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Sure! You might want to check out http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-u

by kalid 14y ago

Sure! You might want to check out http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://betterexplained.com/articles/intuitive-understanding-... I like this question because it really works your intuition. The basics: x^y means "grow at x, for y units of time". I see "2^3" as "grow at 2x for 3 units of time". Having a base of i means your "growth" is a rotation at 90 degrees, no scaling. So i^(1/2) means a 45 degree rotation, i^3 means a 270 rotation, etc. Raising this to the i power (or 1/i power, which is -i) means the growth that was originally purely rotational is now rotated. So instead of growing at i, you are growing at (i * 1/i = 1). So, we should expect a positive real number, greater than 1 since our growth is positive. How long do we actually grow for? Well, the base of "i" is really e^(i* pi/2), which means "Start at 1.0 and rotate continuously for pi/2 seconds". We've now modified this to "Start at 1.0 and grow at 1.0 for pi/2 seconds", which is e^(pi/2). So the answer is i^(1/i) = e^(pi/2) ~ 4.8 It's a bit tough with text-only, read the above article for more diagrams.

yequalsx 14y ago
Note that 2575.97 also has the property that raised to the ith power gives i. There are infinitely many such numbers.
- diminish 14y ago
  or 0.008983291.. Are there countably infinite solutions or otherwise? See infinity discussion few days ago at HN http://news.ycombinator.com/item?id=4526049 http://news.ycombinator.com/item?id=4526049 PS: Just curious, i am not a mathematician.
  - btilly 14y ago
    It is countably infinite. Let's try to find them all. A relatively simple way to understand this is that i^i = e^(i * log(i)) for every possible log of i. So all we need to do is understand what values log(i) could have (there are actually many), and then we can work it out. But log(z) just undoes e^z, so we need to understand e^z. Now let's work backwards. If z = x + y i with x and y real, then x tells us the absolute value of e^z and y tells us the angle. The absolute value of i is 1, so any possible solution to log(i) has real part 0. The angle that we want to wind up with is 90 degrees, or pi/2. Therefore y can be ..., -3.5 pi, -1.5 pi, .5 pi, 2.5 pi, 4.5 pi, ... . Therefore log(i) has to be one of 1.5 pi i, -.5 pi i, -2.5 pi i, -4.5 pi i, ... . Now i^i is e^(i log(i)) so it can be any of ..., e^(3.5 pi), e^(1.5 pi), e^(-.5 pi), e^(-2.5 pi), e^(-4.5 pi), ... . Unless I've made a trivial calculation error, that is the whole list.
    - yogrish 14y ago
      simple explanation of PI: http://en.wikipedia.org/wiki/File:Pi-unrolled-720.gif http://en.wikipedia.org/wiki/File:Pi-unrolled-720.gif

by kalid 14y ago

yequalsx 14y ago

Note that 2575.97 also has the property that raised to the ith power gives i. There are infinitely many such numbers.

diminish 14y ago
or 0.008983291.. Are there countably infinite solutions or otherwise? See infinity discussion few days ago at HN http://news.ycombinator.com/item?id=4526049 http://news.ycombinator.com/item?id=4526049 PS: Just curious, i am not a mathematician.
- btilly 14y ago
  It is countably infinite. Let's try to find them all. A relatively simple way to understand this is that i^i = e^(i * log(i)) for every possible log of i. So all we need to do is understand what values log(i) could have (there are actually many), and then we can work it out. But log(z) just undoes e^z, so we need to understand e^z. Now let's work backwards. If z = x + y i with x and y real, then x tells us the absolute value of e^z and y tells us the angle. The absolute value of i is 1, so any possible solution to log(i) has real part 0. The angle that we want to wind up with is 90 degrees, or pi/2. Therefore y can be ..., -3.5 pi, -1.5 pi, .5 pi, 2.5 pi, 4.5 pi, ... . Therefore log(i) has to be one of 1.5 pi i, -.5 pi i, -2.5 pi i, -4.5 pi i, ... . Now i^i is e^(i log(i)) so it can be any of ..., e^(3.5 pi), e^(1.5 pi), e^(-.5 pi), e^(-2.5 pi), e^(-4.5 pi), ... . Unless I've made a trivial calculation error, that is the whole list.
  - yogrish 14y ago
    simple explanation of PI: http://en.wikipedia.org/wiki/File:Pi-unrolled-720.gif http://en.wikipedia.org/wiki/File:Pi-unrolled-720.gif