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?
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$\begingroup$They mean differently.
$\color{green}{\Large\bullet}$ Absolute value of $x = |x|$ and is equal to $x$ if $x \geq 0$ or is equal to $-x$ if $x < 0$.
$\color{green}{\Large\bullet}$ Modulo, usually refers to the type of arithmetic called modulo arithmetic. For example, because $13 = 4\times 3 + 1$, we write $13\ \equiv\ 1\ (\textrm{mod}\ 3)$. In common mathematical language, it is taken as "$13$ is congruent to $1$ modulo $3$".
$\color{green}{\Large\bullet}$ Modulus refers to the magnitude/length of a vector.
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How about “An introduction to the theory of Numbers – by Niven Zuckerman” and “Pure Mathematics I & II by F. Gerrish”?
Those names in question 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.
$\endgroup$ 2 $\begingroup$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 for some history of the many uses of "modulus" and its variants.
$\endgroup$ 1 $\begingroup$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.
$\endgroup$ 2 $\begingroup$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.
$\endgroup$ 2 $\begingroup$We should keep the distinction between modulus for absolute value and modulo for modular arithmetic. Most of the literature do not keep this distinction.
$\endgroup$ $\begingroup$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.
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