
What are imaginary numbers? - arcatek
http://math.stackexchange.com/questions/199676/what-are-imaginary-numbers
======
T-hawk
Here is an even better discussion on the same topic, and the HN thread from
last year: [http://betterexplained.com/articles/a-visual-intuitive-
guide...](http://betterexplained.com/articles/a-visual-intuitive-guide-to-
imaginary-numbers/) <https://news.ycombinator.com/item?id=2712575>

One great conclusion from this approach is how intuitive it becomes to
understand the square root of _i_. I always thought you'd need another
dimension to describe that, and another dimension for the square root of
_that_ unit, and so on.

But think of i as a 90° rotation, so applying it twice (squaring it) results
in 180° which is -1. Then √i is a 45° rotation so applying it twice results in
90° which is i. Sure enough, this works out. The unit vector at 45° is 0.5√2 +
0.5√2 * i. Follow the rules of complex arithmetic to square that and you do
indeed get i.

A rotation of -135° applied twice gives the same result. Square the unit
vector of -0.5√2 + -0.5√2 * i and you also get i. We've arrived back at the
axiom that all numbers have two square roots of opposite signs.

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.

~~~
diminish
One more question; what is the "i"th root of i? Please try to use the angle
metaphor :)

~~~
kalid
Sure! You might want to check out
[http://betterexplained.com/articles/intuitive-
understanding-...](http://betterexplained.com/articles/intuitive-
understanding-of-eulers-formula/)

I like this question because it really works your intuition.

The basics: x^y means "grow at x, for y units of time". I see "2^3" as "grow
at 2x for 3 units of time".

Having a base of i means your "growth" is a rotation at 90 degrees, no
scaling. So i^(1/2) means a 45 degree rotation, i^3 means a 270 rotation, etc.

Raising this to the i power (or 1/i power, which is -i) means the growth that
was _originally_ purely rotational is now rotated. So instead of growing at i,
you are growing at (i * 1/i = 1). So, we should expect a _positive real
number, greater than 1_ since our growth is positive.

How long do we actually grow for? Well, the base of "i" is really e^(i* pi/2),
which means "Start at 1.0 and rotate continuously for pi/2 seconds". We've now
modified this to "Start at 1.0 and grow at 1.0 for pi/2 seconds", which is
e^(pi/2).

So the answer is i^(1/i) = e^(pi/2) ~ 4.8

It's a bit tough with text-only, read the above article for more diagrams.

~~~
yequalsx
Note that 2575.97 also has the property that raised to the ith power gives i.
There are infinitely many such numbers.

~~~
diminish
or 0.008983291.. Are there countably infinite solutions or otherwise? See
infinity discussion few days ago at HN
<http://news.ycombinator.com/item?id=4526049>

PS: Just curious, i am not a mathematician.

~~~
btilly
It is countably infinite. Let's try to find them all.

A relatively simple way to understand this is that i^i = e^(i * log(i)) for
every possible log of i. So all we need to do is understand what values log(i)
could have (there are actually many), and then we can work it out. But log(z)
just undoes e^z, so we need to understand e^z.

Now let's work backwards. If z = x + y i with x and y real, then x tells us
the absolute value of e^z and y tells us the angle. The absolute value of i is
1, so any possible solution to log(i) has real part 0. The angle that we want
to wind up with is 90 degrees, or pi/2. Therefore y can be ..., -3.5 pi, -1.5
pi, .5 pi, 2.5 pi, 4.5 pi, ... .

Therefore log(i) has to be one of 1.5 pi i, -.5 pi i, -2.5 pi i, -4.5 pi i,
... .

Now i^i is e^(i log(i)) so it can be any of ..., e^(3.5 pi), e^(1.5 pi),
e^(-.5 pi), e^(-2.5 pi), e^(-4.5 pi), ... .

Unless I've made a trivial calculation error, that is the whole list.

~~~
yogrish
simple explanation of PI: <http://en.wikipedia.org/wiki/File:Pi-
unrolled-720.gif>

------
jacobolus
Everyone should learn some of the tools of Geometric Algebra sometime in high
school, and it would save a whole lot of confusion.

