
What are the complex numbers, really? - ColinWright
http://robjlow.blogspot.com/2017/06/what-is-this-thing-called-i.html
======
sxp
I'm wary of anyone who tries to explain complex numbers without pictures. The
best explanation of complex numbers that I've ever seen is
[http://acko.net/blog/how-to-fold-a-julia-fractal/](http://acko.net/blog/how-
to-fold-a-julia-fractal/). I wish that explaination existed when I was
struggling with my Electrical Engineering classes in college.

~~~
Grue3
I'd rather be wary of explaining mathematical concepts with pictures. Pictures
are known to be highly misleading and force the reader to imagine a thing in a
certain way. For example, most pictures of triangles have all 3 angles being
acute. This leads people into forgetting the case when one of the angles is
obtuse.

~~~
wglb
Ok, so how would you explain venn diagrams without pictures? How would you
describe Mandelbrot sets without pictures? How would you explain hyperbolic
parabaloids without pictures? How would you explain conic sections without
pictures? How would you explain a trapezoid without pictures? How would you
describe a sine wave without pictures? Eversion of a sphere? How would you
illustrate the calculation of the slope of a curve when talking about
derivatives without pictures? How about a cycloid? How would you show the
representation of a signal in both time domain and frequency domain?

~~~
Grue3
>How would you describe Mandelbrot sets without pictures?

It's literally easier to describe Mandelbrot set without the picture than with
it. The set of complex numbers c where recursive equation "z_n = z_(n-1)^2 +
c, z_0 = 0" does not diverge.

In fact, the picture of a Mandelbrot set is actively misleading! You think you
can see the set, but it's a fractal! It's infinitely complex in a way that
cannot be shown in a picture.

>Eversion of a sphere?

Pretty sure that one was figured out way before anyone was able to visualize
it. Same with Banach-Tarski paradox. These things resist visual intuition in
the first place, so it's much safer to rely on equations when dealing with
problems like these.

~~~
wglb
If you read of the early work that Mandelbrot did on this equation, you will
recall that despite having those symbols in front of him, he had to plot it
out, crudely at first on a line printer, before he fully understood the
implications.

Thus pictures are not only tutorial but a fundamental necessity for
understanding mathematics.

------
harpocrates
I think this actually misses the biggest point. Complex numbers are special
because they are an algebraic closure of the real numbers (what makes the real
numbers interesting should be more obvious), and because Zorn's lemma tells us
that such a closure is unique (up to isomorphism). In other words, complex
numbers are precisely what you need to add to real numbers in order to always
be able to solve polynomial equations.

TL;DR: complex numbers are the unique algebraic extension to the real numbers

~~~
akvadrako
What do you mean by closure?

~~~
nabla9
What others said, but more abstract and simple way to say it:

Algebra is set of objects and operations on them. Operation is function that
gets number of objects as argument and maps it to some other thing

A set has closure under an operation(s) if using those operations on the set
members always gives you a member in the same set.

Example:

Natural numbers (0,1,2, ...) are closed under addition and multiplication,
because you can't add two numbers and get something that is not natural
number. Natural numbers are not closed under subtraction or division because
you can get a result that is not a natural number.

~~~
katastic
>A set has closure under an operation(s) if using those operations on the set
members always gives you a member in the same set.

THANK YOU. That line was key and just clicked for me. I'm actually liking
everyone's completely-different ways of explaining the same things. When one,
or five, are confusing, the sixth may be key to filling in the holes.

------
mathgenius
Rob talks about how complex numbers are just another kind of "fake number" on
top of negative numbers, fractions, square roots... And this is a good thing
to point out. But there is one further thing to note, which is the inherent
ambiguity between i and -i. We can't tell these apart. John Conway says this
well, when asked about the square root of -1 his reply is "which square root
of -1 do you mean?" (paraphrasing.)

This ambiguity is where Galois theory starts. Analogous "theories of
ambiguity" arise all throughout mathematics (Galois connections). This is
really fascinating stuff, with deep connections to quantum physics...

~~~
dsacco
_> Rob talks about how complex numbers are just another kind of "fake number"
on top of negative numbers, fractions, square roots...And this is a good thing
to point out._

I agree. I think complex numbers are best explained in the context of
motivations for number systems (and I made a comment explaining complex
numbers this way about a month ago[1]). This way takes longer than just
answering the question, “what’s a complex number?”, but it also builds a
better intuition of why complex numbers aren’t silly (or at least, why they’re
only as silly as anything that isn’t a natural number). If you can get a
student to be okay with the idea that we define new number systems to resolve
problems in prior ones, you can get them to be okay with a construction of
complex numbers from the reals, just as they’ll accept a construction of the
irrationals from the rationals, and integers from the naturals, etc.

