
Complex numbers have never been so intuitive to me - AndrewMoffat
http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/
======
T-hawk
Wow, I _got_ this too.

Here's my followup question and answer. We know i is the square root of -1,
but what is the square root of _-i_ ? I've always thought that you'd need
another dimension to describe that, and another dimension for the square root
of _that_ unit, and so on.

But no. (Let's tackle sqrt(i) as a simpler case first.) We can answer sqrt(i)
in terms of rotation as described in the article. i is a rotation from the
unit vector by 90°, so applying that twice turns 1 into -1. What operation
applied twice would result in i? This just clicked: a 45° rotation. Thus the
unit vector at a 45° angle is the square root of i: 0.5√2 + 0.5√2 * i.

The mathematical approach bears that out. Follow the rules of complex number
arithmetic to square 0.5√2 + 0.5√2 * i (multiply it by itself) and you do
indeed get i.

And we can solve sqrt(-i) the same way. -i is a 270° rotation from the unit
vector 1. So the square root of -i is a 135° rotation, or -0.5√2 + 0.5√2 * i.

Finally, a 270° rotation is equivalent to a -90° rotation. So -45° should also
be a square root of -i, and indeed it is. Multiplying 0.5√2 + -0.5√2 * i by
itself also gives you -i. We've arrived back at the axiom that all numbers
have two square roots of opposite signs. 135° and -45° are the same vector
pointing in exactly opposite directions.

Last question: What's the _cube_ root of i? Easy: a 30° rotation. The 30° unit
vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.

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Deestan
So real numbers are 1-dimensional, and complex numbers are 2-dimensional.
Going along the same lines, we also have Quaternions, 4-dimensional numbers:
<http://en.wikipedia.org/wiki/Quaternion> Further again we have 8-dimensional
Octonions, and 16-dimensional Sedenions.

I'm curious as to why we don't have any useful numbers for the non-power-of-2
dimensions. E.g. 3-dimensional numbers.

~~~
beza1e1
I wonder why mathematicians use this funny i-notation? You could write complex
numbers as vectors, e.g. 2+3i <=> (2,3). Of course, it is more terse to write
i instead of (0,1) and this seems to matter a lot to mathematicians.

~~~
kalid
I think it'd be awesome to have some sort of "decimal" notation where 2 + 3i
was encapsulated as a single entity, something like 2_3. We don't write 2 + .3
when we mean 2.3, and this distinction makes it seem like a complex number is
"less put together" than a real one.

~~~
zem
in a sense it is - it involves magnitudes along two orthogonal directions.
there is no qualitative difference between 2 and 0.3 in this particular
context.

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aristus
"It’s a testament to our mental potential that today’s children are expected
to understand ideas that once confounded ancient mathematicians."

Ex-fucking-actly. Math is hard. Compsci is hard. But if it _remains_ hard,
then we adults have failed to do our job.

~~~
kalid
Yes -- things are hard, now, because we don't have the right models.
Multiplication is hard when you're working with Roman numerals. Is it a
problem with multiplication, or our thinking?

What's funny is that we think it stops there. "Oh, we made made multiplication
easy, and negatives, and decimals, but Calculus... well that needs to remain
difficult forever and ever.".

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natemartin
This is a fantastic explanation. I've always had trouble "getting" imaginary
numbers.... even though I've had to use the fairly often as an Electrical
Engineer. This is the first time they've made intuitive sense to me.

~~~
fastfinner
^Same here, I have taken countless tests using imaginary numbers, but it never
went beyond a really pointless exercise in my mind. _Now_ I get it.

~~~
checker
It's a shame I paid for all that college and knowing this would have greatly
expanded my understanding of a lot of what I learned in my math classes.

EDIT: I guess I learned it in high school though, and the math teacher
probably didn't know this either.

Point being, everyone should come across this at some point. It would be
beneficial to many math students.

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iwwr
In the same vein, you can have split-complex numbers which represent the 2d
hyperbolic plane. Also, there are the 4-dimensional generalizations like
quaternions, split-quaternions or other related algebra systems.

What's amazing about these systems is that there is usually an Euler relation
that holds. Example: e^(i*t) = cosh(t) + sinh(t) for the split-complex.