<http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf>

~~~
tgb
I like the linked paper a lot so far (still reading it) even though I disagree
that everyone should learn this just so that physics majors and similar can
avoid some amount of confusion.

~~~
jacobolus
Not just physics students. Anyone who has to deal with geometry should learn
it. Anyone who would otherwise study trigonometry, linear algebra, vector
calculus, complex analysis, projective geometry, &c. should learn it first.
It’s a much better set of mathematical tools for understanding spatial
relationships and transformations than the alternatives for many if not most
purposes.

For anyone dealing with computer vision, graphics, or modeling type fields, I
recommend the more recent book “Geometric Algebra for Computer Science”.
<http://www.geometricalgebra.net/>

------
shou4577
I have mixed feelings about this explanation.

On the one hand, the mathematician inside of me is raging "this is neither
specific, nor rigorous!" In my opinion, this definition is as close to useless
as a mathematical definition can be, since the person who has this (and only
this) will be hard pressed to answer any other questions involving complex
numbers. For example, what is multiplication by i+1? Without further
understanding, I don't think such a question can be answered.

On the other hand, the teacher in my head is celebrating. This person managed
to take a concept normally feared and hated by the population at large and
made it interesting and understandable. The response is well thought out, and
is much more engaging to the reader than any response that I could come up
with. In short, this explanation is more likely to make the person interested
in mathematics, while a more technical definition would probably just make
them zone out.

What do other people here think about this? Is the clarity and interest worth
the loss of utility? Am I totally off base?

~~~
egsmith
I would argue something is better than nothing. And if this gets people to
start using the tool then maybe they will start to hit the cases where the
loss of utility matters, and then this can be the source of inspiration to
actually tough out the technical inspiration. Probably a long shot in general
but it worked that way for me.

------
fchollet
The title statement is meaningless. "Imaginary numbers" are not "multiply" by
"90°". What does it even mean to "multiply" by an angle?

Correct statement: "*i is equivalent to a 90° rotation in some contexts."
Which is kind of obvious to anybody who has done some maths or physics at some
point?

~~~
walrus
Trivially, this result is clearly obvious.

~~~
Tipzntrix
I swear the only time I ever heard trivial in a lecture was once I got into
university. It's a whole new ball game; high school teachers wouldn't dare say
trivial just in case they had someone who was behind.

------
jostmey
Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason.
We could construct number lines that work differently, so that imaginary
numbers never appear. Such alternative number lines would still allow us to
solve the exact same physics and engineering problems. Sure, the computations
would work differently, but the way we would measure and use the initial
conditions in our equations would be different too. In the end, the predicted
outcome would still be the same.

Imaginary numbers are an artifact of how our number line is constructed. We
could construct alternative number lines where imaginary numbers do no exists.
The computations involving such alternative number lines would be different,
but the outcome would be the same.

~~~
ColinWright
This sounds profound, but is wrong on so many levels. In a sense everything
about mathematics is arbitrary, but there's a consistency and structure that
makes such a statement unhelpful and misleading.

Consider.

If you're content with the counting numbers then we can construct the negative
numbers. These have the specific property that when added to the positive
number of the same size we get zero,

But most people are happy with the integers, so move on. We're reasonably
happy with addition, but what is multiplication? If you think of it as
repeated addition you're screwed when you want to multiply by 2 1/2. It's
better to think of it as a scaling. Multiplying by 2 means that you scale
things up to be twice as big. Thus 1 goes to 2, 2 goes to 4, and 5 goes to 10.
Also, -1 goes to -2, -6 goes to -12, and so on.

So what do we mean when we scale by -1? We look at the sequence of scaling by
4, then by 3, then by 2, and so on, each time asking where the number 1 gets
sent.

    
    
        Scale by 4 and 1 -> 4
        Scale by 3 and 1 -> 3
        Scale by 2 and 1 -> 2
        Scale by 1 and 1 -> 1
        Scale by 0 and 1 -> 0
    

Following this progression we see that it's natural in some sense to say that
scaling by -1 means that 1goes to -1. And indeed, 2 goes to -2, and 73 goes to
-73.

