nitpick: Anything in reality that can be modeled with complex numbers can be with real numbers. Complex numbers provide incredible convenience, but that's all. There's no phenomenon that requires complex numbers to explain - Schrodinger's Equation included.
  - teleforce 5y ago
    There are real, complex, quaternion and octonion numbers. Saying that we can just get by only with real numbers is really being dishonest and in denial, it's just like saying we do not need OS and higher level languages because bare metal and assembly are all that we ever need. As a rough analogy, certainly we can travel all over the world on foot by walking and swimming, eventually we will get there but in reality we will not. But the more correct analogy is that we certainly cannot go to space on foot, and higher dimensional complex and hypercomplex numbering systems is like our spacecrafts. Ironicaly the famous gimbal lock problem for spacecraft navigation can be easily solved using quaternion as opposed to disaster waiting to happen if we use only complex numbers [1]. Einstein certainly will have never come up with the infamous equation E= mc^2 and corresponding relativity theory if not because of quaternion numbers [2]. [1]Gimbal lock: https://en.wikipedia.org/wiki/Gimbal_lock https://en.wikipedia.org/wiki/Gimbal_lock [2]Quaternions in University-Level Physics Considering Special Relativity: https://arxiv.org/ftp/physics/papers/0308/0308017.pdf https://arxiv.org/ftp/physics/papers/0308/0308017.pdf
gregfjohnson 5y ago
A Danish cartographer named Caspar Wessel came up with an early formal treatment of complex numbers, in his work "On the Analytical Representation of Direction" (wikipedia has a nice article about him). It was published in an obscure forum, and predates subsequent rediscovery of complex numbers by others. His formulation is IMHO beautiful, intuitive, and compelling. He did it in terms of directions on a map, replacing the "sign" of a conventional real number with a "direction" or "compass heading". So, one might say, "the nearest Starbucks is two blocks east and one block north". He was simply using what became known as the polar form of complex numbers. One can follow intuition and define reasonable notions of addition and multiplication by real values. But what of multiplying two "Directions"? Wessel derived what multiplication must mean, and went further in deriving a large number of identities involving his newly discovered "directional numbers". If you pick a specific important pair of directional numbers, the multiplicative identity (call it "1") and a number 90 degrees away from it (call it "i"), it is convenient to represent any directional number as a the sum of scalar multiples of these two numbers. Then, one considers the simple formula "(i + 1)(i - 1) = i^2 - 1". A straightforward geometrical argument demonstrates that i^2 must be equal to -1. ("Show HN": gregfjohnson.com/complex)
lxe 5y ago
I remember just playing with my good old TI-83 Plus Silver Edition, plugging a bunch of symbols together, like e, and i, etc... when I suddenly got the e^i*pi = -1. How in the world do these completely unrelated (to me, at the time, at least) constants ended up with -1? It was mind blowing. This made me actually interested in math.
- vishnugupta 5y ago
  Thanks to the way math is taught in India I still have very little understanding of so many of these concepts though I'm very good at remembering formulae/steps, mechanically calculating them, and applying them to real-world problems. It wasn't until I stumbled upon Better Explained[1] I began to appreciate the need to understand math at a deeper/common-sensical level. Now I make it an effort to teach math to my son using examples, analogies so that he gets concepts like positional numbering system, why we need it, its advantages etc., https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...
- pjbk 5y ago
  If you realize where e comes from, which is compounding and what we call an exponential process, being that "the current rate of change is proportional to the present value", then the exponential of a complex number is no different to the exponent of a real number. In the case of one-dimensional real numbers you assign the slope of the function to the current value. The complex case is exactly the same, but if you think instead in the two-dimensional Argand plane and complex algebra, the slope is the tangent to a circle proportional to the angle at that radius (which curve is the one that at every point the slope is equal to its complex value?). Therefore with the rules of complex arithmetic the tangent provides the "curling" effect in the curvature of a periodic circle, while in the real case you get the common compounding shape of the exponential curve. The relation to sine and cosine is just the projection into direct(real)/quadrature(imaginary) components in either fixed or intrinsic coordinates. Same when you expand the exponential into its complex power series. BTW, this is actually the essence of infinitesimal transformations in continuous groups and the exponential map, which generalizes this concept to other types of numbers or abstract objects (i.e. Lie group theory).
zsmi 5y ago
I really enjoy looking into the history of mathematics and physics. I think it gives one a much better appreciation of why things are defined the way they are, and also the limitations of those definitions. There is a really great book on the history of imaginary numbers. The history mostly focuses on how i was used to help solve algebra problems, so definitely one should be comfortable with high school algebra to get something from the text, but I don't think one needs much more math than that for the first half of the book. The second half gets more into how various use cases developed, in those chapters basic college level calculus would be a major plus. I read it more than 10 years ago though so no promises. :) An Imaginary Tale: The Story of √-1 Paul J. Nahin https://press.princeton.edu/books/paperback/9780691169248/an-imaginary-tale https://press.princeton.edu/books/paperback/9780691169248/an...
