Absolute Value of a Complex Number

High School

Definition

Also known as the modulus of a complex number in the form of a+bi. It tells you the distance between the origin and a point (a,b) that's on a complex plane. It is a measure of distance from 0 measured on the complex number plane, rather than using a number line as per usual absolute values.

Worked examples

\(|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

Square the real part and imaginary part, add them, then take the square root.

\(|{-6 + 8i}| = \sqrt{({-6})^2 + 8^2} = \sqrt{36 + 64} = 10\)

Negative real parts are squared too, making them positive inside the radical.

\(|5i| = \sqrt{0^2 + 5^2} = 5\)

For a purely imaginary number, the real part is zero.

Common mistakes

\(|3 + 4i| = 3 + 4 = 7\) → \(|3 + 4i| = \sqrt{3^2 + 4^2} = 5\) You must use the distance formula, not simple addition of the parts.
\(|{-6 + 8i}| = \sqrt{({-6})^2 + 8^2} = \sqrt{-36 + 64}\) → \(\sqrt{({-6})^2 + 8^2} = \sqrt{36 + 64}\) Squaring a negative number gives a positive result.
\(|a + bi| = |a| + |b|i\) → \(|a + bi| = \sqrt{a^2 + b^2}\) The modulus is a single real number, not a complex expression.

Where you'll use it next

You'll use the absolute value of complex numbers when multiplying and dividing in polar form, solving polynomial equations with complex roots, and analyzing magnitude in engineering applications like electrical circuits and signal processing.

Found in 1 StudyPug lesson

Angle and Absolute Value of a Complex Number

Grade 12 Math

Discover how to plot a complex number, measure its distance from the origin (modulus), and find the angle it makes with the real axis (argument).

12th Grade12th