How to graph complex numbers in the complex plane

A complex number has two parts, so we can show it as a point with two coordinates. The real part tells us how far to move horizontally, and the imaginary part tells us how far to move vertically. Let’s use those directions to plot numbers and their sums.

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Read the two axes

The complex number $a+bi$ corresponds to the point $(a,b)$. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. For $3-2i$, move three units right and two units down.

A pure imaginary number such as $2i$ lies on the vertical axis. A real number such as $-1$ lies on the horizontal axis.

Plot three complex numbers

Complex numbers and their coordinates

Number	Point
$3+2i$	$(3,2)$
$-1-3i$	$(-1,-3)$
$1.5i$	$(0,1.5)$

Blue: 3 + 2i. Purple: −1 − 3i. Coral: 1.5i. Each point uses its real part horizontally and its imaginary part vertically.

Find a sum before plotting it

$\begin{array}{l}(-4+3i)+(2-6i)=(-4+2)+(3-6)i\\ =-2-3i\end{array}$

The sum is the point $(-2,-3)$, two units left and three units down.

Find a difference before plotting it

$\begin{array}{l}1-(5-3i)=(1-5)+(0+3)i\\ =-4+3i\end{array}$

The difference is $(-4,3)$, four units left and three units up. If the original numbers are shown only on a graph, read their two coordinates first, then combine the real and imaginary parts in the same way.

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