vishnugupta 5y ago
While I enjoyed the history part towards the end it got super hand wavy; especially as it went from describing "i" as a rotation to e^ix; Better Explained does a fantastic job of explaining these from the first principles. https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid... https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...
- wyager 5y ago
  Macdonald's Linear and Geometric Algebra is an excellent GA text which not only geometrically derives complex numbers (and rotation via their exponentiation), but also quaternions.
6gvONxR4sf7o 5y ago
My favorite mental model for imaginary numbers is that they are placeholders or stepping stones. A speed of (30 + i*10) miles per hour is nonsense, but sometimes in the course of calculating a nice real 40 mph, you might take a detour through some steps including complex numbers. Having them handy sure will make the derivation easier. Same as negative numbers. I can’t be negative six feet tall, but negative feet (as a magnitude) make the calculations way easier. You can always reformulate these derivations into derivations that don’t use e.g. imaginary numbers, but there’s no need to. If I were apointed Senior Exectutive Math Concept Namer For Humanity, I’d call them something like “algebraic placeholders” or “algebraic closure stepping stones” or something in that vein instead of “imaginary numbers.”
- jasonwatkinspdx 5y ago
  Yeah, this is the intuition I've come to, particularly after reading Quantum Computing Since Democritus. If you'll let me be slightly hand wavy, you can get to Quantum Mechanics by generalizing probabilities in two steps: allow them to be negative, then allow them to be complex. Negative probabilities are quite unintuitive but seeing them as an accounting placeholder similar to how negative numbers are vs counting numbers helps it all click imo. The other big thing is that if you want algebraic closure you need complex numbers or something isomorphic to them.
echopurity 5y ago
This video is epistemological trash.
dexwiz 5y ago
Real numbers are for straight number lines, and imaginary numbers are for circular number lines. That is about all there is to it. It's really a shame we introduce imaginary numbers as quirky work around for negative square roots. I would say that is more of a side effect. Multiplying positive numbers always results in positive numbers, but due to the circular nature of imaginary numbers you can get positive or negative results. Complex roots are like saying this equation has no solution in a traditional Cartesian/Euclidean world, but does have solutions in a rotational one. I think we if introduced imaginary numbers as rotations initially, the seemingly magical things like Euler's Identify and Formula would look much more trivial. EDIT: I would also say the lie that we live in a 3+1D world (X, Y, Z + Time), helps confuse the mater. We probably live in a 6D + 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time. Just as you can conceive of a flatland 2D universe without a Z axis, you could conceive of a 3D world without rotational orientation.
- mmcdermott 5y ago
  I really like this explanation. The negative square roots explanation never sat right with me in high school and I think this touches on why.
  - CamperBob2 5y ago
    The way I think about it: - To express direction in Cartesian space, you need signed numbers - To express 2D rotation, you need imaginary numbers - To express 3D rotation, you "need" quaternions (in the sense that they're the smallest way to represent an arbitrary rotation)
- dls2016 5y ago
  > introduced imaginary numbers as rotations initially How would you do this without developing the exponential first?
  - zwkrt 5y ago
    Well any complex number a + bi can be described as a transformation matrix |a -b| |b a| So you could start by just asking what it means to transform the plane
rspk 5y ago
There is also a great playlist by Welch Labs called "Imaginary Number are Real" that I really like: https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
gkfasdfasdf 5y ago
I really appreciated the demonstration of solving the equations with shapes. It was like watching algebra's origin story.
amai 5y ago
Imaginary numbers were invented too early, that is why the usual explanations following the historical development are so mysterious. The notion of Sqrt(-1) does not help to explain what they actually are. In fact imaginary "numbers" are not numbers. They cannot be ordered by size. Imaginary "numbers" are also not vectors in the geometrical sense. This is because you can multiply two geometrical vectors and get a scalar as result (scalar product). But if you multiply two imaginary "numbers" you get another imaginary "number", so this is different to a geometrical vector. In fact imaginary "numbers" are transformations or operators, because they multiply like operators/matrices and they act on geometrical vectors like rotation matrices. Group theory makes all that very clear: Imaginary "numbers" are just the elements of the group SO(2), the group of rotation operations in 2D (multiplied with a scaling factor). However the invention of group theory started only with Galois in 1832 when trying to solve not the cubic, but the quintic. And the powerful notation of matrices as is ubiquitous nowadays was only invented in 1913 (by Cuthbert Edmund Cullis, see https://en.wikipedia.org/wiki/Matrix_(mathematics)#History https://en.wikipedia.org/wiki/Matrix_(mathematics)#History) Unfortunately mathematicians (and YouTubers) don't stop to teach the confusing outdated notions of imaginary/complex numbers rooted in the 18th and 19th century. And unfortunately this video belongs to the many videos just repeating the same old stuff over and over again.