Scaling by -1 sends something to the same distance on the other side of zero.

So where does -1 get sent under a scaling of -1? It gets sent the same
distance the other side of zero. -1 gets sent to 1.

Therefore it makes sense to say that -1 scaled by -1 is 1.

(-1) * (-1) = 1

Wecan use this to ask about the square root of -1. What geometric operation
can we perform on the number line, such that doing it twice is the same as
multiplying by -1?

An answer is to rotate anti-clockwise by 90 degrees. Another answer is to
rotate clockwise by 90 degrees.

Pursue this, and you start to construct the Agrand diagram, and the complex
numbers.

~~~
fexl
Beautiful exposition. Even in the late 1700s, many mathematicians rejected the
use of mere _negative_ numbers, viewing them as anomalies which indicated that
one had phrased a problem wrong to begin with. On the other hand, Euler
understood everything very well and even calmly explained how to take
logarithms of complex numbers, which bewildered most of his contemporaries.

Someone once said that a lot of confusion could have been avoided if, instead
of the terms positive, negative, and imaginary, they had instead used the
terms forward, backward, and lateral.

~~~
d0m
Thanks for this citation. Forward, backward and lateral.. for some reason,
this is what make the most sense to me in all these explanations of complex
numbers. I guess you could also go upward/downward. And then, in a fourth or
even nth dimension.

------
yason
The obligatory, if you're still scratching your head thinking about complex
numbers: [http://betterexplained.com/articles/a-visual-intuitive-
guide...](http://betterexplained.com/articles/a-visual-intuitive-guide-to-
imaginary-numbers/)

------
Jun8
I am humbled by the clarity of this answer!

The problem with textbook for all levels (other than them being outrageously
expensive) is that they contain many level of abstraction, or what one may
call self-censure, presumably in order not to scare kids off. One level is the
general consensus of how should a topic should be taught, another layer is the
author's view of how it should be taught In practice there's the third layer,
where the teacher presents them in a certain way. This results in a long chain
of Simon Says where the final, safe stuff that's taught, for the benefit of
the students, mind you, may become very detached from the reality and
excitement of the topic. Unfortunately, students always sense this and they
tune it out, leading to _so_ many people not liking math, physics, signal
processing, you name it.

The joy of access to a person who is infinitely (compared to you)
knowledgeable in atopic and is willing to interface you on multiple levels and
telling you as it is is enormous. Best known such example, of course, is
Feynman but physics SE and some other boards come close.

~~~
egiva
I agree 100% that textbooks have too many levels of abstraction added over
time. There's a really interesting answer given on this exact same subject by
Bill Gates at the Aspen Festival - check it out here:
<http://www.youtube.com/watch?v=Iqf3rvg742g>

------
friendlyfrog
Imaginary numbers are defined. Imaginary numbers are not rotations or anything
else that I keep hearing. Those are all properties of the fact that we define
an imaginary number to be z = a + bi, where a and b are real numbers, and i^2
= -1. That's it.

If someone asks what is an imaginary number the correct answer is "z = a+bi,
where a and b are real numbers, and i^2 = -1".

~~~
austintaylor
You're thinking of complex numbers.

------
stiff
A complex "number" (don't think of it as of a number! think of it like you
would think of a vector, group, ring or any other abstract structure) is just
an ordered pair of real numbers that behaves in a certain predefined way when
being added to another complex number or multiplied by it. For an
introduction, to avoid unnecessary confusion, it is best to write such
"numbers" as ordered pairs using the notation: (a,b). The definitions for the
operations are the following:

    
    
      (a,b)+(c,d) = (a+c,b+d)
      (a,b)-(c,d) = (a-c,b-d)
      (a,b)*(c,d) = (ac-bd,bc+ad)
    

It is useful to have separate names for each part of a complex number, so the
a in (a,b) is called the real part, and the b the imaginary part, but for now
think about those names as completely devoid of any meaning. Now, observe that
under the above definition:

    
    
