What are imaginary numbers? Well, wonder no more. This is the most solid explanation I could find online, and it's really good.
http://math.stackexchange.com/questions/199676/what-are-imaginary-numbers

- anonymous

- katieb

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- bahrom7893

yooo you're in luck.. I'm taking a course in complex variables.

- bahrom7893

uhmm I don't have a scanner at home, but msg me your email, I'll scan my notes tomorrow and email them to you.

- anonymous

What are imaginary numbers?
Not real......

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## More answers

- bahrom7893

basically an imaginary number's an ordered pair

- anonymous

its when you try to take the square root of a negative number

- anonymous

There is only one imaginary number, i, defined by i^2 = -1

- bahrom7893

not really... a+bi are all imaginary.. a is the real part, b is the imaginary part.

- anonymous

b is real

- anonymous

you guys are confusing this student, lol

- anonymous

Stir,stir......

- bahrom7893

if a=0, the number is called pure imaginary.. Idk otherwise my professor was lying to me. No we're not, everyone knows that i is imaginary.

- anonymous

i is appended to the real number system by fiat......

- anonymous

lol

- bahrom7893

which is defined by:
(a,b)(c,d) = (ac-bd, ad+bc)

- anonymous

this conversation is really good for this student, keep it up guys

- bahrom7893

i meant i is an ordered pair, (0,1)

- anonymous

Then you make up an imaginary plane to go with this imagijnary number.....

- anonymous

The link I provided in the OP is a solid construction of how we (us normal humans, and bahrom7893) conjectured the Platonic existence of imaginary numbers. For further reading, check out "Mathematics: Its Content, Methods and Meaning" by A.D. Aleksandrov, et al.

- anonymous

Then the student says "Does it work in 3D?"

- anonymous

lol,

- bahrom7893

i^2 = i*i = (0,1)(0,1) = (0*0-1*1,0*1+1*0)=(-1,0)... that's where i^2 = -1 came from.

- bahrom7893

I was like WOOOOOOOOOOOOOOWW

- anonymous

The real wow comes from http://en.wikipedia.org/wiki/Euler%27s_identity .

- bahrom7893

yea that too.. i need to read its proof

- anonymous

it seems the questions stater already knew much info but wanted to see how we explain complex numbers, lol

- anonymous

the sqrt of -1 came from messing about with negative logs....

- bahrom7893

it seems that you don't know badrefs lol

- bahrom7893

never heard of that one estudier..

- anonymous

That's because I'm on very intermittently. I can't get to know everyone around here.

- anonymous

Yeah, logarithms provided the construction of imaginaries. Let me pull one up.

- bahrom7893

my professor lied to me.... :/

- anonymous

Not necessarily. I learned something else in math. In physics we constructed Platonic imaginaries through logs.

- anonymous

Because in physics we were more concerned with the "existence" of things.

- anonymous

@bahrom7893 - that's cos complex analysis is for pure mathematicians (they have to justify their existence, y'see:-)

- anonymous

That explains it. I can't find the reference right now, sorry. :<

- anonymous

Let me see if I can dig it up.....

- anonymous

Save us @estudier !

- anonymous

This might be it
http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2046%20e%20pi%20and%20i.pdf

- anonymous

"...They were perplexed because they had equally convincing (and flawed) arguments to "prove" that ln(-x) = ln(x)..."

- anonymous

"They" being Bernoulli and Euler

- anonymous

Anyway, the point is ln(-1) = pi*i

- anonymous

I'll try to post something as to how you can get a quantity that evaluates to -1 without all the hoopla.....

- anonymous

A really amusing question from Math Underflow http://math.stackexchange.com/questions/202172/why-is-i-0-498015668-0-154949828i
What does \(i!\) evaluate to?

- anonymous

Take a pair of vectors uv with the normal rules for multiplication etc and so write
uv = 1/2(uv+vu) + 1/2(uv-vu)
So that first term is basically u dot v and we'll call the second one u (wedge) v.
uv = u.v + u wedge v
vu = u.v - u wedge v
Multiply these two
uvvu = (u.v9^2 -(u wedge v)^2
Since vv = |v|^2 -> (u wedge v)^2 =-|u|^2|v|^2sin^2 theta
So whatever u wedge v is, it's square is a negative scalar

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