nabla9 5y ago

That was one of the most interesting 20 min history lessons I have ever watched.

VHRanger 5y ago
Veritasium always makes great stuff
satori99 5y ago
I wish I was taught this history when I was taught the math. My schooling provided next to no historical context when learning mathematics. My math teachers just taught the math, and my history teachers were probably largely unaware of math history.

timonoko 5y ago

Ouch. There is some logic and sanity in e^ix.

coremoff 5y ago
you might like this: https://www.youtube.com/watch?v=mvmuCPvRoWQ https://www.youtube.com/watch?v=mvmuCPvRoWQ
mode80 5y ago
you would like this: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...

motohagiography 5y ago

If we still had math duels, my life would have taken a very different trajectory. We need these.

adamnemecek 5y ago

I think that i is not interesting because i^2 = -1 but because i * conjugate(i) = 1.

jonsen 5y ago
It's the same thing i * i = -1 <=> -i * i = 1
- adamnemecek 5y ago
  I know but the semantics are different.
  - jonsen 5y ago
    In what way does multiplying an equation with -1 change its semantics?
    - adamnemecek 5y ago
      It obviates the unit circle and more importantly, for split complex numbers, you’ll have j*conjugate(j) = -1 and the -1 obviates the connection between split-complex numbers and reflection.

VHRanger 5y ago

I'd say mathematical concepts are "discovered" rather than "invented". Even something like complex numbers.

mxwsn 5y ago
I wonder if one can place mathematical concepts on a spectrum from discovery to invention? To me, the pythagorean theorem feels much more like a discovery of a "hidden" eternal truth that was once beyond our grasp. But to me, complex numbers seem more like a notational choice, more akin to an invention. To put it another way, I would expect aliens to have the pythagorean theorem, but perhaps different notation for complex numbers, like representing them with matrices and computing over them with linear algebra. Such a representation would reduce the distinction of the "imaginary axis" which is really not that special. I may be naive; I haven't worked or thought with complex numbers since completing my basic education.
- gizmo686 5y ago
  Definitions are invented, theorems are discovered. The Pythagorean theorem is a discovery about Euclidean geometry. But in order for that discovery to be meaningful, one must first invent Euclidean geometry, or at least something sufficiently similar to Euclidean geometry.
count 5y ago
Pure math concepts like complex numbers are not naturally existing, just waiting for us to find them, they're human-defined tools to describe things. Like new words, they're invented. They wouldn't be there without us, as they are, for the most part, artifacts of our cognition.
- pohl 5y ago
  If that were the case, would you expect another civilization on some rock in different galaxy to arrive at entirely different concepts, or ones isomorphic to our own?

