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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 + j sin θ), r is the absolute value (or modulus) 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

Wikipedia's Absolute Value

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.

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