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badreferences

  • 3 years ago

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

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  1. bahrom7893
    • 3 years ago
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    yooo you're in luck.. I'm taking a course in complex variables.

  2. bahrom7893
    • 3 years ago
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    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.

  3. estudier
    • 3 years ago
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    What are imaginary numbers? Not real......

  4. bahrom7893
    • 3 years ago
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    basically an imaginary number's an ordered pair

  5. heedcom
    • 3 years ago
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    its when you try to take the square root of a negative number

  6. estudier
    • 3 years ago
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    There is only one imaginary number, i, defined by i^2 = -1

  7. bahrom7893
    • 3 years ago
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    not really... a+bi are all imaginary.. a is the real part, b is the imaginary part.

  8. estudier
    • 3 years ago
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    b is real

  9. heedcom
    • 3 years ago
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    you guys are confusing this student, lol

  10. estudier
    • 3 years ago
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    Stir,stir......

  11. bahrom7893
    • 3 years ago
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    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.

  12. estudier
    • 3 years ago
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    i is appended to the real number system by fiat......

  13. heedcom
    • 3 years ago
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    lol

  14. bahrom7893
    • 3 years ago
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    which is defined by: (a,b)(c,d) = (ac-bd, ad+bc)

  15. heedcom
    • 3 years ago
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    this conversation is really good for this student, keep it up guys

  16. bahrom7893
    • 3 years ago
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    i meant i is an ordered pair, (0,1)

  17. estudier
    • 3 years ago
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    Then you make up an imaginary plane to go with this imagijnary number.....

  18. badreferences
    • 3 years ago
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    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.

  19. estudier
    • 3 years ago
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    Then the student says "Does it work in 3D?"

  20. heedcom
    • 3 years ago
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    lol,

  21. bahrom7893
    • 3 years ago
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    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.

  22. bahrom7893
    • 3 years ago
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    I was like WOOOOOOOOOOOOOOWW

  23. badreferences
    • 3 years ago
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    The real wow comes from http://en.wikipedia.org/wiki/Euler%27s_identity .

  24. bahrom7893
    • 3 years ago
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    yea that too.. i need to read its proof

  25. heedcom
    • 3 years ago
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    it seems the questions stater already knew much info but wanted to see how we explain complex numbers, lol

  26. estudier
    • 3 years ago
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    the sqrt of -1 came from messing about with negative logs....

  27. bahrom7893
    • 3 years ago
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    it seems that you don't know badrefs lol

  28. bahrom7893
    • 3 years ago
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    never heard of that one estudier..

  29. badreferences
    • 3 years ago
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    That's because I'm on very intermittently. I can't get to know everyone around here.

  30. badreferences
    • 3 years ago
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    Yeah, logarithms provided the construction of imaginaries. Let me pull one up.

  31. bahrom7893
    • 3 years ago
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    my professor lied to me.... :/

  32. badreferences
    • 3 years ago
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    Not necessarily. I learned something else in math. In physics we constructed Platonic imaginaries through logs.

  33. badreferences
    • 3 years ago
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    Because in physics we were more concerned with the "existence" of things.

  34. estudier
    • 3 years ago
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    @bahrom7893 - that's cos complex analysis is for pure mathematicians (they have to justify their existence, y'see:-)

  35. badreferences
    • 3 years ago
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    That explains it. I can't find the reference right now, sorry. :<

  36. estudier
    • 3 years ago
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    Let me see if I can dig it up.....

  37. badreferences
    • 3 years ago
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    Save us @estudier !

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

  39. estudier
    • 3 years ago
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    "...They were perplexed because they had equally convincing (and flawed) arguments to "prove" that ln(-x) = ln(x)..."

  40. estudier
    • 3 years ago
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    "They" being Bernoulli and Euler

  41. estudier
    • 3 years ago
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    Anyway, the point is ln(-1) = pi*i

  42. estudier
    • 3 years ago
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    I'll try to post something as to how you can get a quantity that evaluates to -1 without all the hoopla.....

  43. badreferences
    • 3 years ago
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    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?

  44. estudier
    • 3 years ago
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    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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