      (a,0)+(c,0) = (a+c,0)
      (a,0)-(c,0) = (a-c,0)
      (a,0)*(c,0) = (ac,0)
    

But those are, if you consider only the real parts of the complex numbers,
ordinary operations on the real numbers! An example consequence of this is
that we can take some equation concerning real numbers like:

    
    
      2*x + 5 = 21
    

and write it down in terms of complex "numbers":

    
    
      (2,0)*x + (5,0) = (21,0)
    

Since as we have seen pairs of the form (a,0) behave just like real numbers,
we have not changed the meaning of the equation, hence we are free to solve it
using the rules of complex algebra and if we happen to arrive at another
number of the form (a,0), we can take out the real part of it, plug it into
the original equation in terms of real numbers and it is certain to be a valid
solution.

This is one of the two properties that makes the use of complex "numbers"
fruitful. At the other one we arrive if we now look at "numbers" that are NOT
of the form (a,0), for example at a curious property of (0,1):

    
    
      (0,1)*(0,1) = (-1,0)
    

So, in the domain of complex "numbers", the "number" that corresponds to the
real number -1, happens to have the equivalent of what we for real numbers
call the "square root". We just talk about the "square root", but it is a
different operation when we are talking about complex numbers.

Those two properties combined allowed mathematicians to tackle some problems
that previously did not have a solution. One example is the problem of finding
a solution to cubic equations. The math here gets more complicated, but
basically it turns out that by writing cubic equations in real numbers in
complex numbers instead, you can find general formulas in terms of complex
numbers for finding all the possible solutions, and as we have discussed if
applying such a formula in the end yields a number of the form (a,0), it is
guaranteed to be a valid solution for the original real equation. Google for
"cubic equations cardano" to see the details.

Now, this going back and forth between complex and real numbers is so useful,
that for the purpose of brevity mathematicians sacrificed intelligibility and
introduced sort of a shorthand notation of the form: a + bi, so instead of
writing (0,1) as we did above, we just write i, instead of (5,0) we just write
5, and instead of (1,2) we write 1 + 2i. This is purely a trick, there is
nothing magical about the "i", it is just a "dummy" variable that allows
convenient carrying out of the operations with pairs described above in the
manner reassembling ordinary high-school algebra we all know and love.

All this is maybe a bit elementary, but I think this is the part most people
fail to understand and because of this start treating complex numbers as
something mysterious. There is in fact nothing mysterious about them, you have
to boil every application you see of them to the above and then you will get a
clear understanding of what is happening and why they are useful. Points on
the plane happen to be a model for complex numbers with rotation corresponding
to multiplication and so forth, this is of course very interesting, but I feel
an introduction to the topic should start with what I have just tried to
explain.

~~~
btilly
I saw, and experienced, this approach in school and see no value in it for
improving your understanding complex numbers. The point of the approach is to
teach students about abstraction and formalism, and complex numbers happen to
be a convenient place to do it. But the formalism is a barrier for building up
a more convenient mental model about what is really going on.

Before disputing this, in any calculation that you've ever done by hand with
complex numbers, do you naturally write it as (a, b) or a + bi? I always do
the latter, and it saves me both time and conceptual effort.

And a random note. If you go on past advanced Calculus, you'll encounter two
subjects that take Calculus and go back to the basic foundations and build
them up. The first is real analysis, for which you have to learn all of the
ways that things fail to work out like you would want them to. The other is
complex analysis, where you wind up learning all of the ways that everything
has to work out amazingly perfectly.

The difference between the two subjects is that "differentiable" in the
2-dimensional structure of complex numbers is a far, far stronger condition
than "differentiable" is for the real numbers. Indeed there actually exist
functions that you can construct which are infinitely differentiable
everywhere in the real numbers, but for which on no interval can you extend
them to a function that is differentiable in the complex plane.

~~~
kahirsch
I disagree with this for several reasons.

First, I find that having more different ways to look at a problem, the better
I am able to deal with it. I can look at an equation algebraically, or as a
graph, e.g. I can use rectangular coordinates or polar coordinates. I can look
at complex numbers as abstract entities or as points in a plane.

