The modulus and the conjugate

Complex numbers, grade 12 (5 units) Lesson 4 of 18Learning

The modulus and the conjugate

The modulus

The modulus of $\displaystyle {z}={a+bi}$ is the distance of the point $\displaystyle (a,b)$ from the origin – by Pythagoras’ theorem:

\displaystyle |z|

\displaystyle =

\displaystyle \sqrt{a^{2}+b^{2}}

$\displaystyle {|3-4i|}={\sqrt{9+16}}={5}$ .
$\displaystyle {|-2i|}={2}$ , and $\displaystyle {|-7|}={7}$ : for a real number this is the absolute value you know.
$\displaystyle {z\cdot\bar{z}}={a^2+b^2}$ , so $\displaystyle {z\cdot\bar{z}}={|z|^2}$ .

The complex numbers have no order, but their moduli are real numbers, and those can be compared: $\displaystyle |3+4i|>|1+i|$ .

The distance between two points

$\displaystyle z_1-z_2$ is the vector from $\displaystyle z_2$ to $\displaystyle z_1$ , so $\displaystyle |z_1-z_2|$ is its length – the distance between the points:

\displaystyle |z_{1}-z_{2}|

\displaystyle =

\displaystyle \sqrt{(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}}

The distance between $\displaystyle 2+3i$ and $\displaystyle -1-i$ : $\displaystyle {|(2+3i)-(-1-i)|}={|3+4i|}={5}$ .

Properties of the conjugate and the modulus

These properties are proved by substituting $\displaystyle {z}={a+bi}$ :

$\displaystyle {z+\bar{z}}={2a}={2\operatorname{Re}(z)}$ – a real number.
$\displaystyle {z-\bar{z}}={2bi}={2i\operatorname{Im}(z)}$ – a purely imaginary number, or 0.
$\displaystyle {\overline{z_1+z_2}}={\bar{z}_1+\bar{z}_2}$ and $\displaystyle {\overline{z_1\cdot z_2}}={\bar{z}_1\cdot\bar{z}_2}$ .
$\displaystyle {\bar{\bar{z}}}={z}$ and $\displaystyle {|\bar{z}|}={|z|}$ .
$\displaystyle {|z_1\cdot z_2|}={|z_1|\cdot|z_2|}$ and $\displaystyle {\left|\frac{z_1}{z_2}\right|}={\frac{|z_1|}{|z_2|}}$ .
$\displaystyle z$ is real exactly when $\displaystyle {\bar{z}}={z}$ .

A proof of $\displaystyle {|z_1\cdot z_2|}={|z_1|\cdot|z_2|}$ : $\displaystyle {|z_1z_2|^2}={z_1z_2\cdot\overline{z_1z_2}}={z_1\bar{z}_1\cdot z_2\bar{z}_2}={|z_1|^2|z_2|^2}$ , and neither side is negative.

The triangle inequality

In the parallelogram whose vertices are $\displaystyle 0$ , $\displaystyle z_1$ , $\displaystyle z_2$ and $\displaystyle z_1+z_2$ , the sides from the origin have lengths $\displaystyle |z_1|$ and $\displaystyle |z_2|$ , and the diagonal has length $\displaystyle |z_1+z_2|$ . In a triangle a side is shorter than the sum of the other two, so:

\displaystyle |z_{1}+z_{2}|\le|z_{1}|+|z_{2}|

Equality holds only when $\displaystyle z_1$ and $\displaystyle z_2$ point the same way from the origin (or one of them is 0).

Equations with

\displaystyle \bar{z}

and

\displaystyle |z|

In an equation that holds $\displaystyle \bar{z}$ or $\displaystyle |z|$ , substitute $\displaystyle {z}={a+bi}$ and compare the real parts and the imaginary parts. The example below solves $\displaystyle {|z|+z}={2+8i}$ :

$\displaystyle {\sqrt{a^2+b^2}+a+bi}={2+8i}$ .
The imaginary parts: $\displaystyle {b}={8}$ .
The real parts: $\displaystyle {\sqrt{a^2+64}}={2-a}$ . Squaring: $\displaystyle {a^2+64}={4-4a+a^2}$ , so $\displaystyle {a}={-15}$ .
Check: $\displaystyle {2-a}={17\geq0}$ , and $\displaystyle {\sqrt{225+64}}={17}$ . The solution: $\displaystyle {z}={-15+8i}$ .

Exercise 1

Calculate.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

In the Gauss plane $\displaystyle {A}={2+i}$ , $\displaystyle {B}={5+5i}$ and $\displaystyle {C}={-2+4i}$ .
Calculate the lengths of the sides of the triangle $\displaystyle ABC$ .

$\displaystyle {AB}={|B-A|}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle {AC}={|C-A|}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

$\displaystyle {BC}={|C-B|}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What kind of triangle is $\displaystyle ABC$ ? Give reasons.

