# Modulus or absolute value of a complex number?

By Murray Bourne, 15 Mar 2012

On my Polar Form of a Complex Number page in IntMath, I state:

For the complex number

x + yj=r(cos θ +jsin θ),ris theabsolute value(ormodulus) of the complex number.

Reader Sunshine from the Philippines challenged this statement by saying:

absolute value doesn't have the same definition as modulus

I enjoy such feedback because it makes me think more deeply about how I have written the definitions (or perhaps notation) on the site. We need to be clear, precise, and accurate, while making it understandable.

## So, is Sunshine correct?

This was my response to her.

In the case of complex numbers, the terms are generally interchangeable, but I agree this could be sloppy.

These other math resources also use both terms for the same thing:

Cut-The-Knot's Complex Numbers

Mathwords' absolute value of a Complex Number

However, Pauls Online Notes (Lamar University) makes some distinction between modulus of a complex number and absolute value of a real number (the latter is a degenerative case of the former).

But in the sense that "absolute value" means distance from the origin for a real number (on the one-dimensional number line), and "modulus" means distance from the origin for a complex number (on the 2-dimensional complex plane), I don't believe there is a big problem with the interchangeability of the terms. The concept is certainly the same and it doesn't lead to a great deal of confusion.

I would probably not write "the absolute value of a complex number" - it's certainly less common, and prefer "the modulus of a complex number".

Thanks for triggering me to think about it!

Readers, what are your thoughts on this? What does your text-book, or lecture notes say?

See the 6 Comments below.