## Absolute Value of Complex Numbers

Geometrically, the absolute value of a complex number is the number’s distance from the origin in the complex plane.

In the diagram at the left, the complex number 8 + 6i is plotted in the complex plane on an Argand diagram (where the vertical axis is the imaginary axis). For this problem, the distance from the point 8 + 6i to the origin is 10 units. Distance is a positive measure.

Notice the Pythagorean Theorem at work in this problem.

A complex number can be represented by a point, or by a vector from the origin to the point. When thinking of a complex number as a vector, the absolute value of the complex number is simply the length of the vector, called the magnitude.

In the Pythagorean Theorem, c is the hypotenuse and when represented in the coordinate plane, is always positive. This same idea holds true for the distance from the origin in the complex plane. Using the absolute value in the formula will always yield a positive result.

To find the absolute value of a complex number a + bi:

1. Be sure your number is expressed in a + bi form

2. Pick out the coefficients for a and b

3. Substitute into the formula

**Example** 1:

Plot z = 8 + 6i on the complex plane, connect the graph of z to the origin (see graph below), then find | z | by appropriate use of the definition of the absolute value of a complex number.

**Example 2**:

Find the | z | by appropriate use of the Pythagorean Theorem when z = 2 – 3i.

**Example 3**:

If z = – 8 – 15i, find | z |.

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