Britannica Sources used in this answer

How can the distance between complex numbers be calculated in the complex plane?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

In the complex plane, the distance between two complex numbers is calculated using Pythagoras’s theorem from elementary geometry.1

To find this distance, you construct a right triangle where the hypotenuse spans the distance between the two points and the sides are parallel to the coordinate axes. By applying the theorem, the distance between two complex numbers is equivalent to the absolute value (or modulus) of their difference.21

For a complex number $z = a + b i$ , the absolute value is defined as:
$| z | = \sqrt{a^{2} + b^{2}}$
This represents the distance of the number from the origin of the complex plane. When calculating the distance between two distinct complex numbers $z_{1}$ and $z_{2}$ , you find the absolute value of the difference between them, $[2] | z_{1} - z_{2} |$ , which effectively measures the magnitude of the displacement between those two points.21