The Gauss plane

Complex numbers, grade 12 (5 units) Lesson 3 of 18Learning

The Gauss plane

A number as a point

The complex number $\displaystyle {z}={a+bi}$ has two real parts, so it can be marked as the point $\displaystyle (a,b)$ in a coordinate system. This coordinate system is called the Gauss plane:

the $\displaystyle x$ -axis is the real axis: the real numbers are on it;
the $\displaystyle y$ -axis is the imaginary axis: the purely imaginary numbers are on it;
every complex number is one point, and every point is one complex number.

The coordinate system below shows the points of $\displaystyle {z_1}={2+3i}$ , $\displaystyle {z_2}={-3+i}$ , $\displaystyle {z_3}={-2i}$ and $\displaystyle {z_4}={4}$ .

\displaystyle z_1=(2,3)

\displaystyle z_2=(-3,1)

\displaystyle z_3=(0,-2)

\displaystyle z_4=(4,0)

Adding and subtracting as vectors

You can also see $\displaystyle {z}={a+bi}$ as a vector – an arrow from the origin to the point $\displaystyle (a,b)$ . Adding complex numbers is adding their vectors, component by component:

$\displaystyle z_1+z_2$ is the fourth vertex of the parallelogram whose vertices are $\displaystyle 0$ , $\displaystyle z_1$ and $\displaystyle z_2$ : the diagonal that starts at the origin.
$\displaystyle z_1-z_2$ is the vector from $\displaystyle z_2$ to $\displaystyle z_1$ , moved to the origin: $\displaystyle {z_2+(z_1-z_2)}={z_1}$ .

The coordinate system below shows $\displaystyle {z_1}={1+2i}$ , $\displaystyle {z_2}={3-i}$ and the sum $\displaystyle {z_1+z_2}={4+i}$ .

\displaystyle z_1=(1,2)

\displaystyle z_2=(3,-1)

\displaystyle z_1+z_2=(4,1)

\displaystyle O=(0,0)

The conjugate and the opposite

The conjugate and the opposite of $\displaystyle {z}={a+bi}$ are reflections of the point $\displaystyle (a,b)$ :

$\displaystyle {\bar{z}}={a-bi}$ is the point $\displaystyle (a,-b)$ : the reflection of $\displaystyle z$ in the real axis.
$\displaystyle {-z}={-a-bi}$ is the point $\displaystyle (-a,-b)$ : the reflection of $\displaystyle z$ in the origin.
$\displaystyle {-\bar{z}}={-a+bi}$ is the point $\displaystyle (-a,b)$ : the reflection in the imaginary axis.
$\displaystyle kz$ , with $\displaystyle k$ real and positive, is the vector of $\displaystyle z$ stretched $\displaystyle k$ times, in the same direction.

Exercise 1

Given $\displaystyle {z}={3-2i}$ .
Mark the points of $\displaystyle z$ , $\displaystyle \bar{z}$ , $\displaystyle -z$ and $\displaystyle 2z$ in the coordinate system.

Exercise 2

Given $\displaystyle {z_1}={1+2i}$ and $\displaystyle {z_2}={3-i}$ .

Calculate $\displaystyle z_1-z_2$ .

/ – 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

Calculate $\displaystyle 2z_1+z_2$ .

/ – 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

Mark $\displaystyle z_1-z_2$ and $\displaystyle 2z_1+z_2$ in the coordinate system, and check that they fit the construction of adding and subtracting vectors.

\displaystyle z_1=(1,2)

\displaystyle z_2=(3,-1)

Exercise 3

$\displaystyle ABCD$ is a parallelogram. In the Gauss plane $\displaystyle {A}={1+i}$ , $\displaystyle {B}={4+2i}$ and $\displaystyle {C}={5+5i}$ .
Find the complex number of the vertex $\displaystyle D$ .
Hint: $\displaystyle {\overrightarrow{AD}}={\overrightarrow{BC}}$ , so $\displaystyle {D-A}={C-B}$ .

/ – 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

Exercise 4

Given $\displaystyle {z}={a+bi}$ , with $\displaystyle a\neq0$ and $\displaystyle b\neq0$ .
What quadrilateral do the points $\displaystyle z$ , $\displaystyle \bar{z}$ , $\displaystyle -z$ and $\displaystyle -\bar{z}$ form? Give reasons.

Formulas: complex numbers, grade 12 (5 units)

Open as a page

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 $\displaystyle z_0$
$\displaystyle w=z_{0}+(z-z_{0})\operatorname{cis}\theta$