Absolute value of a complex number

The absolute value of a complex number a + bi, also called the modulus of a complex number a + bi, is its distance from the origin on the complex number plane.

On the complex plane, the complex number a + bi is represented as a point (a, b) and the coordinate of the origin is (0, 0).

Absolute value of a complex number

Just find the distance between (0,0) and (a,b)

We can use the distance formula to find the absolute value of a complex number.

$$ Distance = \sqrt{(a-0)^{2}+(b-0)^{2}}$$
$$ Distance = \sqrt{(a)^{2}+(b)^{2}}$$
$$ |a + bi| = \sqrt{(a)^{2}+(b)^{2}}$$

Two examples showing how to find the absolute value of a complex number.

Find the absolute value of the complex numbers 4 - 3i and 6i.

Notice that 6i = 0 + 6i.

$$ |4 - 3i| = \sqrt{(4)^{2}+(-3)^{2}}$$
$$ |4 - 3i| = \sqrt{16 + 9} = 5 $$
$$ |6i| = \sqrt{(0)^{2}+(6)^{2}}$$
$$ |6i| = \sqrt{0 + 36} = 6 $$

The figure below shows the distance for the complex number 4 - 3i in red and the distance for the complex number 6i in blue.

Absolute value of complex numbers

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