# Interpretations of Exponents

I have been reading one of the proofs of Euler's identity, $$e^{i\theta}=\cos(\theta)+i\sin(\theta)$$.

I have always thought that exponents can be interpreted as its base being multiplied its exponent times (i.e. $$3^5$$ multiplying 3 5 times together).

But, this interpretation breaks down when the exponent is not a rational number. ($$2^{1/2}$$ can be interpreted using this logic. $$2^1$$ is 2 multiplied once. $$\left(2^{1/2}\right)^{2}=2^1$$ So $$2^{1/2}$$ is a number that can be multiplied twice to get 2).

Why does this intuition break down when we multiply complex numbers ($$a+b\textbf{i}$$) and irrational numbers? And also is there some other geometric intuition for complex and irrational numbers?

The rate at which $$e^z$$ changes from $$1$$ to something else as $$z$$ changes from $$0$$ to something else, is the same as the rate at which $$z$$ changes. Thus if $$z$$ is changing at a rate of $$i,$$ then so is $$e^z.$$
• Isn't that just derived from the fact that $\frac{d}{dz}\left(e^z\right)=z\cdot e^z$ Jul 27, 2020 at 6:06
• @JadenLee : It's simpler than that: $\dfrac d {dz} e^z = e^z. \qquad$ Jul 27, 2020 at 6:10
• Oh yeah. Differentiating with respect to z doesn't make z multiply. Haha I'm usually used to having $x$ as the variable, so I thought that $z$ was a constant. Jul 27, 2020 at 6:11
For exponents to a complex number’s power, it is as if you are rotating a vector. For example, $$2^{2+i}$$ can’t be broken down to $$2^2\cdot 2^i$$. $$2^2$$ Scales your vector and the $$2^i$$ part rotates your vector 1 radian. For example in the case of $$e^{i\pi}$$, it rotates the unit vector $$\pi \ rad$$ to make $$e^{i\pi} = -1$$
• Oh. So $c\textbf{i}$ rotates $c \ rad$ around the origin? Jul 27, 2020 at 6:07
• Yes. That is exactly what is does and that explains why $e^{i\pi}=-1$