Second, if you look at the history of complex numbers, mathematicians were
just not sure what to make of them, and had no way to have confidence that
what they were doing was even consistent. Being able to interpret them as
point in a plane with intuitive geometric operations gave them a huge boost.

Third, thinking of them this way led to the search for generalizations. Gauss
and Hamilton tried to find a way to do arithmetic in three dimensions, or
prove that it couldn't be done. Hamilton eventually found the four-dimensional
quaternions. And the (ac-bd,bc+ad) definition was generalized to the Cayley-
Dickson construction.

~~~
klodolph
And not long afterwards, Clifford generalized real numbers, complex numbers,
and quaternions into what are now known as "Clifford algebras". Handy stuff.
Certain algebras allow you to express geometric shapes such as points and
lines using very simple equations. Other algebras show how quaternions (for
example) arise naturally as the even subalgebra of Cl0,3(R). The "spacetime
algebra" appears as CL1,3(R), which makes it easy to express special
relativity.

------
scotty79
I tend to think that Real numbers is the first kind of numbers that have
nothing to do with reality. There are no perfect circles, squares or triangles
in reality. Real things can only look like circle if you don't look close
enough. I think same goes for sinusoidal waves and everything else.

All fundamental physical laws that contain e or pi or event sqrt(2) seem
suspicious to me.

------
alok-g
Can someone please also do this for:

1\. Matrices, especially matrix multiplication. Unlike matrix addition,
multiplication is defined in a very weird way. I think I understand where it
is coming from -- defining it that way allows representing and solving linear
equations. More insights, however, would help.

2\. Dot and cross products. E.g., the magnitude of dot product in 3D is
a.b.cos(theta), while for cross product, it is a.b.sin(theta). I never got
sure what happens to other combinations like a new "vector" product whose
magnitude is a.b."cos"(theta)?

3\. How is a set and "belongs-to" operator defined? Most books I have come
across just assume these (and later define natural numbers and addition from
them).

4\. Why is 0.9999... considered to be "equal" to 1. I understand them to be
equal "under the limit", but not without. This seems to be in my way of
understanding Cantor's infinities.

~~~
nhaehnle
I'll try to give some short pointers.

1\. Matrices represent linear functions between finite-dimensional vector
spaces. That is, if you have a function f from V to W that satisfies f(ax +
by) = af(x) + bf(y), then there is a matrix A such that f(x) = Ax, and vice
versa.

Once you understand that, try to figure out what happens to those matrices
when you compose functions. In other words, when you define h(x) = g(f(x))
(assuming that f and g are linear maps that can be composed in this way), then
given the matrices for g and f, what will the matrix for h look like? You will
end up with exactly the rules for matrix multiplication.

The reason that you call this result "multiplication" is simply that it
behaves very much like the multiplication that you are used to from the reals.
In particular, you get a ring on square matrices, with (matrix) addition and
multiplication that satisfy a distributive law.

2\. I personally think that those angles are a bit of a red herring. In
particular, the cross product generalizes in a somewhat more complicated way
to higher dimensions, and then you have to talk about determinants instead of
angles. That would take too much time and space to explain properly here.

3\. Set and the "element-of" relation are not defined in the usual sense. They
are indeed simply assumed, and you just postulate the properties that they
need to satisfy, a.k.a. the axioms of set theory. It's a way of thinking that
takes some getting used to, but as an analogy, try to work through Euclid. He
doesn't define points or lines, either, but only postulates properties that
they need to satisfy.

4\. Because 0.9999... is usually interpreted as a real number, and not as an
infinite sequence of characters. As a real number, 0.9999... has no meaning
except as a limit, and hence they must be equal. As infinite sequences of
characters, 0.9999... and 1 are of course different, but that's not how we
usually interpret them.

