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I come across A Visual, Intuitive Guide to Imaginary Numbers, and find it's difficult for me to understand this section:

Let’s take a look. Suppose I’m on a boat, with a heading of 3 units East for every 4 units North. I want to change my heading 45 degrees counter-clockwise. What’s the new heading?
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Let’s try a simpler approach: we’re on a heading of 3 + 4i (whatever that angle is; we don’t really care), and want to rotate by 45 degrees. Well, 45 degrees is 1 + i (perfect diagonal), so we can multiply by that amount! enter image description here

Per my understanding, rotation should not change the distance from 0, but it's obviously the distance of 3 +　4i isn't same with -1 + 7j. How to understand complex rotation?

2 Answers 2

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They are only interested in the angle. If you want to preserve the magnitude, you must multiply by a complex number of absolute value one; in this example, you should take $(1+i)/\sqrt2$.

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I'd suggest you to familiarize with a polar form of a complex number (see at Wikipedia), defined by an Euler's formula:

$$m(\cos \varphi + i\sin\varphi) = m\,e^{i\varphi}$$

where $m$ is a magnitude of the complex number (i.e. the corresponding vector's length) and $\varphi$ is its argument (i.e. the angle from the real axis).

Then, applying standard laws of multiplication of products and powers:

$$z_1 \cdot z_2 = |z_1|e^{i\varphi_1} \cdot |z_2|e^{i\varphi_2} = |z_1|\,|z_2|\cdot e^{i(\varphi_1 + \varphi_2)}$$

a product of two numbers has a magnitude being a product of magnitudes of the two terms, and an argument being a sum of their arguments.

Your number $(1+i)$ has a magnitude $\sqrt{1^2+1^2} = \sqrt 2$ and argument $\arctan \frac 11 = 45^\circ$, hence in multiplication of $(3+4i)$ by $(1+i)$ the vector representing a product is longer than that of $(3+4i)$ by a factor of $\sqrt 2$ and rotated by $45^\circ$.