
A Visual, Intuitive Guide to Imaginary Numbers (2007) - mgdo
https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
======
kowdermeister
Once I renamed imaginary numbers to 2D numbers internally it made more sense.

Do you know any other examples in math where fixing terrible naming makes the
concept easier to digest?

~~~
westoncb
I think there's definitely something to your suggestion here. For instance,
there is no result to sqrt(-1) on the real line, but it becomes possible in
the complex plane because you have an additional degree of freedom.

One qualifier I'd add though is that they're 'asymmetric' 2D numbers. The real
vs imaginary axes have different behaviors. If I multiply by the positive
unitary value on the imaginary axis (i.e. 'i'), I get CCW rotation by 90
degrees; with the negative imaginary unit I get CW rotation; positive _real_
unit, no change occurs; negative real unit, rotation by 180 degrees.

Actually that's something I wonder about—could there be a 'symmetric' version
of that? Maybe something where the real axis behaves more like the imaginary
axis, but maybe some signs are flipped or something.

I guess an important aspect of how using the complex plane is beneficial is
the fact that it combines these disparate elements though—we can start with
something real, do certain transformations to it in our more broadly capable
complex plane, then bring it back into the real line.

~~~
yiyus
You want geometric algebra. In 2D GA, you would have two unitary vectors, i
and j, such that i * i = 1 and j * j = 1. The (non-commutative) product of
between them (or their division) would be a bivector: i * j = ij = - j * i.
You rotate 90 degrees the i and j vectors using the bivector.

The good thing about GA is that the same concept can be easily extended to 3D
(quaternions), and in fact to 4D and nD.

~~~
westoncb
That does sound pretty close to what I had in mind. Are i and j from your
example associated with orthogonal axes in a plane?

~~~
yiyus
Yes, they are. i would be the unitary vector in the horizontal direction, x,
and j in the vertical one, y.

In fact, geometric algebras are very general, and the one I briefly sketched
is not the only possible interpretation. If you are interested, you can find
many good introductions online, directed at different audiences. You can also
search for the term "Clifford algebras" if you are interested in a more formal
approach.

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dang
Discussed in 2011:
[https://news.ycombinator.com/item?id=2712575](https://news.ycombinator.com/item?id=2712575)

And all the way back in 2007, long enough ago that nickb posted it:
[https://news.ycombinator.com/item?id=91811](https://news.ycombinator.com/item?id=91811).

~~~
bdamm
Who was nickb?

~~~
uryga
[https://news.ycombinator.com/item?id=152428](https://news.ycombinator.com/item?id=152428)

~~~
SilasX
They're arguing about whether it's Paul Graham but I need to read a lot of
comments to determine whether that's the consensus or it's a joke.

~~~
dang
[https://news.ycombinator.com/item?id=152477](https://news.ycombinator.com/item?id=152477)

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jatsign
I really enjoyed this video series on imaginary numbers. Went through the
history as well as the math:

[https://www.youtube.com/watch?v=T647CGsuOVU](https://www.youtube.com/watch?v=T647CGsuOVU)

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weiming
> Focusing on relationships, not mechanical formulas.

The focus on formula memorization in schools is tragic. Once upon a time, I
too have learned everything about imaginary numbers ... everything, other than
why the heck they are actually useful. Can do all the calculations, don't know
why I am doing them.

Are there other great math textbooks/websites (Calculus level and higher,
Stats, Linear Algebra, etc.) that try to do this better? For someone older
than school level who wants to learn again.

~~~
kevin_thibedeau
They are only really useful in the science and engineering fields that need
them. That is why they remain opaque to most people. High school level
pedagogy doesn't deal effectively with the useful applications of these
concepts (matrices are also poorly introduced).

Most of our modern technology would not be achievable without the use of
complex numbers as a tool.

~~~
tangentspace
I always liked how Feynman dealt with complex numbers in 'QED: The Strange
Theory of Light and Matter'.