If you think that this is in your way when understanding Cantor's infinities,
perhaps you should try to use the diagonalisation argument on infinite bit
strings instead of on real numbers. That way, those kinds of subtleties simply
do not arise.

~~~
alok-g
Here is what I am confused about with regards to diagonalisation:

Start with binary non-negative integers: 000 001 010 011 100 101 ... (goes to
infinity)

This set now includes all possible bit strings of infinite length since the
way these are iteratively generated includes all possibilities.

This is also an enumerable set by definition.

Let's now reverse the bits and put them after a decimal. These are just real
numbers now going from zero to one. (This step is actually unnecessary I
think.) .000 .100 .010 .110 .001 .101 .011 ...

This must be enumerable set too.

Using diagonalisation argument, .111111 is never to be found in this set. This
is exactly where I am stuck. This number comes into the set from flipping of
infinity in the original set, which includes all possible stings of infinite
length.

I immediately read into your message on treating these as bit strings instead
of real numbers. I still am stuck though. (Do you also see a connection to
0.99999... by the way?)

~~~
ColinWright

      > Start with binary non-negative integers:
      > 000 001 010 011 100 101 ... (goes to infinity)
    
      > This set now includes all possible bit strings of
      > infinite length
    

No, it only contains the strings of finite length. There are infinitely many
of them, but each one stops after a while. In particular, then n^th one only
has log2(n) places before it then becomes all 0s.

    
    
      > This is also an enumerable set by definition.
    

Yes.

    
    
      > Let's now reverse the bits and put them after a decimal.
      > These are just real numbers now going from zero to one.
      > ... .000 .100 .010 .110 .001 .101 .011 ...
    

But not all of them, since these are only those numbers that have a finite
number of 1's in them.

    
    
      > This must be enumerable set too.
    

Yes it is.

    
    
      > Using diagonalisation argument, .111111 is never to be
      > found in this set.
    

Irrelevant.

    
    
      > This is exactly where I am stuck. This number comes
      > into the set from flipping of infinity in the original set,
    

"Infinity" was never in your original set. And if it was, it wouldn't be
produced by the diagonalisation argument.

    
    
      > which includes all possible strings of infinite length.
    

No it doesn't.

~~~
alok-g
For the first set, I meant to write:

[Prepend each string with infinite zeroes]

...000

...001

...010

...011

...100

...101

...000

Now all bit strings here have infinite length.

The set is still enumerable since this is just binary encoding mapping to the
set {0, 1, 2, 3, ...}

The question still is if it covers all possible bit strings of infinite
length.

For units place, we covered both zero and one. For (n+1)th place, we cover
both zero and one together with all combinations for the first (n) bits. As n
-> infinite, all possibilities get covered.

The question is if 111111111... is also there in this set. But isn't it there
too?

~~~
ColinWright

      > For the first set, I meant to write:
      > [Prepend each string with infinite zeroes]
      > ...000
      > ...001
      > ...010
    

...

If there are infinitely many zeros on the front, you can't actually append
anything. That doesn't end up being well-defined.

(Well, actually, there are transfinite ordinals, but that would confuse the
issue. It's not what you mean, and it doesn't help)

    
    
      > Now all bit strings here have infinite length.
    

If you want to talk about an infinite "decimal" string, you need to talk about
the things that come after the decimal point, in order. As such, they come in
order, and you can't have infinitely many zeros and then a finite string on
the end.

    
    
      > The string is still enumerable since this is just binary
      > encoding mapping to the set {0, 1, 2, 3, ...}
    

You need to be more careful about how you actually define the strings. Strings
have a start, then they go on one place by one place.

    
    
      > The question still is if it covers all possible bit strings of
      > infinite length.
    

Well, you haven't actually properly defined strings, but even so, no.
Everything you have starts with a zero.

    
    
      > For units place, we covered both zero and one.
    

No, you don't seem to have.

    
    
      > For (n+1)th place, we cover both zero and one
      > together with all combinations for the first (n) bits.
      > As n -> infinite, all possibilities get covered.
    

No, because as soon as you have a one in your expansion the string is finite,
so not all possibilities are covered.

    
    
      > The question is if 111111111... is also there in this set.
      > But isn't it there too?
    