I find that a really rigorous (albeit...spartan) first pass for understanding
complex numbers can be developed by reading through Chapter 1 of Rudin’s
_Principles of Mathematical Analysis._ He doesn’t go quite as far as
developing Peano arithmetic from first principles, but he does build up the
number systems successively, beginning with the natural numbers if I recall
correctly. That was the first book I read where I felt like I really
understood what complex numbers were, because up until that point I was only
dealing with them algebraically using an explicit _i_ in the form _a_ \+ _bi_.
Rudin’s development of the complex plane and representation of real and
complex numbers as points ( _a_ , _b_ ) is much better for intuition, in my
experience, and it lights the way to an even deeper intuition of what complex
numbers are in the context of Euclidean spaces later on.

That said, Rudin is terse to the point that many would call him pretentious,
so if there is another book that does the same thing with better exposition,
that might be a better choice...

 _> But there is one further thing to note, which is the inherent ambiguity
between i and -i. We can't tell these apart. John Conway says this well, when
asked about the square root of -1 his reply is "which square root of -1 do you
mean?" (paraphrasing.)_

It’s been a while since I did this exercise, but if I recall correctly this
ambiguity is part of the proof that we cannot order the complex field, right?
We end up with an absurdity because _i_ should be greater than - _i_ , but _i_
^3 = - _i_ , which evidently resists coherent ordering. And similarly _i_ x
_i_ = -1, but - _i_ x _i_ = 1.

____________________________

1\.
[https://news.ycombinator.com/item?id=15729381](https://news.ycombinator.com/item?id=15729381)

~~~
jeeyoungk
The ambiguity comes from the definition of i and -i from algebraic
perspective. They are two roots of sqrt(-1) and they have indistinguishable
properties other than the fact that they are different. I.e. if we live in an
alternate world where the complex plane is flipped (j = -i) then there is no
way to distinguish this world between the original.

It’s similar to the inherent ambiguity between left and right - the only
defining characteristics of them are they are opposite to each other.

Note that 1 and -1 have different mathematical properties (in most cases - in
Z/2 they would be equal) that allow us to distinguish. 1 * x = x but -1 * x =
-x.

~~~
zardo
1 * x = x but -1 * x = -x.

This is also true for x == i.

------
jonsen
There's a phenomenon where I think you can sense the physical reality of
complex numbers. When to propagating waves meet, and they are of opposite
phase, they cancel out. At the meeting point the combined amplitude becomes
zero. Despite of apparently total cancelation the two waves emerge undisturbed
on the other side of the "silent" point. A wave can be described with rotating
complex numbers of constant modulus. The two waves will each have its own
modulus intact even at the point of cancelation. So the sum-to-zero real value
is not really real.

------
techno_modus
> So what are the complex numbers, _really_?

If by _reality_ we mean the real world (not mathematics or some abstract
model) then the article does not actually answer the question what the complex
numbers are really -- they describe what complex numbers mean mathematically
(geometrically, algebraically, number theory).

Here are some interpretations of imaginary and real parts in terms of the real
(physical) world:

o Time vs. space. This means that they have to be measured using different
constituents with different properties. For example, space is reversible and
periodic (we can return to the same point where we were before many times
while it is not possible to return to previous point in time).

o Periodic vs. monotonic behavior. For example, rotation frequencies or
oscillators (electric or physical) vs. linear motion.

~~~
jonsen
Well the article does mention:

 _If beauty doesn 't motivate you sufficiently, then take solace in the fact
that the properties of some differential equations which arise in physics and
engineering are determined by polynomials you can build out of them, and the
systems modelled by these differential equations are stable if the real part
of all the roots are negative. Google for control theory to see lots more._

and

 _This turns out to be a useful way of thinking about alternating current, and
you can analyse AC circuits using complex arithmetic in much the same way as
you use real algebra to analyse DC circuits._

and ... did you read it all?

