Absolute value graph

Question

What happens to the absolute value of a complex number $a+bi$ if it is multiplied by $i$? Describe what happens to its graph.

Well doesn't it just go back and forth between two numbers? I don't really know though. If this is missing context please comment before voting to close.

Answer 1

Let $z=a+bi$. Then $\left|z\right| = \sqrt{a^2+b^2}$. Multiplying by $i$ we get $zi = ai + bi^2 = -b + ai$ and hence $\left|iz\right| = \sqrt{b^2 + a^2} = \left|z\right|$. Thus, the absolute value does not change when we multiply by $i$. As for the second question, I'm not aware of any sense in which a complex number has 'a graph'.

Answer 2

Hint: $(a+bi)i=ai+bi^2$. What is $i^2$?

Answer 3

$$ \arg(z) \rightarrow \arg(zi) \equiv \arg(z e^{i\pi/2}) \equiv \arg(z) + \pi/2 $$

so "the graph" rotates $\pi/2$ counterclockwise.

$$||z|| \rightarrow ||zi|| \equiv ||z e^{i\pi/2}|| \equiv ||z|| $$

Answer 4

1. When $\text{z}\in\mathbb{C}$: $$\left|\text{z}\right|=\left|\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right|=\sqrt{\Re^2\left[\text{z}\right]+\Im^2\left[\text{z}\right]}$$
2. When $\text{z}\in\mathbb{C}$: $$\left|\text{z}i\right|=\left|\left(\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right)i\right|=\left|\Re\left[\text{z}\right]i-\Im\left[\text{z}\right]\right|=\sqrt{\Re^2\left[\text{z}\right]+\Im^2\left[\text{z}\right]}$$

So, it does not change at all.

Answer 5

If $z = x + yi$ then $zi = (x + yi)i = xi + yi^2 = xi - y$, considering $i^2 = -1$. You can see that the real part of $z$ is the imaginary part in $zi$ and the imaginary part in $z$ is the real part in $zi$. On a graph you could see this as "switching the axis" where the value on the x-axis becomes the value on the y-axis and analogous for the value on the y-axis. However, $|z| = |zi|$, because $|z| = \sqrt{x^2 + y^2}$ and $|zi| = |xi - y| = \sqrt{x^2 + (-y)^2} = \sqrt{x^2 + y^2}$.