New Zealand
Level 8 - NCEA Level 3

# Complex Moduli

Lesson

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

Remember!

$r=\left|z\right|=\left|x+iy\right|=\sqrt{x^2+y^2}$r=|z|=|x+iy|=x2+y2

##### Example

If $z=-2+2i$z=2+2i and $w=4-i$w=4i 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|=22.

 $|w|$|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) $|z-w|$|zw|

 $|z-w|$|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.

1. What is the complex number represented by the point?

What is the height of the point?

3. What is the horizontal distance of the point from the origin?

4. What is the length of the hypotenuse?

##### QUESTION 2

Find the modulus of $-8+3i$8+3i.

##### QUESTION 3

If $z=12-16i$z=1216i and $w=3-4i$w=34i, find:

1. $\left|z\right|$|z|

2. $\left|w\right|$|w|

3. $\left|z+w\right|$|z+w|

4. $\left|z\right|+\left|w\right|$|z|+|w|

### Outcomes

#### M8-9

Manipulate complex numbers and present them graphically

#### 91577

Apply the algebra of complex numbers in solving problems