# Calculating the absolute value of a complex number - am I right?

To calculate the absolute value of a complex number u must use the following formular $(a^2+b^2)^½$=|a+bi|

So for instance with -4-5i would have the absolute value $((-4)^2+(-5)^2)^½$=$(16+25)^½$=$\sqrt(41)$

But how come it is so?

## 2 Answers

In short, that's how it's defined. Geometrically, it might be more intuitive if you identity each complex number $a + bi$ with a vector of the form $(a,\ b)$. The length of the vector is then given by the familiar Pythagorean theorem from which the norm of the complex number is derived.

Much of the intuition regarding complex numbers comes from their interpretation as essentially $2$-dimensional analogues of real numbers. In particular, this definition is especially natural when taking advantage of the "built in" polar coordinate representation.

• Thanks, it makes alot more sense with that describition than just using a "random" formular :) – Alek Oliver Oct 31 '12 at 7:09

Think of the absolute value as the distance from zero. If you represent your number $z=a+bi$ in the complex plane what is the distance of $z$ from zero? Use Pythagorean theorem.