Complex Moduli
The modulus of a complex number $x+yi$x+yi is the distance of the vector created from the origin to the point $\left(x,y\right)$(x,y). In other words, it is the size of the number.
The modulus is denoted using many different notations in various texts, websites, countries and schools. Here are a list of various notations.
- the letter $r$r
- $modz$modz
- $|z|$|z|
- $|x+iy|$|x+iy|
In this diagram, the point $P$P has been plotted. It corresponds to the complex number $3+4i$3+4i.
We can see the horizontal distance is $3$3, (the $x$x value) and the vertical distance is $4$4, (the $y$y value). The modulus is the length of the hypotenuse, (which is the length of the vector $P$P)
This distance can be found using Pythagoras' theorem.
| $r^2$r2 | $=$= | $x^2+y^2$x2+y2 |
| $r$r | $=$= | $\sqrt{x^2+y^2}$√x2+y2 |
| $=$= | $\sqrt{3^2+4^2}$√32+42 | |
| $=$= | $5$5 |
$r=\left|z\right|=\left|x+iy\right|=\sqrt{x^2+y^2}$r=|z|=|x+iy|=√x2+y2
Example
If $z=-2+2i$z=−2+2i and $w=4-i$w=4−i then find
a) $|z|$|z|
| $|z|$|z| | $=$= | $|-2+2i|$|−2+2i| |
| $=$= | $\sqrt{(-2^2)+2^2}$√(−22)+22 | |
| $=$= | $\sqrt{8}$√8 | |
| $=$= | $2\sqrt{2}$2√2 |
b) $|z|^2$|z|2
| $|z|^2$|z|2 | $=$= | $|-2+2i|^2$|−2+2i|2 |
| $=$= | $\left(\sqrt{8}\right)^2$(√8)2 | |
| $=$= | $8$8 |
c) $|z^2|$|z2|
| $|z^2|$|z2| | $=$= | $\left|\left(-2+2i\right)^2\right|$|(−2+2i)2| |
| $=$= | $\left|\left(-2+2i\right)\left(-2+2i\right)\right|$|(−2+2i)(−2+2i)| | |
| $=$= | $\left|-4-4i-4i+4i^2\right|$|−4−4i−4i+4i2| | |
| $=$= | $\left|-8-8i\right|$|−8−8i| | |
| $=$= | $\sqrt{-8^2+-8^2}$√−82+−82 | |
| $=$= | $\sqrt{64+64}$√64+64 | |
| $=$= | $\sqrt{128}$√128 | |
| $=$= | $8\sqrt{2}$8√2 |
d) $|z|-|w|$|z|−|w|
From part (a), we know that $|z|=2\sqrt{2}$|z|=2√2.
| $|w|$|w| | $=$= | $\sqrt{4^2+-1^2}$√42+−12 |
| $=$= | $\sqrt{16+1}$√16+1 | |
| $=$= | $\sqrt{17}$√17 | |
| Therefore: | ||
| $|z|-|w|$|z|−|w| | $=$= | $2\sqrt{2}-\sqrt{17}$2√2−√17 |
which is approximately $1.29$1.29.
e) $|z-w|$|z−w|
| $|z-w|$|z−w| | $=$= | $\left|\left(-2-2i\right)-\left(4-i\right)\right|$|(−2−2i)−(4−i)| |
| $=$= | $\left|-2-4-2i+i\right|$|−2−4−2i+i| | |
| $=$= | $\left|-6-i\right|$|−6−i| | |
| $=$= | $\sqrt{(-6)^2+(-1)^2}$√(−6)2+(−1)2 | |
| $=$= | $\sqrt{37}$√37 |
More Worked Examples
QUESTION 1
Consider the point on the Argand diagram.
What is the complex number represented by the point?
- Loading Graph...
What is the height of the point?
What is the horizontal distance of the point from the origin?
What is the length of the hypotenuse?
QUESTION 2
Find the modulus of $-8+3i$−8+3i.
QUESTION 3
If $z=12-16i$z=12−16i and $w=3-4i$w=3−4i, find:
$\left|z\right|$|z|
$\left|w\right|$|w|
$\left|z+w\right|$|z+w|
$\left|z\right|+\left|w\right|$|z|+|w|