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vain
i wish someone had explained this to me about 15 years ago. i had sought, and
settled on the unsatisfying explanation that it is a conventional notation
with useful ways to do coordinate geometry. non essential, but a way of doing
it. though i have looked at euler's identity with awe, it's mostly been a
mystical sort of awe.

I hate to admit it but i had started using a line of argument to certain
theist friends, that if god helps you, as a concept, no need to be bothered,
think no more of it than a concept such as the the mathematical concept of i,
its a number that does not exist but has real consequences. now I feel stupid
for doubting the existence of i.

I keep trying to relearn my fundamentals, and this article does that
beautifully. i would otherwise have died a disbeliever.

~~~
AndrewMoffat
> though i have looked at euler's identity with awe, it's mostly been a
> mystical sort of awe.

betterexplained also has a _fantastic_ visual explanation of euler's identity:
[http://betterexplained.com/articles/intuitive-
understanding-...](http://betterexplained.com/articles/intuitive-
understanding-of-eulers-formula/)

~~~
kalid
Thanks so much for helping share this! :)

My goal is to get all these concepts out of awe (which I had too) into a real
sense of "Ah, I get it!". It doesn't help anyone to memorize incantations.

Two other insights I love:

* e as continuous growth: [http://betterexplained.com/articles/an-intuitive-guide-to-ex...](http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/) * radians as the perspective of the mover: [http://betterexplained.com/articles/intuitive-guide-to-angle...](http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/)

I realized that I didn't _really_ get what e and radians were about -- I
memorized them, but I wasn't comfortable. Once you have the right analogies,
Euler's formula starts making sense (without resorting to "Oh, take the Taylor
series expansion of each and see how they match up", which is essentially an
incantation).

~~~
AndrewMoffat
No problem, I'm happy to share something that genuinely helped me. I love what
you're doing, please keep doing it :)

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mvzink
Wow. I feel like if I had been taught this before trigonometry, high school
would have been a breeze.

~~~
CallMeV
It was for me. But then, I had a good maths tutor. Ironically, I cannot recall
his name. But the lessons he taught stayed with me, so at least he's
remembered for something.

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cormullion
If you'd like some sexy French graphics to accompany that, there's a good
Dimensions episode devoted to imaginary numbers. I found it on YouTube at

<http://www.youtube.com/watch?v=egIPnwcJuZ8>

But the main website is

<http://www.dimensions-math.org/Dim_download_E.htm>

And I think it's chapter 5.

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baddox
Given a complex number a + bi, the square root of a^2 + b^2 is called the
norm. The norm shouldn't be thought of as a measure of the "size" of a complex
number, since complex numbers are not well-ordered. It makes little sense to
say that 2 + 3i is equal to 3 + 2i.

~~~
jules
Why wouldn't you be able to call it size? Equal size doesn't imply equality
elsewhere in mathematics or elsewhere in the world (Jimmy and Paul are the
same height, so they must be the same person?).

~~~
baddox
Two people of equal height aren't the same person, so it would be odd for an
article to claim that the way to convert a person to a real number would be to
simply measure their height, and even odder to call that the "size" of a
person.

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dkokelley
It's unfortunate that our schools in general haven't been able to convey with
as much clarity and passion this concept. I suspect that any passion or
enthusiasm for a subject quickly gets destroyed when it's turned into a job
(particularly a job in a system run by bureaucrats).

This is why I am excited about programs like Khan Academy. One of the things
he's been able to do that has eluded most public schooling is explain a
concept simply, and with enthusiasm.

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PetrolMan
Why do I feel like my teachers intentionally made math harder than it needed
to be? Some concepts just lend themselves to simple visual representations and
just never seem to be taught that way...

~~~
lotharbot
Very often, teachers don't understand the concepts very well either.

~~~
kalid
Exactly. And ironically, we expect that to be the norm. "Oh, nobody gets
imaginary numbers, let's just memorize it and move on."

To me, that's a _huge_ canary in the coal mine! Why aren't we stopping here
and making sure we get it? It's like reading a sentence, not understanding the
key vocabulary word, and moving on. Yes, you "read" it but did you get
anything from it?

