Integers - Division of Integers and its Properties

Division as Inverse Operation: Division is the mathematical inverse of multiplication. For every multiplication statement for non-zero integers, there are two related division statements. For example, if $2 \times (-3) = -6$, then $(-6) \div (-3) = 2$ and $(-6) \div 2 = -3$.

Division of Integers with Like Signs: When a positive integer is divided by a positive integer, or a negative integer is divided by a negative integer, the quotient is always a positive integer. Conceptually, if you think of a number line, dividing a total negative distance into groups of negative steps results in a positive number of groups.

Division of Integers with Different Signs: When a positive integer is divided by a negative integer, or a negative integer is divided by a positive integer, the quotient is always a negative integer. This can be visualized as taking a negative total and dividing it into equal positive parts, which naturally stay on the negative side of the number line.

Division by Zero and One: Any integer $a$ divided by $1$ gives the same integer $a$. However, division of any integer by zero is undefined because it is impossible to divide a quantity into zero parts. Conversely, zero divided by any non-zero integer $a$ is always $0$.

Properties of Division (Closure and Commutative): Unlike addition and multiplication, division of integers does not satisfy the closure property because the quotient of two integers may not always be an integer (e.g., $5 \div 2 = 2.5$). It is also not commutative, meaning the order of numbers matters: $a \div b$ is not equal to $b \div a$.

Associative Property of Division: The associative property does not hold for the division of integers. This means that for any three integers $a$, $b$, and $c$, the grouping of terms changes the result: $(a \div b) \div c \neq a \div (b \div c)$. If you visualize this as nested operations, changing the 'inner' group changes the scale of the final division.