Why did Euler use e to represent complex numbers?

From Euler we've learned that $z=re^{i\theta}$.

And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$.

Why must we use e to represent these numbers correctly? It seems that I could arbitrarily choose a different exponent $z=r\pi^{i\theta}$ and get the same size for $z$ as I did before: $|z|^2=r\pi^{i\theta}\times r\pi^{-i\theta}=r^2$

What did I miss?

Answer

If we wish to express $\pi^{i\theta}$ as a series then we have:

$$\pi^{i\theta} = e^{i\ln(\pi)\theta} = \sum_{n=0}^\infty i^n \frac{(\ln(\pi)\theta)^n}{n!} = \cos(\ln(\pi)\theta)+i\sin(\ln(\pi)\theta).$$

Calculating precisely $\ln(N)$ for $N \in \mathbb{N}$ can be difficult, not to mention $\ln(\pi)$. This would add more complications than it would be worth. Moreover, $\pi^{i\theta}$ has period $2\pi/\ln(\pi)$, which is not compatible with polar coordinates.

On the other hand, since we can write $$e^{i\theta} = \cos(\theta) + i \sin(\theta),$$ we can express $e^{i\theta}$ by calculating the already well known trigonometric functions.

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I would like to add that the use of $e^{i\theta}$ is because of the nice representation found by Euler. If you were to approach the polar representation for the first time, you would approach it more like this:

Let $z=x+iy$ be a complex number, which we can visualize as a vector in $\mathbb{R}^2$, $z=(x,y)$. The magnitude of $z$ is $\|z\|= \sqrt{x^2+y^2}$. We can write the real part as $x=\|z\| \cos(\theta)$ where $\theta$ is the angle formed between the real axis and the vector at the origin. Similarly $y=\|z\| \sin(\theta)$. Thus $$z= \|z\|\cos(\theta)+i \|z\|\sin(\theta) = \|z\|(\cos(\theta)+i\sin(\theta)).$$

Until now, our reasoning was completely geometric. Independently we can work out the expression, due to Euler, $e^{i\theta} = \cos(\theta)+i\sin(\theta)$. This now naturally leads to $$z=\|z\|e^{i\theta}.$$ If it turned out that $\pi^{i\theta} = \cos(\theta)+i\sin(\theta)$ then we would use that instead. However, we know that this is not the case.

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I would also like to point out that there is an intuitive reason to think that $e^{i\theta}$ should be of the form $\cos(\theta)+i\sin(\theta)$.

Notice that if we write $f(\theta) = e^{i\theta} = u(\theta)+iv(\theta)$, then $$f''(\theta) = i^2 f(\theta) = - f(\theta).$$

Hence $$u''(\theta) = -u(\theta) \text{ and } v''(\theta) = - v(\theta).$$

Thus from differential equations, we can express $u$ and $v$ as a linear combination of $\sin(\theta)$ and $\cos(\theta)$.

This motivates the investigation into the series of the exponential function. From this perspective, it is not surprising to discover $\cos(\theta)$ and $\sin(\theta)$ inside the series for $e^{i\theta}$.

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One final edit: If we let $A$ and $B$ be complex numbers, then my previous statement can be expressed as: $$e^{i\theta} = A\cos(\theta)+B\sin(\theta)$$

Setting $\theta=0$ we see that $e^{0}=1=A\cdot 1 = A$. And $\theta = \pi/2$ yields $e^{i\pi/2} = B$.

Therefore, $$e^{i\theta} = \cos(\theta) + e^{i\pi/2} \sin(\theta).$$ What is left is to determine $e^{i\pi/2}$. Since $e^{i\theta}$ is $2\pi$ periodic, $e^{0}=e^{i2\pi}$. Thus we can see that $(e^{i\pi/2})^4 -1 = 0$, which means $e^{i\pi/2}$ satisfies the polynomial $x^4-1=0$. Thus $e^{i\pi/2} = \pm 1 \text{ or } \pm i$.

Taking the derivative of both sides of $e^{i\theta} = \cos(\theta) + e^{i\pi/2} \sin(\theta)$ we find: $$ie^{i\theta} = -\sin(\theta) + e^{i\pi/2} \cos(\theta)$$ and therefore by setting $\theta = 0$ we have: $$i = e^{i\pi/2}.$$ Thus we conclude $$e^{i\theta} = \cos(\theta)+i\sin(\theta).$$ All without Taylor series.