You defined the set - tell me where it is. Even leaving alone the fact that
these aren't proper strings, it doesn't appear to be there.

~~~
alok-g
>>> [Prepend each string with infinite zeroes] > ...000 > ...001 > ...010 >>
If there are infinitely many zeros on the front, you can't actually append
anything.

I am lost. In this first set, I do not have a decimal point anywhere. Why
cannot I have an infinitely many zeros to the left of 1. It will still be just
one when looked at as a number.

>> Strings have a start, then they go on one place by one place

As far as representing it as a string, I may still start from the right and
work towards the left.

I understand your point for the decimal case, infinitely many zeroes on the
right of decimal cannot be followed a finite string. My argument does not
require this however. (I change ...00000011010 from the first set to
.0101100000000... in the second set.)

>> > For units place, we covered both zero and one. >> No, you don't seem to
have.

The units place is the rightmost below. Both zero and one are covered.

...000 ...001

Stating the above for the decimal case, let n=1 be the place right after the
decimal, n=2 to the right of it, and so on. Now for place n=1, the both zero
and one are covered (first two cases below). For n=2 place, again both zero
and one are covered for all possible combinations above for n=1 place (first
four cases below).

0.000000000000... 0.100000000000... 0.010000000000... 0.110000000000...
0.001000000000... 0.101000000000...

Using the mathematical induction argument, all combinations are covered. This
must include 0.1111111111... It sits exactly where (simple) infinity sits in
the enumerable set {0, 1, 2, 3, ... }

~~~
ColinWright
OK, so you're not talking about the usual diagonalisation. I'll try to follow
what you've said and respond as I go.

    
    
      > In this first set, I do not have a decimal point anywhere.
    

OK, fine. But then you talked about flipping them around to come after the
decimal point. When you do that you have only those strings that only have a
finite number of 1s in them.

    
    
      > As far as representing it as a string, I may still start
      > from the right and work towards the left.
    

Yes you can, but that's not what people do when talking about Cantor and
diagonalisation, so it's now completely unclear what you're talking about.

However, you start with finite strings of 0s and 1s, basically the non-
negative whole numbers, represented as binary strings. Note that these are all
finite, and the non-zero parts are still finite, even if you prepend an
infinite number of 0s.

    
    
      > Stating the above for the decimal case, let n=1 be the
      > place right after the decimal, n=2 to the right of it, and
      > so on.
    

See, now you're talking about stuff after the decimal point. I'll continue ...

    
    
      > Now for place n=1, the both zero and one are covered
      > (first two cases below).
    

But for diagonalisation that's irrelevant. We only ask what is the first digit
of the first number.

    
    
      > For n=2 place, again both zero and one are covered for
      > all possible combinations above for n=1 place
    

Again, irrelevant. We only ask what is the 2nd digit of the 2nd number.

    
    
      > 0.000000000000...
      > 0.100000000000...
      > 0.010000000000...
      > 0.110000000000...
      > 0.001000000000...
      > 0.101000000000...
    

So here if we construct the diagonal of this sequence as you've listed it we
have 0.00000... Let me highlight the diagonal for you from this quoted
section:

    
    
      > 0.0xxxxxxxxxxx...
      > 0.x0xxxxxxxxxx...
      > 0.xx0xxxxxxxxx...
      > 0.xxx0xxxxxxxx...
      > 0.xxxx0xxxxxxx...
      > 0.xxxxx0xxxxxx...
    

If we now flip this we get 0.11111....

Now observe that all of the strings you have contain only a finite number of
1s. That means that 0.11111... is not in your sequence.

    
    
      > Using the mathematical induction argument,
    

You haven't made an induction argument.

    
    
      > ... all combinations are covered.
    

For each place, both possibilities are eventually covered. but for each
sequence that you give, it is eventually all zeros.

    
    
      > This must include 0.1111111111... 
    

No, it doesn't.

    
    
      > It sits exactly where (simple) infinity sits in
      > the enumerable set {0, 1, 2, 3, ... }
    

Infinity does not fit in that set, and 0.1111... is not in your defined set of
numbers.

