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Things that are equal to 1

  1. Feb 23, 2013 #1
    Sort of a fun question, I think. Or perhaps silly. I was just thinking about all the different things in mathematics that happen to equal one. Such as

    (sin x)^2 + (cos x)^2 = 1

    in probability P(s) = 1

    The radius of a unit circle: x^2 + y^2 = 1

    Obviously I don't mean things like 3-2.

    Since "Things which are equal to the same thing are equal to each other." any of these things, regardless of the branch of mathematics they are part of, would be equal to each other.

    So then P(s) = (sin x)^2 + (cos x)^2

    I have no idea what that would actually mean, of course. But it's interesting (to me) to think about.

    What else?

    -Dave K
     
  2. jcsd
  3. Feb 23, 2013 #2
    Last edited by a moderator: May 6, 2017
  4. Feb 23, 2013 #3
  5. Feb 23, 2013 #4
    1=1, 1-1+1=1, 1-1+1-1+1=1, ...

    One could come up with an infinite list of things equal to 1. :)
     
  6. Feb 23, 2013 #5
    Which is why I said I didn't mean stuff like that... :tongue:
     
  7. Feb 23, 2013 #6
    How about the area of one quarter period of the sine function?
     
  8. Feb 23, 2013 #7
    ah, so integral from 0 to pi/2 of sinx

    I gotta start learning how to do Latex here.
     
  9. Feb 23, 2013 #8
    :) Not too hard. This would be, using fairly simple syntax, \int_0^{\frac{\pi}{2}}\sin(x)\ dx, surrounded with [ itex ] tags, or [ tex ] tags for better formatting but requiring their own line. (The other thing you should probably know is that spaces don't mean anything in LaTeX unless a \ is right before them.)
     
  10. Feb 23, 2013 #9
    Right, and if we use 2pi (or tau :) ) in place of pi we get

    [itex]e^{i2\pi} = 1 [/itex]
     
  11. Feb 23, 2013 #10
  12. Feb 23, 2013 #11
    Workin on it. Thanks. We should have a LaTex practice thread.
     
  13. Feb 25, 2013 #12
    [itex](2i((1/2)!)))^2)/(-\pi)[/itex]
     
    Last edited: Feb 26, 2013
  14. Feb 25, 2013 #13
    This is going to be pedantic, but (1/2)! is not defined. The factorial (how you use it) is only defined for nonnegative natural numbers. You should use the Gamma function and not the factorial.
     
  15. Feb 26, 2013 #14
    opps!
     
    Last edited: Feb 26, 2013
  16. Feb 26, 2013 #15
    Still, we can use the Gamma function. That's cool.
     
  17. Feb 26, 2013 #16
    Yeah, for sure. The Gamma function is one of the coolest functions in mathematics!
     
  18. Feb 26, 2013 #17

    Mute

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    The TI-83 I used in high school interpreted (-1/2)! to be ##\sqrt{\pi}##. I remember being pretty surprised when I found out about that (this was before I knew about the Gamma function)! I think that may have been the only negative value for which the factorial function returned an actual answer, though, so it was probably specially programmed in.
     
  19. Feb 26, 2013 #18
    You mean ##\sqrt{\pi}/2## ? Wolframalpha shows it as that... but it shows Gamma for (-1/3)! as ##-\gamma (4/3)##
     
  20. Feb 26, 2013 #19

    Mute

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    Nope, notice there's a minus sign: "(-1/2)!", if interpreted as ##\Gamma(1-1/2)##, is equal to ##\sqrt{\pi}##. "(+1/2)!", if interpreted as ##\Gamma(1+1/2)##, is equal to ##\sqrt{\pi}/2##.
     
  21. Feb 26, 2013 #20
    Hmm, I guess I don't understand Gamma...

    The results of -(1/2)! and (-1/2)! are different as you indicate.

    Gamma takes precedence in the order of operations?
    I'll take a look at Gamma.
     
    Last edited: Feb 26, 2013
  22. Feb 26, 2013 #21

    Mute

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    'Gamma' is a function, so "##\Gamma(x)##" is the same sort of notation as "f(x)", except that f(x) is a general notation for a function while ##\Gamma(x)## generally refers to a specific function defined in terms of an integral (and the analytic continuation if we consider complex number inputs to the function).

    It can be shown that for x = n, where n is an integer, ##\Gamma(n+1) = n!##. The trick with the non-integer factorials comes from abusing this notation in the case where x is not an integer, i.e., writing ##\Gamma(x+1) = x!##. From this it may be easier to see why (-x)! is different from -(x!).

    Edit: to keep this post somewhat on the actual topic, one of the forms of 1 that I use often enough is introducing ##1 = z^\ast/z^\ast## when I want to rewrite a complex number ##1/z## in a more convenient form with the imaginary and real parts readily obvious:

    $$\frac{1}{z} = \frac{1}{z}\times 1 = \frac{1}{z} \frac{z^\ast}{z^\ast} = \frac{z^\ast}{|z|^2}.$$
     
  23. Feb 27, 2013 #22
    Cool discussion, and thanks for humoring me. :)

    -Dave K
     
  24. Feb 27, 2013 #23

    mfb

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    $$0.\bar{9}$$
    $$\lim_{n \to \infty} \sqrt[n]{n}$$
    More general, for every real a:
    $$\lim_{n \to \infty} \sqrt[n]{n^a}
     
  25. Feb 27, 2013 #24
    -e^(i*pi*2k) where k is an integer
     
  26. Feb 27, 2013 #25
    Are you sure about that - ??
     
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