4 ms·
It still works with irrational exponents (start with fractional numbers and work towards that). It also works with imaginary exponents. It works because the pow
by dlss 15y ago
It still works with irrational exponents (start with fractional numbers and work towards that). It also works with imaginary exponents. It works because the power notation is short hand. But that wasn't the point of my reply. The point was that stating things multiple ways assists us in understanding. Does this not match your experience?
- xyzzyz 15y agoIt still works with irrational exponents (start with fractional numbers and work towards that). So why e^pi * e^e = e^(pi + e)? Yes, it follows from the fact that it works for rational numbers, but in order infer this, you'd need to prove the continuity of exponential function, which is nontrivial at best. Of course, if you define a^x to be the unique continuous function f: R -> R, such that f(1) = a and f(a)f(b) = f(a+b), as soon as you proved the existence and uniqueness of this function, this follows straight from definition. There are also different definitions of exponential functions, like exp(x) = lim n->inf (1+x/n)^n, or exp(x) = sum_{n=0}^inf x^n/n! . How easy it is to prove now that exp(pi)exp(e) = exp(pi+e) ?