Imaginary numbers

Originally posted by sonhouse
That is a mere description, not any kind of deep understanding. We call this unknown quantity i and have logic that proves it works, thats about all we can say about it. Not exactly a fundamental understanding.

But there is nothing to understand about i other than it is the square root of -1. While Pi is quite a strange, curious number, it is not i itself that is strange, it is more the consequences of i that are curious, if that makes sense?

Originally posted by FabianFnas
No sarcasm intended. I just thought your wordings was quite humourus. Perhaps I should blame my lack of linguistic skills.

Indeed.

Originally posted by Scyhte
You're not the only one to find the wordings fun - maybe it's a nordic thingy then. Personally I can't wait to get to use complex numbers in math, our next course should include that. Too bad we don't get to use complex numbers in physics until in university 🙁

On another note what use do graphs with complex numbers have in electricity calculations? I alwa ...[text shortened]... s much simpler and thus they would have no point in representing things with them in graphs.

Imaginary numbers are used in electronics as a way of simplifying trig. calculations which would otherwise get tedious.

Originally posted by flexmore
it does not imply that it lies on the real number line

Indeed. But complex (complex as a + bi no just i) numbers have a geometrical representation. They can be represented as points in the complex plane where the x-axis defines the real part(a) and y-axis defines the imaginary part(b - the coefficient of i) or they can be interpreted as vectors. That's somehow connected to getting all the formulas for trig form of complex numbers. I have studied complex numbers only briefly so I suggest you read the wikipedia article - http://en.wikipedia.org/wiki/Complex_numbers - it's quite interesting. I'll probably have an intro of complex functions this year in high school but that's still some time ahead.

Originally posted by Swlabr
But there is nothing to understand about i other than it is the square root of -1. While Pi is quite a strange, curious number, it is not i itself that is strange, it is more the consequences of i that are curious, if that makes sense?

It may also be a symptom of something missing from our mathematical logic that requires special cases for some operations. Maybe there is some underlying math principle or technique that would bring complex numbers to the real number line without resorting to 'cheating🙂'

Originally posted by PinkFloyd
No it was a real question; expecting a sober, non-sarcastic answer.

"Real" in math is an antonym to "imaginary". Thus the humor.

Originally posted by sonhouse
It may also be a symptom of something missing from our mathematical logic that requires special cases for some operations. Maybe there is some underlying math principle or technique that would bring complex numbers to the real number line without resorting to 'cheating🙂'

It can be seen as a result of obsession: Then need for "i" comes from our obsession to know where things came from. Not only do we want know what -4x-4 is but we also want to know what "y" can be so that yxy is equal to -4. What thing can evolve itself into -4? This is just our obsession with evolution.

Surely you would not prefer the intelligent design arguement: -4 just is. -4 was created by god, and did not evolve from some other number multiplying itself.

Originally posted by flexmore
It can be seen as a result of obsession: Then need for "i" comes from our obsession to know where things came from. Not only do we want know what -4x-4 is but we also want to know what "y" can be so that yxy is equal to -4. What thing can evolve itself into -4? This is just our obsession with evolution.

Surely you would not prefer the intelligent desig ...[text shortened]... ust is. -4 was created by god, and did not evolve from some other number multiplying itself.

I was making the argument maybe our math was not developed well enough to include complex numbers in a more direct way. I didn't think I was bringing a spiritual vector into it. Shouldn't that be in the spiritual forum?

Originally posted by PinkFloyd
What does the Q mean in your formula? I'd like to try it.

There's not really anything to try; I've just given a definition, but I will elaborate.

Q is the field of rational numbers, and it's standard to construct Q[x], which consists of all polynomials with coefficients in Q. Polynomials can be added componentwise and multiplied the way you were taught in school, and with this addition and multiplication, Q[x] becomes a commutative ring with a 1.

Now declare that two polynomials in Q[x] are equivalent if and only if their difference is a multiple (in the Q[x] multiplication) of x^2 + 1. This partitions Q[x] into equivalence classes (these are called the cosets of the principal ideal generated by x^2 + 1, if you feel like Wikipediating).

Now we call the set of these equivalence classes Q[x]/(x^2 + 1). We can define multiplication and addition in here, too, by letting this new set "inherit" the operations from Q[x]. Explicitly, the sum and product of the cosets containing p(x) and q(x) are the cosets containing p(x) + q(x) and p(x)q(x), respectively. It's not hard to show that these operations are well-defined, and that Q[x]/(x^2 + 1) becomes a commutative ring, with the coset containing 1 as its 1 and the zero coset (i.e. the set of polynomials divisible by x^2 + 1) as its zero.

Now do a little algebra to show that there are no polynomials in Q[x] whose product is x^2 + 1. From this it follows that Q[x]/(x^2 +1) is actually a field -- we have multiplicative inverses in here!

There are now a couple of equivalent ways to define i with this machinery. First, the field we just built is a 2-dimensional vector space over Q, and Q embeds in it, so it has a Q-basis consisting of 1 and something not in Q. We can pick the other basis element to have minimal polynomial x^2 + 1, by that construction, so we call it i, i.e. the field we constructed just now is Q extended by a root of x^2 + 1 which we just "imagined" into existence. We haven't brought up order yet, so right now there is nothing distinguishing i and -i, since they are both roots -- either one works.

Now make the real vector space with 1 and i ans a basis, form the subspace generated by i, knock out 0, and you have the imaginary numbers.