Exercise 3

Solve. Substitute $\displaystyle {z}={a+bi}$ , and compare the real parts and the imaginary parts.

Exercise 4

Decide whether the statement is always true, and explain – by algebra or in the Gauss plane.

$\displaystyle {\overline{z_1-z_2}}={\bar{z}_1-\bar{z}_2}$

$\displaystyle {|z_1+z_2|}={|z_1|+|z_2|}$

$\displaystyle {z\cdot\bar{z}}={z^2}$

Formulas: complex numbers, grade 12 (5 units)

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The algebraic form and the four operations

The number $\displaystyle i$
$\displaystyle i^{2}=-1$
The algebraic form
$\displaystyle z=a+bi$ $\displaystyle a=\operatorname{Re}(z),\ b=\operatorname{Im}(z)$
Equality of complex numbers
$\displaystyle a+bi=c+di\iff a=c,\ b=d$
Multiplying
$\displaystyle (a+bi)(c+di)$ $\displaystyle =$ $\displaystyle (ac-bd)+(ad+bc)i$
The conjugate
$\displaystyle \bar{z}=a-bi$
A number times its conjugate
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle a^{2}+b^{2}$
Dividing – times the denominator’s conjugate
$\displaystyle \frac{z_{1}}{z_{2}}$ $\displaystyle =$ $\displaystyle \frac{z_{1}\cdot\bar{z}_{2}}{z_{2}\cdot\bar{z}_{2}}$

The modulus and the conjugate

The modulus – the distance from the origin
$\displaystyle |z|$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}+b^{2}}$
The distance between two points
$\displaystyle |z_{1}-z_{2}|$ $\displaystyle =$ $\displaystyle \sqrt{(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}}$
The product with the conjugate and the modulus
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle |z|^{2}$
The modulus of a product
$\displaystyle |z_{1}\cdot z_{2}|$ $\displaystyle =$ $\displaystyle |z_{1}|\cdot|z_{2}|$
The conjugate of a sum
$\displaystyle \overline{z_{1}+z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}+\bar{z}_{2}$
The conjugate of a product
$\displaystyle \overline{z_{1}\cdot z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}\cdot\bar{z}_{2}$
A number plus its conjugate
$\displaystyle z+\bar{z}$ $\displaystyle =$ $\displaystyle 2\operatorname{Re}(z)$
The triangle inequality
$\displaystyle |z_{1}+z_{2}|\le|z_{1}|+|z_{2}|$

The quadratic equation

The quadratic formula
$\displaystyle z_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
The sum of the roots
$\displaystyle z_{1}+z_{2}$ $\displaystyle =$ $\displaystyle -\frac{b}{a}$
The product of the roots
$\displaystyle z_{1}\cdot z_{2}$ $\displaystyle =$ $\displaystyle \frac{c}{a}$

The polar form (angles in radians)

The polar form
$\displaystyle z=r(\cos\theta+i\sin\theta)$ $\displaystyle r=|z|$
The notation cis
$\displaystyle r\operatorname{cis}\theta$ $\displaystyle =$ $\displaystyle r(\cos\theta+i\sin\theta)$
The real part
$\displaystyle a=r\cos\theta$
The imaginary part
$\displaystyle b=r\sin\theta$
The angle – by the quadrant of the point
$\displaystyle \tan\theta=\frac{b}{a}$ $\displaystyle a\neq0$
A product in polar form
$\displaystyle z_{1}\cdot z_{2}=r_{1}\cdot r_{2}\left[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})\right]$
A quotient in polar form
$\displaystyle \frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\operatorname{cis}(\theta_{1}-\theta_{2})$
De Moivre’s theorem
$\displaystyle \left[R(\cos\varphi+i\sin\varphi)\right]^{n}$ $\displaystyle =$ $\displaystyle R^{n}(\cos n\varphi+i\sin n\varphi)$
The solutions of $\displaystyle {z^n}={R(\cos\varphi+i\sin\varphi)}$
$\displaystyle z_{k}=\sqrt[n]{R}\left[\cos\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)\right]$ $\displaystyle z^{n}=R(\cos\varphi+i\sin\varphi),\ k=0,1,\ldots,n-1$

Loci and rotation

The circle with centre $\displaystyle z_0$ and radius $\displaystyle R$
$\displaystyle |z-z_{0}|=R$
The perpendicular bisector of the segment $\displaystyle z_1z_2$
$\displaystyle |z-z_{1}|=|z-z_{2}|$
Apollonius’ circle
$\displaystyle |z-z_{1}|=m|z-z_{2}|$ $\displaystyle m>0,\ m\neq1$
Rotating by $\displaystyle \theta$ about $\displaystyle z_0$
$\displaystyle w=z_{0}+(z-z_{0})\operatorname{cis}\theta$