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 4 Comments below.

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4 Comments on “Modulus or absolute value of a complex number?”

  1. Alexander Bogomolny says:

    The terminology has to do with the expansion of functions from the real domain to the complex. x², x∈R, and z², z∈C, are designated as “x squared” and “z squared” – so there is no difference in the nomenclature. For |x| and |z|, both expression could be called “absolute value” or “modulus”, regardless of the domain of definition. However, for the real x, |x| is seldom referred to as “modulus”, but this is only a matter of convention. For the complex |z|, both terms are being used, albeit “modulus” is more common.

    Classical “The advanced geometry of plane curves and their applications” by C. Zwikker (p. 12) points to |z| as the absolute value or the modulus of the vector! In D. Pedoe’s “Geometry” it’s “modulus”; “absolute value” is not mentioned at all. David Darling in “The universal book of mathematics” defines |z| as “absolute value” or “magnitude”.

  2. Philip Petrov says:

    I think your comment on the case is correct Murray. Many people never think of that “the absolute value” is actually “the distance to the origin”…

  3. Rene says:

    I hate to say “modulus”. It seems to be the English term for |z|. But it sounds outdated, old. Would you ever use mod(z) for this? Certainly not.

    Be aware, that the term “absolute value” is also not sensible, since it translates to the value with the sign removed (latin absolute). But it the clearer term. Then, some people define the “sign” of a complex number as z/|z|, which makes perfect sense together with the “absolute” value.

    Math, indeed, is not about words, but about useful concepts.

  4. Rina Sengupta says:

    I agree with Rene that math is all about useful concepts but the proper useful ‘words’ are also necessary for correct communications.

    Modulus of a complex number is mostly used…

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