------
joe_the_user
Aside from people's very worthwhile answers describing the complex number
system, I think it is worth mentioning that the use of the term "imaginary" is
an unfortunate historical remnant. In experience, a lot of the average student
confusion comes from their trying to get their head around the naive meaning
of imaginary.

Now that modern mathematics understands that all number systems are more or
less games with axioms, we know that no part of a number system is really more
imaginary than any other part. "Imaginary" might better be termed "augmented"
- we can augment the "real" number system by adding an element "i" which we
say is equal to the square root of -1.

And it is just as ironic that the "real" number field itself has perhaps as
many weird elements as say the rational "imaginary numbers" (pi, e, Theta etc)

~~~
zvrba
Also ironic is that real numbers may not be "real" (i.e., exist) at all. The
uncountable part which is THE part that completes rationals to continuity
cannot be described or generated in any way since there are only COUNTABLY
many different computer programs (or mathematical formulas).

------
radicalbyte
The best explanation I've seen yet is one Hobbes gave to Calvin:

<http://www.gocomics.com/calvinandhobbes/2009/02/18/>

------
zvrba
From the OP's question:

> When I tried to calculate the square root of -1 on my calculator, it gave me
> an error.

This was a red flag to me. This person takes the answer from a calculator as
the ultimate truth, whereas it (obviously) is not. I wonder how he would have
reacted had he had an advanced calculator (HP50g or some TI) and got "i" as
the answer instead of an error.

> To this day I do not understand imaginary numbers. It makes no sense to me
> at all.

Coming from a mixed EE/CS background, I had a lot of math and math-heavy
classes in the first 2.5 years (linear algebra, real, vector and complex
analysis, discrete math, systems theory)..

What I observed during the 5-yr master's degree was that the people who were
seeking "meaning" in math courses were the same people who 1) had the most
problems with math exams, 2) had the least ability to _apply_ the math to
other domains in order to do something useful.

What worked for me was to accept that math is manipulation of abstract symbols
according to some rules, combined with some ingenuity (i.e., when you have to
consider external information in addition to the rules).

In the case of complex numbers, you introduce a new abstract symbol called "i"
with the property that i X i=-1. Other rules that you (hopefully) learned
earlier still apply. So computing (x+y)X(x+z) or (2+3i)X(2-6i) is in essence
the same process except that you replace i^2 with -1 when you encounter it.

I actually enjoyed doing math on that level: as a manipulation of abstract
rules and invention of new rules (theorems).

In simple EE, inducntances and capacitances are modeled as pure imaginary
"resistances"). However, I somehow figured out how to make use of math (not
only complex numbers) in other domains. I made use of math in electronics for
drawing Bode plots, information theory, systems theory, and even economics!
(In economics, I was reading a lengthy chunk of text with some summations,
only to realize that, in actuality, integration was described and everything
could be summarized to few sentences explaining the meaning of the involved
integral. On the exam, I gave MY explanation -- based on the integral --
instead of the lecturer's, and I passed!)

To me the problem seemed to be that lecturers in other subjects resorted to
awkward, special-cased explanations instead of showing us how to model the
problems at hand with the math we have already learned.

------
luser001
I don't think I quite followed the step where he/she writes "i^4 = 1", where
they are relating rotation and the natural numbers. The RHS is theoretically
the concept "identity under rotation". Why should it be be the same as the
natural number 1?

Maybe I missed something in the explanation. Can somebody explain?

~~~
T-hawk
There is an implicit start from the unit vector 1. i (really 1 * i) is a 90°
rotation of that. i^4 is four 90° rotations. Rotate anything four times and
you've rotated it 360°, ending up where you started at 1.

~~~
luser001
Ok, thx. I think I understand now.

------
grandalf
I have started reading the story of i:

[http://www.amazon.com/An-Imaginary-Tale-Story-
square/dp/0691...](http://www.amazon.com/An-Imaginary-Tale-Story-
square/dp/0691027951)

------
erichocean
Complex numbers are cool, but automatic differentiation using _dual numbers_
is even cooler.

------
math33
Isn't 4 rotations 4i? Why is it i^4

~~~
Evbn
Rotation is multiplication. Translation is addition.

