How to Find the Distance Between Two Complex Numbers

You can calculate the distance between two complex numbers in the same manner as you would when working with vectors. Given the two complex numbers $z_{1}$ and $z_{2}$ , you can calculate the distance between them as the norm $r = | z_{1} - z_{2} |$ of the difference $z_{3} = z_{1} - z_{2}$ . Since you’re only interested in the distance between $z_{1}$ and $z_{2}$ , it makes no difference whether you choose to calculate the norm of $z_{3} = z_{1} - z_{2}$ or $z_{4} = z_{2} - z_{1}$ .

Example 1

Find the distance between $z_{1} = 5 + 2 i$ and $z_{2} = 1 - i$

In order to calculate the distance between the numbers, you first have to calculate $z_{3} = z_{1} - z_{2}$ . This is done element-wise by looking at the real and the imaginary parts separately:

\begin{array}{l} z_{3} & = (5 + 2 i) - (1 - i) \\ = (5 - 1) + (2 - (- 1)) i \\ = 4 + 3 i . \end{array}

z_{3} = (5 + 2 i) - (1 - i) = (5 - 1) + (2 - (- 1)) i = 4 + 3 i .

You can now calculate the distance between

z_{1}

and

z_{2}

as the norm of

z_{3}

\begin{array}{l} r_{1} & = | z_{3} | \\ = \sqrt{4^{2} + 3^{2}} \\ = \sqrt{16 + 9} \\ = \sqrt{25} \\ = 5 . \end{array}

r_{1} = | z_{3} | = \sqrt{4^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5 .

Thus, the distance between

z_{1}

and

z_{2}

equals

5

. If you want, you may also calculate the difference

z_{4} = z_{2} - z_{1}

\begin{array}{l} z_{4} & = (1 - i) - (5 + 2 i) \\ = (1 - 5) + (- 1 - 2) i \\ = - 4 - 3 i . \end{array}

z_{4} = (1 - i) - (5 + 2 i) = (1 - 5) + (- 1 - 2) i = - 4 - 3 i .

Even though

z_{3}

and

z_{4}

are not the same numbers, they do have the same norm:

\begin{array}{l} r_{2} & = | z_{4} | \\ = \sqrt{{(- 4)}^{2} + {(- 3)}^{2}} \\ = \sqrt{16 + 9} \\ = \sqrt{25} \\ = 5 . \end{array}

r_{2} = | z_{4} | = \sqrt{{(- 4)}^{2} + {(- 3)}^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5 .

In other words, you may freely choose which difference to use when finding the distance between two complex numbers.

By using the complex plane, you can visualize the distance between two complex numbers geometrically. You can draw the numbers used in Example 1 in the complex plane and use the Pythagorean theorem to calculate the distance between them:

The distance between two complex numbers in the complex plane.

The distance between two complex numbers can be used to sketch areas in the complex plane:

Example 2

Sketch the set ${z \in ℂ : | z + 1 | \leq 1}$ of complex numbers $z$ such that $| z + 1 | \leq 1$

In this exercise, your task is to sketch all complex numbers $z$ whose distance to $- 1$ is less than or equal to $1$ . In the complex plane this is a closed disk centered at $- 1$ with a radius of $1$ :

Sketched area of the complex plane.

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