With which number should I multiply $z = a + i b$ to rotate it by 45 degrees

Question

​Multiplying​ ​a​ ​non​ ​zero​ ​complex​ ​number​ ​$z$​ ​by​ the imaginary unit $i$​ ​is​ ​equivalent​ ​to​ ​a ​counter​ ​clockwise​ 90​ ​degrees​ rotation ​in​ ​the​ ​complex​ ​plane.​ Instead, a ​counter​ ​clockwise​ ​rotation​ ​by​ ​45 degrees​ ​results​ ​from​ ​multiplying​ ​$z$​ ​by which number?

Answer 1

$z = a + ib = |z| e^{i\theta}$

Then if we take $w = e^{i\frac{\pi}{4}} = cos(\frac{\pi}{4}) + i sin(\frac{\pi}{4})$

$zw = |z|e^{i(\theta + \frac{\pi}{4})}$

In other words, thats rotate $z$ $45$ degrees

Answer 2

Although the correct answer has already been given, namely $\frac{\sqrt2}2(1+i)$, I think it is worth pointing out, that provided you agree to replace the word "rotation" with "similarity", it is a safe bet that the most natural answer would be $1+i$ : Let $s:\mathbb{C}=\mathbb{R}^2 \to \mathbb{C}, z=x+iy \mapsto u+iy=(1+i)z=(1+i)(x+iy)=(x-y,x+y)$.

Then $s(\mathbb{Z}\times\mathbb{Z})\subset \mathbb{Z}\times\mathbb{Z}$. Let $\{1,i\}$ the canonical basis of $\mathbb{R}^2$. Then the matrix of $s$ is $\begin{bmatrix}1 &-1 \\1 & 1\end{bmatrix} $. You can illustrate $s$ with drawings such that :