What the hell were they thinking?

Discussion in 'Physics & Math' started by Mr. Hamtastic, Sep 9, 2008.

  1. Mr. Hamtastic whackawhackado! Registered Senior Member

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    square root of negative 1=i.

    What? I'm no mathemetician, but do you guys really use something frequently that is 'imaginary'?
     
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  3. rpenner Fully Wired Valued Senior Member

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    You do if you want to be able to find roots to equations like \(x^2 + 1 = 0\) or relate the exponential function to cosine and sine.
     
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  5. Mr. Hamtastic whackawhackado! Registered Senior Member

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    what? why would you want to find those? (how do you do the squared thing?)
    why can't x=1?
     
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  7. Janus58 Valued Senior Member

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    Not only in mathematics, but in electronics complex numbers (numbers with both real and imaginary components) are used when working with AC circuits.
     
  8. Mr. Hamtastic whackawhackado! Registered Senior Member

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    What!? So what is going on with my electricity is partly IMAGINARY!?
     
  9. Vkothii Banned Banned

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    Actually, all numbers are imaginary, but some of them seem to be real fundamental things (that we imagine are really there). Pi, or e, for example are fundamental like that; i is too: if negative numbers exist, i must exist.
     
  10. kevinalm Registered Senior Member

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    993
    No, but certain aspects of manipulating vectors can be treated as if you were multipling by i. Somewhat oversimplifying, multiplying a vector by i means something like, "at right angles to".

    Also, there is a very usefull math formula, developed by Euler, in which multiplying by i effectively differentiates the formula. Really neat for some electronics and harmonic motion problems.

    I wouldn't get overly excited by the term 'imaginary'. It's really just a nifty math tool.
     
  11. D H Some other guy Valued Senior Member

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    Beat me to the punch. Mr. Hamtastic, would it make you feel better if you called them nifty numbers rather than imaginary numbers?

    Every extension of the concept of number has been met with some resistance. For a long, long time, negative solutions to equations were viewed as invalid. The concept of negative numbers wasn't well developed until just before Newton's time. Numbers that cannot be expressed as the ratio of two integers -- irrational! The name for the symbol in front of the two here -- \(^{\surd} ^2\) is the "surd", short for absurd. If the square root of two is irrational, the square root of minus one is doubly so. In the end, the names "negative", "irrational", "imaginary", and "complex" are just names. Don't take them for having any more meaning than that.

    Besides, have you ever seen a one in the real world? Not one cow, or one chair -- just a plain, naked one? All numbers are, in a sense, imaginary. We use them, very successfully, to model the real world. That does not mean they are "real". Treating them as such is an example of mistaking the map for the territory.
     
  12. rpenner Fully Wired Valued Senior Member

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    500 years ago, \(x + 1 = 0\) was an equation that couldn't be solved until the negative numbers were invented. But then the question as to what \(x^2 = -1\) meant, which led to the complex numbers and many useful results before the introduction of tensors into physics. But since the complex numbers are "algebraically complete," there is no need to invent new numbers to solve \(x^2 = i\).
     
  13. §outh§tar is feeling caustic Registered Senior Member

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    Don't forget the time when 0 wasn't around. 0 was invented. Hell, every number was invented. But in a country where an evolution denying 'Walmart everymom' can galvanize the electorate, it isn't surprising that our history is forgotten.

    Eh? Who injected that blatantly political piece of resentment?

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  14. §outh§tar is feeling caustic Registered Senior Member

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    Yes. Everything is in your mind, including your mind.
     
  15. temur man of no words Registered Senior Member

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    You skipped transcendental numbers

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  16. Walter L. Wagner Cosmic Truth Seeker Valued Senior Member

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    e^(Pi X i) = -1

    Always has, always will.
     
  17. AlphaNumeric Fully ionized Registered Senior Member

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    Yes. Haven't you ever felt confined by being forced to work in a field which isn't algebraicly closed?

    I mean, hello?!

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  18. rpenner Fully Wired Valued Senior Member

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    \(e^{i \pi} + 1 = 0\) is somewhat prettier. But the point of the discussion was that these are invented human symbols for invented human meanings.

    us.metamath.org/mpegif/eulerid.html
    Here is an implied link which traces these concepts (exponential function, pi, complex and real numbers which include 0 and 1) to a foundation in ZFC-set theory.
     
  19. James R Just this guy, you know? Staff Member

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    The "imaginary" thing is just a name. An "imaginary number" is just like a "natural number" or a "rational number". The term "imaginary" doesn't mean these numbers "aren't real" or anything like that (although, confusingly, there is another class of numbers known as "real numbers").

    In the equation \(x^2 + 1 = 0\)?

    Try it! Let x = 1. Then, x squared is 1 times 1, which equals 1. And 1 + 1 = 0...

    No! 1 + 1 = 2, doesn't it? So, x = 1 doesn't work as a "solution" to the problem.

    What works is x = i, where \(i = \sqrt{-1}\).

    Then i squared is -1, and -1 + 1 = 0.
     
  20. Mr. Hamtastic whackawhackado! Registered Senior Member

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    wow. I'm a simple math kind of guy. All this higher math makes me a bit light-headed. I think this concept may simply be beyond me. Thanks for your answers, though.
     
  21. §outh§tar is feeling caustic Registered Senior Member

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    That must be how the president feels about the multi-trillion dollar debt when he talks to them elitist 'out-of-touch' economists.

    WHAT! Who keeps injecting that into the conversation!?!?

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    :shrug:
     
  22. AlphaNumeric Fully ionized Registered Senior Member

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    I don't mean this in a vitriolic way, so don't read it as such, but why did you post this thread if you admit you're a 'simple math kind of guy'. For instance, I'm a 'simple language kind of guy' but I don't question things like the requirement for the concept of 'adverbs'. Or I'm a 'simple history kind of guy' but I don't question people researching 13th century feudal China. They have their reasons.

    And in the case of complex numbers, a quick Wiki will tell you how important and far ranging they are. For instance, \(e^{ix} = \cos x + i \sin x\), so in just one expression you can encompass both trig functions. You can also solve any quadratic, where (as Read Only explained) previously something like \(x^{2}+1=0\) was unsolvable. I do work in the area of algebraic geometry, where all the methods need me to have a framework which allows for such polynomials to have solutions.

    There's plenty of areas of maths where, using previous knowledge, you can only go so far and then you hit a wall. Mathematical development is often someone saying "Suppose [this] solves [that]. What else does [this] do?". So if \(i = \sqrt{-1}\) allows you to solve \(x^{2}+1=0\), what else can you do with it? Someone once asked "If I invent the concept of an a and b such that a*b=-b*a, what can I build from that?". 50 years later someone said "The quantum mechanics of the electron!". Much of maths looks crazy and "WTF is the point of that!" but you'd be hard pressed to find an area of maths with zero applications to something in the real world.
     
  23. §outh§tar is feeling caustic Registered Senior Member

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    Logic?

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