Definition:Remainder

Definition

Let $a, b \in \Z$ be integers such that $b \ne 0$.

From the Division Theorem, we have that:

$\forall a, b \in \Z, b \ne 0: \exists_1 q, r \in \Z: a = q b + r, 0 \le r < \size b$

The value $r$ is defined as the remainder of $a$ on division by $b$, or the remainder of $\dfrac a b$.

Real Arguments

When $x, y \in \R$ the remainder is still defined:

The remainder of $x$ on division by $y$ is defined as the value of $r$ in the expression:

$\forall x, y \in \R, y \ne 0: \exists! q \in \Z, r \in \R: x = q y + r, 0 \le r < \size y$

From the definition of the Modulo Operation:

$x \bmod y := x - y \floor {\dfrac x y}$

it can be seen that the remainder of $x$ on division by $y$ is defined as:

$r = x \bmod y$

Also known as

A remainder as defined here is also known as a principal remainder.

Also see

• Definition:Integer Division
• Definition:Quotient (Integer Division)

• Definition:Remainder of Division over Euclidean Domain

• Results about remainders can be found here.

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 11.1$: The division algorithm
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): division
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): remainder: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): division
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): remainder: 1.