mrtedbear 5y ago

Veritasium is fantastic, I can't think of many current or past TV shows that are of a similar quality to his videos. A while back he did a video of his own startup story, which was very cool: https://www.youtube.com/watch?v=S1tFT4smd6E https://www.youtube.com/watch?v=S1tFT4smd6E

travisgriggs 5y ago

I love Veritasium. Derek has done a number of videos that are awesome like this. The videos are great and educational, but he also manages to subtly weave in an almost spiritual subtext to them as well. For example, in this one, you learn about imaginary numbers in some cool ways, history, how awesome e^ix is, etc, but also this yen-yang balance between embracing not only what is real but taking a chance on what is imaginary. Some of my other favorites: - https://m.youtube.com/watch?v=rhgwIhB58PA https://m.youtube.com/watch?v=rhgwIhB58PA - https://m.youtube.com/watch?v=HeQX2HjkcNo https://m.youtube.com/watch?v=HeQX2HjkcNo - https://m.youtube.com/watch?v=OxGsU8oIWjY https://m.youtube.com/watch?v=OxGsU8oIWjY - https://m.youtube.com/watch?v=3LopI4YeC4I https://m.youtube.com/watch?v=3LopI4YeC4I - https://m.youtube.com/watch?v=pTn6Ewhb27k https://m.youtube.com/watch?v=pTn6Ewhb27k I think he’s going to lose the $10,000 bet on the car though.

kzrdude 5y ago
I didn't click through all (mobile, too slow) so don't know if it's in there, but one favourite i have is the one where he has an ice cube, a metal block and a plastic block and ask people on the street which block is colder. It teaches a lot about what temperature vs feels cold vs heat is.
lxe 5y ago
Some of popular educational/informational channels and podcasts make this patronizing assumption about the audience, watering everything down to make it "accessible". Veritasium doesn't do this, yet maintains huge popularity.
agumonkey 5y ago
That's my favorite video of him. This one ties history, education, simple geometry, advanced abstractions in an entertaining script. Pretty high bar.
jasonwatkinspdx 5y ago
> I think he’s going to lose the $10,000 bet on the car though. Are you referring to the wind powered propeller car bet or is there a new one?

cool_dude85 5y ago

I've never liked the framing of "imaginary" numbers as "not reflecting reality" or somehow being less "real" or something. It's just an ordered pair of numbers with a convenient extension of multiplication that preserves nice properties of the reals. I think it's because of the name. If you called them double numbers or paired numbers nobody would say that.

amai 5y ago
The problem is not the term "imaginary", the problem is the term "number. Unlike any other numbers imaginary numbers cannot be ordered by size. So in that sense they are not numbers at all. In fact they represent rotations in two dimensions. They should be better called rotation operators instead of imaginary numbers.
mathnmusic 5y ago
Gauss complained about the name too. He suggested using "Direct, inverse and lateral numbers" instead of "positive, negative and imaginary numbers". Imaginary numbers are not pairs. Complex numbers are. Continuing Gauss, I'd rename "complex numbers" as "planar numbers".
- Natsu 5y ago
  I kinda think of them as 'orthogonal' numbers.
  - carlmr 5y ago
    That I think is the best alternative name I've heard yet. Since you describe two orthogonal dimensions. Also vector numbers could be somewhat of a useful name, since they behave like a 2D vector (or even higher dimensional vectors for e.g. quaternions)
    - ogogmad 5y ago
      "Vector numbers" for something purely 2D seems dodgy to me. There are lots of unital algebras, even in 2 and 4 dimensions, and none of them should hog the name "vector numbers" as they all have equal entitlement to the name.

ralusek 5y ago

> Imaginary numbers exist on a dimension perpendicular to the real number line. Correct me if I'm wrong, but I don't think that imaginary numbers exist in a dimension with any sort of fixed spatial relation to real numbers. Isn't the fact that we chose to graph the imaginary axis perpendicular to the real axis on the complex plane simply arbitrary? Or maybe better described as "a functional way to visually plot the negative root component of a number." My understanding is that visualizing the imaginary portion of complex numbers on a spatial plane with perpendicular axes also just happened to get a convenient 90 degree rotation by multiplying by i, and rotations like this just happen to show up all over nature. For example, the wave function they mention in the video, where they say it has an "imaginary component," seems like complete bullshit to me. You could write a program with real numbers in order to graph that wave...in fact I guarantee that's how a technical artist visualized it. In fact, it's the same with the Mandelbrot set, which supposedly "exists on the complex plane." No...it doesn't. It just happens that if you describe the arbitrary rotation operation that is performed in order to define whether or not a coordinate falls within the Mandelbrot set or not, the rotation from the way we've chosen to graph the complex plane means the equation can be written very concisely like this.