He focuses on the intuitive concept of a particle having a spinny arrow
attached, the arrow rotates as the particle flies through space. He only
casually mentions that this is in fact a complex number, whereas the bulk of
the text focuses on developing intuition around arrows.

I read that book in high school, and it certainly influenced the direction I
took in university. It helped to understand that the physical universe often
appears to behave in extremely non-intuitive ways, but using mathematics we
can develop a model that transforms the phenomenon into something that
actually does make intuitive sense.

I think some of the harder concepts in math are difficult because they act
like stepping stones into aspects of our world that just don't make sense
based on day-to-day experiences. But modern technology depends on this!
Pedagogy is improving, but it still lags advances in technology.

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ThrustVectoring
I always find the "algebraic closure" approach to be the best bet for
explaining complex numbers. Much like how having negative numbers means that
you can subtract any two numbers (closure under addition), having complex
numbers means you can find the roots of any polynomial. If you don't have
complex numbers, something like `x^2 + 1` has no real roots, and you have a
problem.

The really nice part of this explanation is that it tells you why complex
numbers show up everywhere. It turns out that it's rather straightforward to
find physical real-world problems with input parameters that are coefficients
to polynomials and behavior that depends on the roots of those polynomials.
Take a slinky or another other harmonic oscillator - when you model it with a
differential equation, the polynomial coefficients are how heavy the slinky
is, how much speed-dependent resistance there is, and how springy it is.
Factoring the polynomial gives you the behavior over time, and it pretty much
always has some sort of behavior, so the roots should be _some_ kind of
number.

~~~
throwaway080383
From the POV of this explanation, it's then quite fortuitous that the
complexes are just two-dimensional over the reals, and can thus be easily
visualized. That is, as soon as you adjoin the roots of x^2 + 1, and close
under field operations, you actually get the roots of _all_ polynomials.

~~~
ginnungagap
There is a deep result stating that for a field F there are only three
possibilities for what the dimension of its algebraic closure can be as an
F-vector space: it can be 1, if F is algebraically closed, it can be 2, as
happens for R and other real closed fields, or it can be infinite, as it
happens for Q, but there is no other option!

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forinti
I wonder if these wonderful resources we have nowadays will result in more
kids getting into science and maths. It would have been fantastic to have had
this when I was in school (I kept an interest in maths in spite of my
teachers' efforts).

~~~
tangentspace
Many people have recognized the need for improvement in mathematics education,
and I think it really is evolving in positive directions. I worked at DreamBox
Learning for a few years, they produce an adaptive math learning program for
elementary schools (and gradually reaching higher levels) which as been very
popular with children.

The kind of math that was traditionally taught in schools is still relevant
and important, but I think we can leverage modern visual and interactive media
to help children develop a broader class of mathematical reasoning skills,
which includes much, much more than a bunch of rules, symbols, and rote
procedures.

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samfisher83
I think I first learned imaginary numbers in algebra 2 which was like 6th or
7th grade. I know they use x^2=1 as an example, but I don't know how useful
that is. I think they really need to teach trig before they teach imaginary
number. They also need to teach polar coordinates with imaginary numbers. Most
all of its uses are for solving trig problems. You use it in EE and ME to
solve sinusoidal problems, but I don't know how i=sqrt(-1) helps you
understand what its used for. I think euler formula is most elegant formula in
math. How did the guy even come up with it.

~~~
gjm11
I don't know how Euler came up with it, but here's a fairly intuitive way:
think about circular motion. If something moves around a circle at constant
speed, then its velocity vector is perpendicular to the vector from 0 to the
moving thing. Once you have the idea (which _isn 't_ terribly difficult) that
complex numbers live on a plane and multiplication by _i_ is rotation through
a right angle, this gives you the differential equation d _x_ /d _t_ = _ix_.
Solving that is easy: _x_ = exp( _it_ ). Since it's also easy (by definition
of the trig functions) to see that _x_ = cos _t_ \+ i sin _t_ , we're done.

