Key Points

The additive inverse of an integer $a$ is $-a$. The sum of an integer and its additive inverse is always zero. For example, $a + (-a) = 0$.

The product of two integers with the same sign is positive. The product of two integers with different signs is negative. For example, $(-a) \times (-b) = ab$ and $a \times (-b) = -ab$.

The quotient of two integers with the same sign is positive. The quotient of two integers with different signs is negative. For example, $(-a) \div (-b) = a \div b$ and $a \div (-b) = -(a \div b)$.

Subtracting an integer is equivalent to adding its additive inverse. For any two integers $a$ and $b$, the operation $a - b$ is the same as $a + (-b)$.

Addition and multiplication are commutative for integers, which means the order of numbers does not affect the result. For any integers $a$ and $b$, $a + b = b + a$ and $a \times b = b \times a$.

Addition and multiplication are associative for integers, which means the grouping of numbers does not affect the result. For any integers $a, b, c$, we have $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.

Multiplication distributes over addition for integers. For any three integers $a, b,$ and $c$, this property is stated as $a \times (b + c) = (a \times b) + (a \times c)$.

The additive identity for integers is 0, because adding 0 to any integer does not change its value ($a + 0 = a$). The multiplicative identity is 1, because multiplying any integer by 1 does not change its value ($a \times 1 = a$).

The product of any integer and zero is always zero. For any integer $a$, the rule is $a \times 0 = 0$.

Multiplying an integer by $-1$ results in its additive inverse. For any integer $a$, we have $a \times (-1) = -a$.

If the number of negative integers in a product is even, the result is a positive integer. If the number of negative integers is odd, the result is a negative integer.

Zero divided by any non-zero integer is zero ($0 \div a = 0$ for $a \neq 0$). Division by zero is undefined and not possible.