
Learning to Love Complex Numbers - adbge
http://jeremykun.com/2014/05/26/learning-to-love-complex-numbers/
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gohrt
Related:

Kalid Azad's oft-posted article:
[http://betterexplained.com/articles/a-visual-intuitive-
guide...](http://betterexplained.com/articles/a-visual-intuitive-guide-to-
imaginary-numbers/)

Tristn Needham's book _Visual Complex Analysis_
[http://usf.usfca.edu/vca//](http://usf.usfca.edu/vca//)

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gballan
There are interactive plots, along the lines of VCA, here
[http://puzlet.com/m/b00dd](http://puzlet.com/m/b00dd) (I am a dev).

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bcbrown
I've loved complex numbers since first learning about them, and my first
significant programming experience was writing these types of programs for my
TI-89 graphing calculator in high school in the late 90s. This is a good
introduction.

Another good equation to play around with is x_next = a x (1 - x), for a from
0 to 4. You'll find that x settles to a single value for low a, eventually
bifurcating as you increase a, then bifurcating again. At one point, though,
it splits into a cycle of three values, then an erratic distribution. Someone
once wrote a paper on how "period three implies chaos".

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darkmighty
aka logistic map

[http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif](http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif)

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bcbrown
Yes, but the visualizations I programmed started with
[http://polygeek.com/images/chaos/fig_01_BifurcationGraph.png](http://polygeek.com/images/chaos/fig_01_BifurcationGraph.png)

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zercool
Some fantastic visualizations to accompany his essay. I would love an ipython
notebook version that I could download and play with.

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makmanalp
Great article! I've always intuited the rules with which we operate on
imaginary numbers as a hack - we don't know what to do with i other than
squaring it, so we avoid doing anything with it and algebra our way around the
issue.

Example: (i + 3) + (i + 3) == i^2 + 6i + 9 == 6i + 8 is about as logical to me
as (x + 3) + (x + 3) == x^2 + 6x + 9. Sure, if we knew what the heck x was
this operation would have been easier, but since we don't, we just use FOIL to
work around it. In the former, somehow we know what i^2 is but not i, so we
can reduce a little further but not completely.

Of course, I don't know how sound this gutfeel impression is.

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DanielRibeiro
It is very sound. You can think of i as being a variable 'x', and therefore,
you can see Complex numbers are polynomials over real numbers (denoted as
R[x]). But you can simplify the polynomials by using the fact that x __2 = 1
for this construction (since x = i on this situation).

This is the exact construction of Complex numbers as an algebraic extension
over the field of Real numbers[1]

What is more amazing: there is no such way to do the same on the complex
numbers. This is because the Complex numbers form an Algebraically closed
field[2]

[1]
[http://en.wikipedia.org/wiki/Field_extension#Examples](http://en.wikipedia.org/wiki/Field_extension#Examples)

[2]
[http://en.wikipedia.org/wiki/Algebraically_closed_field](http://en.wikipedia.org/wiki/Algebraically_closed_field)

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pavelrub
>One fact we all remember about numbers is that squaring a number gives you
something non-negative. 7^2 = 49, (-2)^2 = 4, 0^2 = 0, and so on. But it
certainly doesn’t have to be this way. What if we got sick of that stupid fact
and decided to invent a new number whose square was negative?

This presentation of complex numbers makes the whole article flawed and
useless in my opinion. When I first learned about complex numbers, they were
presented in a very similar fashion: let's just invent a number i such that
i^2 = -1. This is inane: we have a multiplication operation that we are all
familiar with and we know exactly what it does, and then somebody tells us
that we can use it on some "imaginary" thing (??) such that it times itself
equals -1? How is anybody supposed to make any sense of it? It's like saying:
let's invent a "number" j such that j-j=the letter z. What does it even mean?
Nothing! it's gibberish, and a similar definition of "i" is also gibberish. We
cannot make sense, under our normal understanding of multiplication and under
our normal understanding of numbers as including only the real line, of how
can something times itself be -1, and neither should we, because there are
much better ways to present the whole thing from the beginning.

The correct way to present complex numbers is either in the context of
abstract algebra - where there is a very obvious question of whether we can
embed the real line as a field inside the real plane, or simply present them
geometrically without going into fields. It is simply wrong, in my opinion, to
present i as something we "invent" so that i^2=-1 (why would anybody do
that??), and then go on and say that _after_ we have invented this, there are
ways to imagine this geometrically. No! If you want to talk about geometry,
then _define_ i geometrically, then _extend_ the definition of multiplication
geometrically to the plane, and then it becomes _clear_ that i^2 = -1, and
there is no mystery about anything.

Edit: it should also be noted that historically, complex numbers didn't come
into existence because somebody decided on a whim to "invent" a number i such
that i^2=-1. Rather, it was a result of the fact that cubic equations such as
x^3=15x+4 clearly had a solution (for example x=4), but using the cubic
formula to solve them resulted in weird terms such as sqrt(-121). Bombelli, in
the 16th century, decided to try and compute with those terms anyway, and
through this process eventually succeeded in producing the right results
(x=4), so it eventually became clear that the roots of negative numbers aren't
just gibberish: they interacted somehow with the reals, and there was some way
to "make them work" to produce real results, though the full realization of
what was happening probably came much later.

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Grue3
>This is inane: we have a multiplication operation that we are all familiar
with and we know exactly what it does, and then somebody tells us that we can
use it on some "imaginary" thing (??) such that it times itself equals -1?

How is this more inane than defining 0, or negative numbers? Those didn't
always exist either, somebody invented them for various purposes. When a
person learns arithmetic, negative numbers are introduced in a similar way. By
the time a person learns about complex numbers, they already know how to deal
with variables in equations, so i*i=-1 shouldn't be too hard to comprehend.

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pavelrub
The fact remains that complex numbers are confusing to an extremely large
number of people, and negative numbers and 0 aren't. This is probably because
0 can be given direct meaning in terms of everyday applications: I don't have
money = I have 0 dollars. I have 0 dollars and I owe you 5 dollars = I have -5
dollars. But what is i dollars? The switch from real to complex is much more
complicated than the switch from the naturals to integers or from the integers
to the reals. Algebraically, it is also a completely different kind of
extension.

i^2 = -1 _should_ be hard to comprehend, and people _should_ be suspicious
when things are presented this way. If your experience with equations make it
seem easy, I would argue that you don't fully comprehend the significance of
such a definition (for example - lets invent a number j such that 1/j = 0.
Does this also seem easy?).

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j2kun
I would argue that very few people understand the reals any better than they
understand the complex numbers, though they think they do.

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pavelrub
The lack of knowledge people have regarding the reals is different in kind
from the confusion surrounding complex numbers, and has nothing to do with
what is being discussed.

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cwhy
I just hate them.. Why don't we just use vectors and make math easier to
learn?

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darkmighty
Complex numbers are the key to complex analysis which is different than just
real multivariate calculus (it has certain constraints).