AnotherGoodName 5y ago
Every independent variable should be graphed perpendicular. X and Y when graphed are perpendicular. If they are not graphed perpendicular it indicates that a change in X position must change Y (or vice versa) whereas being perpendicular makes it clear that they are independent and you can change one without any thought that you're changing the value of the other. Complex numbers have 2 independent variables. The real and the imaginary parts. You can change one part without being forced to change the other. So you graph them perpendicular. It's not really a convention, it's a rule that each independent variable is perpendicular (or else they aren't independent!). Complex numbers have 2 independent variables.
- ralusek 5y ago
  Well yes, if we want to plot any sort of independent values, we have 3 spatial axes we are familiar with and will use them to graph accordingly. But I'm not going to argue that "housing prices therefore exist on an axes perpendicular to square footage." I just plotted the relationship that way.

pphysch 5y ago

The innovation of "imaginary" numbers is that we can concretely reason about multiple dimensions in a unified manner (unlike classical geometry, which relies on geometric primitives). `i` is not some magic, "imaginary" value, it is an invented syntax that means something like "rotate by tau/4 radians" or "orthogonal to the default vector". It turns out that physical problems are much more accurately modeled in 2 dimensions than in 1 dimension, and even more so in higher dimensions. > Freeman Dyson: “Schrödinger put the square root of minus one into the equation, and suddenly it made sense … the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers.” This quote is emblematic of the mysticism that some mathematicians and academics cannot resist using to advance their careers as public intellectuals. Reality is certainly not based solely in "real numbers" (one dimension), but nor is it based solely in "complex numbers" (two dimensions). The idea that mathematics is "the language of the universe" that can be precisely "discovered" by brilliant minds is a ridiculous notion that only serves the status of the mathematical elite. Mathematics is fundamentally about designing models and abstractions that help us reason about real phenomena with minimal cognitive resources. Everyone does it, and anyone can do it. Disclaimer: I have a degree in mathematics. "All models are wrong, but some are useful" [1] [1] - https://en.wikipedia.org/wiki/All_models_are_wrong https://en.wikipedia.org/wiki/All_models_are_wrong

BeetleB 5y ago
Nitpick: Anything in reality that can be modeled with complex numbers can be with real numbers. Complex numbers provide incredible convenience, but that's all. There's no phenomenon that requires complex numbers to explain - Schrodinger's Equation included.
- teleforce 5y ago
  There are real, complex, quaternion and octonion numbers. Saying that we can just get by only with real numbers is really being dishonest and in denial, it's just like saying we do not need OS and higher level languages because bare metal and assembly are all that we ever need. As a rough analogy, certainly we can travel all over the world on foot by walking and swimming, eventually we will get there but in reality we will not. But the more correct analogy is that we certainly cannot go to space on foot, and higher dimensional complex and hypercomplex numbering systems is like our spacecrafts. Ironicaly the famous gimbal lock problem for spacecraft navigation can be easily solved using quaternion as opposed to disaster waiting to happen if we use only complex numbers [1]. Einstein certainly will have never come up with the infamous equation E= mc^2 and corresponding relativity theory if not because of quaternion numbers [2]. [1]Gimbal lock: https://en.wikipedia.org/wiki/Gimbal_lock https://en.wikipedia.org/wiki/Gimbal_lock [2]Quaternions in University-Level Physics Considering Special Relativity: https://arxiv.org/ftp/physics/papers/0308/0308017.pdf https://arxiv.org/ftp/physics/papers/0308/0308017.pdf

