# What is the difference between Modulus, Absolute value and Modulo?

When referring to the absolute value I have said Modulus. People then correct me and say that that is the operation where you find the remainder. Then I get confused because I've seen people say modulo for the remainder operation. Also if my first statement is correct why do we have two words for absolute value?

• The answer to "what is the difference" is "there isn't even a single similarity." Modulus is a term used for absolute value in complex analysis, and also a term used for the thing-being-divided-by in remainder arithmetic (actually called modular arithmetic). This latter usage extends far beyond in abstract algebra - when we speak of something modulo $I$, or speak of "modding out" by things, we are talking about quotients (quotient groups or quotient rings in particular). – anon Aug 21 '13 at 15:09
• @anon -- Provide as an answer. – Martin F Mar 8 '15 at 17:15

It is an unfortunate fact that mathematical terminology has developed in a haphazard way, and often the same word will be given completely different meanings in different areas of mathematics (sometimes even in the same area, which is worse!). The most extreme example, I think, is "normal".

See http://jeff560.tripod.com/m.html for some history of the many uses of "modulus" and its variants.

• Not really an answer to the question. – Martin F Mar 8 '15 at 17:16

They mean differently.

Absolute value of $x = |x|$ and is equal to $x$ if $x \geq 0$ or is equal to $-x$ if $x < 0$.

Modulo, usually refers to the type of arithmetic called modulo arithmetic.

For example, because $13 = 4\times 3 + 1$, we write $13\ \equiv\ 1\ (\textrm{mod}\ 3)$ ($13$ is congruent to $1$ modulo $3$).

Modulus refers to the magnitude/length of a vector.

How about “An introduction to the theory of Numbers – by Niven Zuckerman” and “Pure Mathematics I & II by F. Gerrish”?

Those names have been commonly used by others and sometimes even interchangeably but, in the books mentioned above, they are clearly and distinctly defined.

The only confusion comes from the “$|…|$” sign, which has been used both for the absolute value of a number and also as the modulus of a vector. Therefore, some used the “$|| … ||$” for the latter to make the meaning distinct. Some don’t even bother when the context is clear or when the readers should be able to distinguish their difference.

• could you cite your source? Most people are saying they are essentially equivalent, but if your distinction is accurate then that would be very useful to know. So a source citation would be extremely helpful. – Hack-R Oct 17 '15 at 17:52
• @Hack-R Extra info has been added. See above. – Mick Oct 18 '15 at 3:08

If $z = a + b \imath$ then $|z| = \sqrt{a^2 + b^2}$ is called its modulus. If you work with both real and complex numbers frequently it is common to misspeak and call the absolute value $|x|$ of a real number $x$ its modulus.

• Misspeak? ${}{}$ – anon Aug 21 '13 at 15:06

As far as I know, I've never seen Absolute Value referred to as modulus, though wikipedia does acknowledge this use. Modulus and modulo both refer to the remainder operation. Generally speaking, "$5 \mod 7$" is read "$5$ modulo $7$," where the modulus is $7$. It's ultimately a matter of which you prefer to use.

• Here is an example of modulus and absolute being used together. – Ali Caglayan Aug 21 '13 at 15:05
• When I was in grad school (but that was in an earlier geological epoch), “modulus” certainly was used for absolute value. But then we didn’t have an operation called “modulo”, all we had was an equivalence relation, of congruence modulo a number, which could occasionally be called the modulus. – Lubin Aug 21 '13 at 16:14

I think there is an issue with naming, but as another poster mentioned it is "a modulo b" where 'b' is the modulus. I was taught the function f(x)=|x| is 'modulus' in mathematics, but whenever we needed to use it in programming or Excel we would use the Absolute (ABS) function or equivalent. The modulo function (a mod b) translated directly into programming etc. I found the naming a bit confusing too, but their usage is distinct enough that they shouldn't get confused. In programming, it's obvious you are going wrong if using 'mod' to change to a positive value, as the function would accept two arguments.