How to find the distance and midpoint between complex numbers

Distance and midpoint work the same way in the complex plane as they do for ordinary coordinate points. The real parts provide the horizontal coordinates, and the imaginary parts provide the vertical coordinates. We’ll use those two directions to measure the distance and locate the halfway point.

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Find the distance from the coordinate differences

For $z_1=a+bi$ and $z_2=c+di$, the horizontal change is $c-a$ and the vertical change is $d-b$. The Pythagorean theorem gives

$D=\sqrt{(c-a{)}^2+(d-b{)}^2}$

Measure the distance between minus one minus three i and three plus two i

Use $z_1=-1-3i$ and $z_2=3+2i$. The changes are $3-(-1)=4$ and $2-(-3)=5$.

$\begin{array}{l}D=\sqrt{4^2+5^2}\\ =\sqrt{41}\end{array}$

The connecting segment is the hypotenuse of a right triangle with horizontal leg 4 and vertical leg 5.

The blue endpoints are −1 − 3i and 3 + 2i. The purple point is their midpoint, 1 − 0.5i. The legs show the horizontal change of 4 and vertical change of 5.

Average the coordinates to find the midpoint

$z_M=\frac{a+c}{2}+\frac{b+d}{2}i$

$\begin{array}{l}{z}_M=\frac{-1+3}{2}+\frac{-3+2}{2}i\\ =1-\frac{1}{2}i\end{array}$

The midpoint has real part $1$ and imaginary part $-1/2$. It lies halfway along the segment joining the two numbers. Equivalently, we can add the complex numbers and divide their sum by 2.

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