gregfjohnson 5y ago

A Danish cartographer named Caspar Wessel came up with an early formal treatment of complex numbers, in his work "On the Analytical Representation of Direction" (wikipedia has a nice article about him). It was published in an obscure forum, and predates subsequent rediscovery of complex numbers by others. His formulation is IMHO beautiful, intuitive, and compelling. He did it in terms of directions on a map, replacing the "sign" of a conventional real number with a "direction" or "compass heading". So, one might say, "the nearest Starbucks is two blocks east and one block north". He was simply using what became known as the polar form of complex numbers. One can follow intuition and define reasonable notions of addition and multiplication by real values. But what of multiplying two "Directions"? Wessel derived what multiplication must mean, and went further in deriving a large number of identities involving his newly discovered "directional numbers". If you pick a specific important pair of directional numbers, the multiplicative identity (call it "1") and a number 90 degrees away from it (call it "i"), it is convenient to represent any directional number as a the sum of scalar multiples of these two numbers. Then, one considers the simple formula "(i + 1)(i - 1) = i^2 - 1". A straightforward geometrical argument demonstrates that i^2 must be equal to -1. ("Show HN": gregfjohnson.com/complex)

lxe 5y ago

I remember just playing with my good old TI-83 Plus Silver Edition, plugging a bunch of symbols together, like e, and i, etc... when I suddenly got the e^i*pi = -1. How in the world do these completely unrelated (to me, at the time, at least) constants ended up with -1? It was mind blowing. This made me actually interested in math.

vishnugupta 5y ago
Thanks to the way math is taught in India I still have very little understanding of so many of these concepts though I'm very good at remembering formulae/steps, mechanically calculating them, and applying them to real-world problems. It wasn't until I stumbled upon Better Explained[1] I began to appreciate the need to understand math at a deeper/common-sensical level. Now I make it an effort to teach math to my son using examples, analogies so that he gets concepts like positional numbering system, why we need it, its advantages etc., https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...
pjbk 5y ago
If you realize where e comes from, which is compounding and what we call an exponential process, being that "the current rate of change is proportional to the present value", then the exponential of a complex number is no different to the exponent of a real number. In the case of one-dimensional real numbers you assign the slope of the function to the current value. The complex case is exactly the same, but if you think instead in the two-dimensional Argand plane and complex algebra, the slope is the tangent to a circle proportional to the angle at that radius (which curve is the one that at every point the slope is equal to its complex value?). Therefore with the rules of complex arithmetic the tangent provides the "curling" effect in the curvature of a periodic circle, while in the real case you get the common compounding shape of the exponential curve. The relation to sine and cosine is just the projection into direct(real)/quadrature(imaginary) components in either fixed or intrinsic coordinates. Same when you expand the exponential into its complex power series. BTW, this is actually the essence of infinitesimal transformations in continuous groups and the exponential map, which generalizes this concept to other types of numbers or abstract objects (i.e. Lie group theory).

zsmi 5y ago

I really enjoy looking into the history of mathematics and physics. I think it gives one a much better appreciation of why things are defined the way they are, and also the limitations of those definitions. There is a really great book on the history of imaginary numbers. The history mostly focuses on how i was used to help solve algebra problems, so definitely one should be comfortable with high school algebra to get something from the text, but I don't think one needs much more math than that for the first half of the book. The second half gets more into how various use cases developed, in those chapters basic college level calculus would be a major plus. I read it more than 10 years ago though so no promises. :) An Imaginary Tale: The Story of √-1 Paul J. Nahin https://press.princeton.edu/books/paperback/9780691169248/an-imaginary-tale https://press.princeton.edu/books/paperback/9780691169248/an...

vishnugupta 5y ago

While I enjoyed the history part towards the end it got super hand wavy; especially as it went from describing "i" as a rotation to e^ix; Better Explained does a fantastic job of explaining these from the first principles. https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid... https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...

wyager 5y ago
Macdonald's Linear and Geometric Algebra is an excellent GA text which not only geometrically derives complex numbers (and rotation via their exponentiation), but also quaternions.