~~~
techno_modus
_This turns out to be a useful way of thinking about alternating current, and
you can analyse AC circuits using complex arithmetic in much the same way as
you use real algebra to analyse DC circuits._

"Userful way of thinking" and "analyse" something is not a real object
strictly speaking - it is a process or phenomenon (no doubt complex numbers
are useful here).

But can you say what kind of real object or property a complex number
represents or measures? For example, an integer can represent the number of
apples and their weight is represented by a real number. What property of a
real object can be expressed, for example, as 50i ?

------
gaze
The phrase “what IS it” has always bugged me in mathematics. The answer to
that is always “what we defined it to be.” The question math answers is “what
does it do.”

~~~
dsacco
I think a lot of the problem in mathematics education is motivation. Students
(and people learning math in general) encounter two problems:

1\. They don’t realize what you’re saying, i.e. number systems and mathematics
in general are built from definitions, not observations, and any definitions
work as long as they’re compatible with what we’ve already proven,

2\. If they realize #1, even implicitly, they still do not know why
mathematical definitions are motivated. A student might say, “Okay so I
understand what complex numbers are, but why do we have them? What’s the point
of all of this?”

In my opinion, new mathematical concepts should be taught by placing them in
the context of the least complicated thing the student already knows, then
motivating them by demonstrating what problem they resolve. In the case of
complex numbers, this can be explained to a student via analogue to the
irrational numbers, which are the set of all numbers _p_ / _q_ where _p_ and
_q_ are positive integers. We “invent” and define irrational numbers to
resolve real world problems involving numbers that cannot be in the integral
or rational systems. Similarly, we define complex numbers to resolve real
world problems involving negative numbers on a plane, and so on...

------
cousin_it
The miracle of complex numbers is that they have two seemingly unrelated but
very nice properties:

1) Every non-constant polynomial has a root

2) Every holomorphic function is analytic

The only connection I know is that (2) implies Liouville's theorem which
implies (1), but it doesn't seem any less miraculous.

~~~
mfukar
Not miraculous at all. The motivation for complex numbers came out of trying
to solve cubic polynomials (Tartaglia, Cardano).

~~~
mbid
What does that have to do with every function being analytic? Anyway, you can
solve every cubic polynomial in the algebraic closure of Q, which is far less
than C, so you don't need to construct the complex numbers for that.

------
qubex
I had the good fortune that my high-school mathematics teacher introduced
complex numbers as “compound numbers”, with two “components” that defined a
point on a plane, one of which was multiplied by a “orthogonal coefficient”
that just happened ( _fabula mirabilis!_ ) to be the square root of minus one
that so vexed us when solving certain types of quadratics.

Totally nonstandard terminology, but pedagogically very sound approach that
was totally at odds with the official syllabus... she just “filled us in” with
‘synonyms’ a while later so that we would not be totally baffled when sitting
our exams (the IB).

It has served me well ever since.

------
dude01
I think this is the best explanation for complex numbers I've ever seen:

    
    
      https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

------
godelski
I think the easiest way to explain complex numbers is to first ask a question,
"How do you multiply two dimensional numbers?" The other person always comes
up with some answer like (a0+b0)x(a1+b1) = a0a1 + b0b1 or the factorization.
You can easily show that this isn't consistent, and usually the student
quickly starts looking for answers to prove you wrong. It becomes quickly easy
to talk about how complex numbers can be used to work in two dimensions, and
being able to explain concepts like closure and fields without using those
words.

There is no need to jump to quaternions (which is 4D, not 3D, and the point
has been missed here), like the author does. There is a much simpler case that
people are more familiar with. The problem with the article is that it does
not gauge its audience well. As another commenter mentioned, pictures help.
With this discussion on two dimensions you can use a lot of great graphics out
there and talk about the beauty of Euler's formula. These topics all
generalize, but it is ludicrous to talk about the generalization first.

------
agnivade
I feel a better explanation of them was given in "Shadows of the mind" by
Roger Penrose where he explains how they play a vital role in Quantum Theory
which impacts real-life behavior.

------
rrauenza
Welch Labs has a wonderful series on youtube on imaginary numbers, "Imaginary
Numbers Are Real":

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

I like to tell my kids that imaginary numbers aren't really imaginary .. that
in many ways they're more real than the Reals.

------
hawktheslayer
Those of us EEs will know it's j.

------
bandrami
They're phase. It means something periodic is happening to something.

~~~
AnimalMuppet
I don't think that's what they _are_. It's merely one of the things they're
_used for_.

------
calebm
tuples