Given the sort of thing Euler was good at, though, it seems just as likely
that he looked at the power series for sin, cos, and exp, and said "aha!".

~~~
kevin_thibedeau
The 2D analogy and use of rotating vectors was alien to Euler and
contemporaries. They really did think of it as an "imaginary" abstraction to
make the math work.

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disqard
I came here expecting to see Steven Wittens' excellent writeup [0] mentioned
in the comments. Since it is not, I feel obligated to mention it.

[0] [https://acko.net/blog/how-to-fold-a-julia-
fractal/](https://acko.net/blog/how-to-fold-a-julia-fractal/)

~~~
mkl
Yes, I think this makes things very intuitive, and I use it (well, the first
slideshow in it) to introduce the idea when I teach complex numbers.

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vinchuco
This has also a great intuitive explanation of why sqrt(-1) is a rotation.

HTTP://greatscottgadgets.com/sdr/6/

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wruza
If thinking like I’m 5, it doesn’t add up. At first, “b times i” means
rotation, but then ‘a+bi’ is a vector of two orthogonal components. I expected
‘bi’ to be an angle in polar coordinates and ‘a’ to be length.

Besides that, why did mathematicians choose this exact representation? Why not
polar, spherical, hyperbolic, Hilbert-like, Minkowski-like? Did anyone explore
on how that could change known problems, like e.g. Riemann-zeta?

~~~
lainga
It still means rotation in "a + bi". If you take "a + b" you get another real
number, but if you take "a + bi", the _b_ component has been rotated by 90
degrees (i), and now it's orthogonal to _a_. Even if we drop complex numbers,
it's not like we write out points in polar coordinates as "5 + 30 degrees" \-
how are you adding a length and an angle together?

~~~
wruza
Yeah, I see now. “times i” is discrete 90 degree rotation itself, not just
‘i’, nor ‘b’. Thanks everyone for making that clear.

This though shows that explanations via analogies or non-strict wording may
confuse one rather than enlighten. I’m not good at math, but once understood
to not search analogies or geometry in things. Instead it is better to “shut
up and calculate”. Not sure if imagining something is required to manage it.
It’s only our brain’s faulty quirk.

[https://m.youtube.com/watch?v=zwAD6dRSVyI](https://m.youtube.com/watch?v=zwAD6dRSVyI)

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carapace
The punchline for complex numbers is in Geometric Algebra...

~~~
ajkjk
I disagree. It suffices to include a rotation operator (or bivectors in
general, if you want to do it in N>2). The 'full' framework of GA is, imo,
foolish, because the geometric product doesn't really have a geometric
interpretation.

~~~
carapace
What? :-)

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bcaa7f3a8bbc
"Does any of this really have to do with the square root of -1? Or do
mathematicians just think they're too cool for regular vectors?"
[https://xkcd.com/2028/](https://xkcd.com/2028/)

~~~
chess19
This is obviously not a completely serious question but it is definitely looks
like a question someone might ask when learning about complex numbers for the
first time.

The answer is completely historical in nature. Imaginary numbers began as
being interpreted as the square root of -1 for the purposes of solving
polynomial equations (hence the name.) Later, their field structure and their
interpretation as vectors-with-multiplication became their primary use but the
name remained.

Mathematicians don't really use "vectors" in the traditional sense like in
physics but deal with abstract vector spaces where a "vector" is simply a
member of a "vector space" which is "a set of things with addition and scalar
multiplication and a few other nice properties".

However, if something needs to be done with vectors in a plane, complex
numbers are extremely useful because scaling and rotation can be represented
as multiplication. Therefore natural operations in the complex numbers often
correspond to natural operations in whatever you are trying to study with
complex numbers.

~~~
mkl
> Mathematicians don't really use "vectors" in the traditional sense like in
> physics but deal with abstract vector spaces

This is not at all true in general: many mathematicians use non-abstract
vectors too. My (maths) PhD, for example, uses vectors throughout but doesn't
mention vector spaces once.

