12.03 Graphing complex numbers
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.
Try this for yourself before checking out the solution.
If $z=1-i$z=1−i, 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?
Can you generalize your result?
(see here for the solution)
More Worked Examples
QUESTION 1
Plot $6+2i$6+2i on the Argand diagram (complex plane).
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QUESTION 2
Consider the following.
Graph the number $-1+2i$−1+2i.
Loading Graph...Evaluate $\left(-1+2i\right)+\left(9+5i\right)$(−1+2i)+(9+5i).
Graph the result of $\left(-1+2i\right)+\left(9+5i\right)$(−1+2i)+(9+5i).
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QUESTION 3
Graph the complex number $-6$−6 as a vector.
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QUESTION 4
What is the complex number represented on the graph?
State the number in rectangular form.