In mathematics, the term "modulus" can refer to different concepts depending on the context in which it is used:

1. Modulus in arithmetic: The modulus refers to the remainder of a division operation. For example, in the expression a mod b, where a and b are integers and b≠0, the result is the remainder when a is divided by b. This operation is often denoted using the '%' symbol in programming languages.

For instance:

17 mod  5=2  because when 17 is divided by 5, the remainder is 2.

−17 mod  5=3 because -17 divided by 5 yields a quotient of -3 with a remainder of 3.

2. Modulus in complex numbers: The modulus of a complex number z =a+bi, where a and b are real numbers and i is the imaginary unit, is a measure of its magnitude or distance from the origin in the complex plane. It's calculated using the formula:

∣z∣=`sqrt(a^2+b^2)`​

For instance, if z=3+4i, then the modulus (or absolute value) of z is ∣z∣=`sqrt(3^2+4^2)​=5`.

3. Modulus in algebra and number theory: The modulus can also represent the size or absolute value of a number, which disregards its sign. For real numbers, the modulus or absolute value of x, denoted as ∣x∣, is defined as:

`|x| = {(x,if,x,ge,0),(x,-,y,=,5):}`

For example, ∣−8∣=8 and ∣5∣=5.

The concept of modulus is versatile and applies across various mathematical disciplines, each with its specific meaning and calculation method.