6gvONxR4sf7o 5y ago

My favorite mental model for imaginary numbers is that they are placeholders or stepping stones. A speed of (30 + i*10) miles per hour is nonsense, but sometimes in the course of calculating a nice real 40 mph, you might take a detour through some steps including complex numbers. Having them handy sure will make the derivation easier. Same as negative numbers. I can’t be negative six feet tall, but negative feet (as a magnitude) make the calculations way easier. You can always reformulate these derivations into derivations that don’t use e.g. imaginary numbers, but there’s no need to. If I were apointed Senior Exectutive Math Concept Namer For Humanity, I’d call them something like “algebraic placeholders” or “algebraic closure stepping stones” or something in that vein instead of “imaginary numbers.”

jasonwatkinspdx 5y ago
Yeah, this is the intuition I've come to, particularly after reading Quantum Computing Since Democritus. If you'll let me be slightly hand wavy, you can get to Quantum Mechanics by generalizing probabilities in two steps: allow them to be negative, then allow them to be complex. Negative probabilities are quite unintuitive but seeing them as an accounting placeholder similar to how negative numbers are vs counting numbers helps it all click imo. The other big thing is that if you want algebraic closure you need complex numbers or something isomorphic to them.

echopurity 5y ago

This video is epistemological trash.

dexwiz 5y ago

Real numbers are for straight number lines, and imaginary numbers are for circular number lines. That is about all there is to it. It's really a shame we introduce imaginary numbers as quirky work around for negative square roots. I would say that is more of a side effect. Multiplying positive numbers always results in positive numbers, but due to the circular nature of imaginary numbers you can get positive or negative results. Complex roots are like saying this equation has no solution in a traditional Cartesian/Euclidean world, but does have solutions in a rotational one. I think we if introduced imaginary numbers as rotations initially, the seemingly magical things like Euler's Identify and Formula would look much more trivial. EDIT: I would also say the lie that we live in a 3+1D world (X, Y, Z + Time), helps confuse the mater. We probably live in a 6D + 1 world: X, Y, Z, Pitch, Yaw, Roll, and Time. Just as you can conceive of a flatland 2D universe without a Z axis, you could conceive of a 3D world without rotational orientation.

mmcdermott 5y ago
I really like this explanation. The negative square roots explanation never sat right with me in high school and I think this touches on why.
- CamperBob2 5y ago
  The way I think about it: - To express direction in Cartesian space, you need signed numbers - To express 2D rotation, you need imaginary numbers - To express 3D rotation, you "need" quaternions (in the sense that they're the smallest way to represent an arbitrary rotation)
dls2016 5y ago
> introduced imaginary numbers as rotations initially How would you do this without developing the exponential first?
- zwkrt 5y ago
  Well any complex number a + bi can be described as a transformation matrix |a -b| |b a| So you could start by just asking what it means to transform the plane

rspk 5y ago

There is also a great playlist by Welch Labs called "Imaginary Number are Real" that I really like: https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

gkfasdfasdf 5y ago

I really appreciated the demonstration of solving the equations with shapes. It was like watching algebra's origin story.

amai 5y ago

Imaginary numbers were invented too early, that is why the usual explanations following the historical development are so mysterious. The notion of Sqrt(-1) does not help to explain what they actually are. In fact imaginary "numbers" are not numbers. They cannot be ordered by size. Imaginary "numbers" are also not vectors in the geometrical sense. This is because you can multiply two geometrical vectors and get a scalar as result (scalar product). But if you multiply two imaginary "numbers" you get another imaginary "number", so this is different to a geometrical vector. In fact imaginary "numbers" are transformations or operators, because they multiply like operators/matrices and they act on geometrical vectors like rotation matrices. Group theory makes all that very clear: Imaginary "numbers" are just the elements of the group SO(2), the group of rotation operations in 2D (multiplied with a scaling factor). However the invention of group theory started only with Galois in 1832 when trying to solve not the cubic, but the quintic. And the powerful notation of matrices as is ubiquitous nowadays was only invented in 1913 (by Cuthbert Edmund Cullis, see https://en.wikipedia.org/wiki/Matrix_(mathematics)#History https://en.wikipedia.org/wiki/Matrix_(mathematics)#History) Unfortunately mathematicians (and YouTubers) don't stop to teach the confusing outdated notions of imaginary/complex numbers rooted in the 18th and 19th century. And unfortunately this video belongs to the many videos just repeating the same old stuff over and over again.