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Modulus Function

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The term modulus is one of those unfortunate terms in mathematics that can mean a few different things, depending on where you’re using it and who the author is. The term modulus function can refer to either the absolute value function or the modulo function.

In calculus, you’ll usually be dealing with the modulus function, which is commonly used as a synonym for the absolute value function in general math (Hennings, 2017; Sherran, 1999). In complex analysis, it means something similar, but not quite the same: the modulus of z is defined as (Britannica, 2020)

“…the real number Square root of √a2 + b2, which corresponds to z’s distance from the origin of the complex plane.”

maximum modulus function

The maximum modulus function from complex analysis (Silverman, 1984):

There are a few more areas of confusion. Most notably that some authors use modulus function as a synonym for the modulo function (which is completely different from the absolute value function). The word modulus, although it’s commonly used to mean the same thing as absolute value, also crops up in modular arithmetic; It refers to the remainder operation returning the remainder of m divided by n. “Modulo” and “modulus” are often used interchangeably, with some authors (e.g. Dannenberg, 2019) calling the remainder operation the modulus function.

Another area that causes confusion: the absolute value bars “| |” are also used to indicate the modulus (or magnitude) of a vector. This is similar to the absolute value, but again, not quite the same.

Despite all of the different definitions, you can usually figure out the author’s intent by considering the big picture of they are writing about.

Why All The Different Definitions for Modulus Function?

There are a staggering number of definitions of the word “modulus”. These developed over time, some completely independent of each other (when people had to communicate by letter or telegraph, if they were able to at all!). A few more (from Jeff Miller’s Earliest Known Uses of Some of the Words of Mathematics):

  • Gauss first wrote about the term “modulo” in Disquisitiones arithmeticae (1801, p. 9)

    “If a number a measure the difference between two numbers b and c, b and c are said to be congruent with respect to a, if not, incongruent; a is called the modulus, and each of the numbers b and c the residue of the other in the first case, the non-residue in the latter case” (Miller, 2020)

  • The Oxford English dictionary defines it as

    “The positive square root of the sum of the squares of the real and imaginary parts of a complex number.”

  • The term modulus means “length of the vector a + bi”, due to Jean Robert Argand (1768-1822) (Cajori 1919, page 265).


Britannica. (2020). Absolute Value.
Cajori, Florian. A History of Mathematics. New York: The Macmillan Co., 1919.
Dannenberg (2019). Modulus (%) Illustrated. Retrieved November 10, 2020 from:
Gauss, F. (2009 Translation). Disquisitiones arithmeticae. Yale University Press, New Haven & London.
Hennings, M. (2017). Cambridge Pre-U Mathematics Coursebook. Cambridge University Press.
Miller, J. Earliest Known Uses of Some of the Words of Mathematics (M).
Oxford English Dictionary. Modulus. Retrieved November 10, 2020 from:
Sherran, P. (2000). Graphical Calculator Support Pack (Complete Advanced Level Mathematics). Nelson Thornes.
Silverman, R. (1984). Introductory Complex Analysis. Dover.

Stephanie Glen. "Modulus Function" From Calculus for the rest of us!

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