Re:Why does e to pi i equal -1? (2015) [video]
by recursive on Monday February 13, 2017 @06:23PM
(#13639820)
Using a matrix as an exponent isn't any more comprehensible than an imaginary number to me. If it works for you, that's great, but it's not much help to me.
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Re:Why does e to pi i equal -1? (2015) [video]
by chombier on Tuesday February 14, 2017 @02:53AM
(#13641979)
Think of exponentiation of some number 'a' as in-between its integer powers: 'a^1.5' is kind-of half-way between 'a' and 'a^2'.
If you plot all the integer powers of 'a', they all belong to a curve and the exponential simply fills-in the gaps for non-integer exponents.
Now, there are many possible ways to fill the gaps but the exponential does it so that a^m * a^n = a^{m+n} holds even for non-integer numbers m and n.
Similarly, if you take integer powers of a complex number, they all lie on some curve and the exponential fills-in the gaps, again turning sums into products. The same works with matrices, and so on.