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yooo you're in luck.. I'm taking a course in complex variables.
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.
What are imaginary numbers? Not real......
basically an imaginary number's an ordered pair
its when you try to take the square root of a negative number
There is only one imaginary number, i, defined by i^2 = -1
not really... a+bi are all imaginary.. a is the real part, b is the imaginary part.
b is real
you guys are confusing this student, lol
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.
i is appended to the real number system by fiat......
which is defined by: (a,b)(c,d) = (ac-bd, ad+bc)
this conversation is really good for this student, keep it up guys
i meant i is an ordered pair, (0,1)
Then you make up an imaginary plane to go with this imagijnary number.....
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.
Then the student says "Does it work in 3D?"
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.
I was like WOOOOOOOOOOOOOOWW
The real wow comes from http://en.wikipedia.org/wiki/Euler%27s_identity .
yea that too.. i need to read its proof
it seems the questions stater already knew much info but wanted to see how we explain complex numbers, lol
the sqrt of -1 came from messing about with negative logs....
it seems that you don't know badrefs lol
never heard of that one estudier..
That's because I'm on very intermittently. I can't get to know everyone around here.
Yeah, logarithms provided the construction of imaginaries. Let me pull one up.
my professor lied to me.... :/
Not necessarily. I learned something else in math. In physics we constructed Platonic imaginaries through logs.
Because in physics we were more concerned with the "existence" of things.
@bahrom7893 - that's cos complex analysis is for pure mathematicians (they have to justify their existence, y'see:-)
That explains it. I can't find the reference right now, sorry. :<
Let me see if I can dig it up.....
Save us @estudier !
This might be it http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2046%20e%20pi%20and%20i.pdf
"...They were perplexed because they had equally convincing (and flawed) arguments to "prove" that ln(-x) = ln(x)..."
"They" being Bernoulli and Euler
Anyway, the point is ln(-1) = pi*i
I'll try to post something as to how you can get a quantity that evaluates to -1 without all the hoopla.....
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?
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