New Zealand
Level 8 - NCEA Level 3

Graphing Complex Numbers - Rectangular Form

Lesson

The Argand Diagram is what we call the plane that will allow us to plot complex numbers.  It is named after the Swiss mathematician Jean Argand (1768 - 1822). Using the $x$x-axis as the real axis, and the $y$y-axis as the imaginary axis, the ordered pairs $\left(a,b\right)$(a,b) reflect complex numbers of the form $a+bi$a+bi.

Plotting points on the plane is as simple as identifying the real and imaginary components from a complex number.

 $z_1=2+3i$z1​=2+3i $\left(a,b\right)=\left(2,3\right)$(a,b)=(2,3) $z_2=12-5i$z2​=12−5i $\left(a,b\right)=\left(12,-5\right)$(a,b)=(12,−5) $z_3=-4-4i$z3​=−4−4i $\left(a,b\right)=\left(-4,-4\right)$(a,b)=(−4,−4) $z_4=7$z4​=7 $\left(a,b\right)=\left(7,0\right)$(a,b)=(7,0) $z_5=-6i$z5​=−6i $\left(a,b\right)=\left(0,-6\right)$(a,b)=(0,−6)

Plotting these points on the Argand diagram would result in the following graph.

There is another way we can display complex number on the plane, and that is as a vector.  The above complex numbers and all be represented as vectors on the plane with initial position $\left(0,0\right)$(0,0) and terminal position at the point $\left(a,b\right)$(a,b) as designated by the values of $a$a and $b$b in the number $a+bi$a+bi.

Activity

Try this for yourself before checking out the solution.

If $z=1-i$z=1i, find $z,z^2,z^3,z^4,z^5,z^6$z,z2,z3,z4,z5,z6

Plot these points on an Argand Diagram.

Is there a geometric pattern?

(see here for the solution)

More Worked Examples

QUESTION 1

Plot $6+2i$6+2i on the Argand diagram (complex plane).

QUESTION 2

Consider the following.

1. Graph the number $-1+2i$1+2i.

2. Evaluate $\left(-1+2i\right)+\left(9+5i\right)$(1+2i)+(9+5i).

3. Graph the result of $\left(-1+2i\right)+\left(9+5i\right)$(1+2i)+(9+5i).

QUESTION 3

Graph the complex number $-6$6 as a vector.

QUESTION 4

What is the complex number represented on the graph?

State the number in rectangular form.

Outcomes

M8-9

Manipulate complex numbers and present them graphically

91577

Apply the algebra of complex numbers in solving problems