Alternatively (and this is what I said in my first post), hunt around in the field we just made for elements which are the roots of polynomials with INTEGER coefficients. These form a ring, and only four of them have multiplicative inverses, namely 1, -1 and two other elements. These form a cyclic group of order four. Any element that generates this group can be called i, and the other generator -i.

The really interesting thing here is that the more essential definition says: "take the algebraic closure of Q. It will be a degree-2 extension of R. It has to contain a root of x^2 + 1 because it is an algebraic closure, so take the subspace generated by this root, knock out 0, and you have the imaginary numbers." The reason this isn't appropriately simple is the second sentence, which is essentially the fundamental theorem of algebra. Now there is a Galois-theoretic proof of this, which uses the construction above and some machinery from group theory. However, most proofs I've seen come from complex analysis, or even wketchy geometry, and all of these proofs rely on a definition of the complex numbers that is in some sense independent of the one given above. It's almost, but not quite, enough to make one able to claim that an analyst and an algebraist mean something different when they write "i".

Originally posted by ChronicLeaky
There's not really anything to try; I've just given a definition, but I will elaborate.

Q is the field of rational numbers, and it's standard to construct Q[x], which consists of all polynomials with coefficients in Q. Polynomials can be added componentwise and multiplied the way you were taught in school, and with this addition and multiplication, n analyst and an algebraist mean something different when they write "i".

Wow, that's quite a post. Are you a mathemetician by any chance? It sure sounds like it.
Can you explain that statement "Q[x] becomes a commutative ring with a 1"?

Originally posted by sonhouse
It may also be a symptom of something missing from our mathematical logic that requires special cases for some operations. Maybe there is some underlying math principle or technique that would bring complex numbers to the real number line without resorting to 'cheating🙂'

But i suppose my point is that there can't be as n^2>0 for all real numbers, rational or irrational (apart from 0, of course). This is the defining property of complex numbers, and so we cannot ever hope to find them in the real numbers.

Originally posted by Swlabr
But i suppose my point is that there can't be as n^2>0 for all real numbers, rational or irrational (apart from 0, of course). This is the defining property of complex numbers, and so we cannot ever hope to find them in the real numbers.

I guess I am just jumping the gun, wondering how imaginaries will be dealt with in a thousand years (that is, assuming the climate, and countries go on well enough for there to be a continuous development of science and math for the next thousand years).

Originally posted by ChronicLeaky
... an analyst and an algebraist mean something different when they write "i".

I have always thought that when an algebraist and an analyst write anything ... they always mean something different.

If there was ever a point at which they meant the same ... then they would align on most of the rest ... they always try their damnest to find a differnt interpretation.

Originally posted by sonhouse
Wow, that's quite a post. Are you a mathemetician by any chance? It sure sounds like it.
Can you explain that statement "Q[x] becomes a commutative ring with a 1"?

in a really good number system: whenever anything is done by one number to another number we will always end up at some place inside the same number system we started in. (imagine if the characters in a film suddenly left the movie!)

in many systems we can easily step outside the boundaries. ... in many commutative rings we can step outside ..(imagine they integers, then dividing 3 by 2 and leaving the integers)

commutative rings are often a stepping stone towards something more complete ...

when simple commutative rings are extended to include the places they step out to they become more complete ... that could potentially expand to become a more complete number system to adequately represent the real universe ...

Originally posted by sonhouse
Wow, that's quite a post. Are you a mathemetician by any chance? It sure sounds like it.
Can you explain that statement "Q[x] becomes a commutative ring with a 1"?

I am but a lowly mathematician-in-training. My plan was to put that post up, which is rather useless unless one has a lot of time to go look up all the terms, and then go through bit-by-bit and explain. This might be a slow process, but I'll start with:

A ring is a set R, together with two functions, called "+" and "*", which each map RxR to R (ie they take as their input an ordered pair from R and output something in R). These functions obey the following rules:

1. r + (s + t) = (r + s) + t
2. There is an element in R called 0 such that 0 + r = r for all r
3. For all R, there is an element called -r such that -r + r =0
4. For all r, s, r + s = s + r

(These rules make R, together with +, into what's called an "abelian group"😉

5. r*(s + t) = r*s + r*t
6. (s + t)*r = s*r + t*r
7. (r*s)*t = r*(s*t)

That's a ring. If, in addition, we have that r*s = s*r for all r and s in R, then we say R is a "commutative ring". If there is an element 1 in R such that 1*r = r*1 = r for all R, we say "R is a ring with identity" or a "ring with a 1".

If R is a commutative ring with a 1 such that:

1. 1 and 0 are different
2. If r*s = 0 then either r or s is 0

we say R is an "integral domain". The rings we'll be dealing with here are all going to be integral domains. You are familiar with at least one integral domain: the integers with normal multiplication and division. To think of some rings that are commutative, have a 1, and are not integral domains, think of calendars and clocks...

To conclude this little clarification, if R is an integral domain with the property that for each r in R, different from 0, there is an s such that r*s = 1, we say R is a field, and this is the most important notion we'll be using.


(I did something similar in an old thread in Debates called "BDN, You're Number is Up!" from perhaps febreuary 2007, if you're interested. That was a thread on a very bare-bones logician's approach to numbers, and it kind of left off right before constructing complex numbers, so I'm kind of picking the whole subject up again, but you needn't read that to read what I'm planning here. However, the basic notions of set theory that I will toss around here are introduced in that thread.)

EDIT flexmore knows what xe is talking about and it is good that xe is here!