Batman (triprc) wrote in algorithms,

What exponent must e be raised to in order to equal zero?
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There is no such exponent, e never equals zero.
As noted above, there's no such value. This is true for any nonzero base. We know from elsewhere that 0 = xy is only possible if x = 0 or y = 0. Well here you're saying 0 = x^k = x*x*..., essentially. So it follows that this is only possible if x=0. So e, and anything else not zero, doesn't work.
I just wanted to respond to you explaining my qualification below. x^k = x*x... (k-times) only when k is a positive integer. There isn't a natural expansion for other values, and exponentiation for all numbers is described by the exponential function. Since I'm no king whiz, I don't know how that function will behave using, say, a complex irrational.
Yes, I was just giving an example to bring out an intuition that should exist from previous experience in math.

There is no value k that will do so, real or complex. The limit of e^x as x approaches negative infinity is 0. That's the best you can do, since negative infinity isn't really a "value" per se, it only has meaning in the limit statement.
To refute the complex number possibility specifically:

Say complex w = xi + y for real x and y.
e^w = e^(xi+y) = e^(xi) * e^y
We know e^y cannot be 0, thus if e^w =0 it must be e^(xi) that is equal to 0.

But by Euler's formula:
e^(xi) = i * sin(x) + cos(x)
0 = i * sin(x) + cos(x) => i * sin(x) = -cos(x)
But neither sin(x) nor cos(x) may have an imaginary value for real x, thus one side of the equation is imaginary and the other is not. Therefore the equation has no solution.

Thus if e^x = 0 lacks a solution for real x, it also lacks one for complex x.
Thanks for the refresher, and I really do mean that.

This sort of thing doesn't come up often in, say, operating systems development, so I haven't touched these topics since college.
Why are you limiting k to equaling a positive integer?
I'm not, it's just easier to relate to simple mathematical ideas about zero that way. I said "essentially" because that's not actually what exponentiation means, but getting into more wouldn't be particularly fruitful.
What exponent? Maybe I've spent too long in Babylon, but this is like asking "how many times must I multiply e against itself to equal 0", and the answer is "there's no answer", because the only number multiplied against itself any number of times, and yield zero, is zero.

Unless there's an imaginary number out there that suffices.
"- infinity" would get you quite close :)
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