4 - The absolute value of a complex number

Let's imagine complex numbers as vectors, because they effectively are so.
The absolute value refers to a point's distance from the origin.

In two dimensions, the x and y axis will form the sides of a right triangle, while the vector itself is the hypothenuse.

Suppose that you have a complex number , and you want to find its absolute value.
This absolute value, the length of this vector, otherwise, the distance of the end point from the origin, will be the hypothenuse of this right triangle.

This is why you learned Pythagora's Theorem.

The real and imaginary parts are the sides of this triangle. Let c be the absolute value, the length of the vector, the length of the hypothenuse of this right triangle.

By Pythagora's Theorem, you know that the hypothenuse c is related to the sides a and b of a right triangle by:

So, how do we find the absolute value?

Simple. Solve. All it takes is one move.

Proper Definition

Let z be a complex number in the form , its absolute value, also called its modulus, the length of the vector representing this complex number, is defined: