Number System - Multiplication and Division of Integers

Rule of Signs for Multiplication: When multiplying two integers, if the signs are the same (both positive or both negative), the product is positive. If the signs are different, the product is negative. This can be visualized as a $2 \times 2$ grid where $(+) \times (+) = (+)$, $(-) \times (-) = (+)$, $(+) \times (-) = (-)$, and $(-) \times (+) = (-)$.

Commutative and Associative Properties: Multiplication of integers is commutative, meaning $a \times b = b \times a$. It is also associative, meaning the grouping of numbers does not change the product: $(a \times b) \times c = a \times (b \times c)$. On a number line, this signifies that the total distance reached by multiple jumps remains the same regardless of the order of operations.

Distributive Property: This property states that multiplication distributes over addition and subtraction: $a \times (b + c) = (a \times b) + (a \times c)$. Visually, this is like calculating the area of a large rectangle by splitting it into two smaller rectangles and adding their areas together.

Multiplicative Identity and Null Property: The integer $1$ is the multiplicative identity because $a \times 1 = a$. Any integer multiplied by $0$ results in $0$. On a coordinate plane or number line, multiplying by zero effectively collapses any value back to the origin.

Rules for Division: Division follows the same sign rules as multiplication. Dividing two integers with like signs results in a positive quotient, while dividing two integers with unlike signs results in a negative quotient. For example, $(-a) \div (-b) = \frac{a}{b}$ and $(-a) \div b = -\frac{a}{b}$.

Properties of Division: Division is neither commutative nor associative ($a \div b \neq b \div a$). Any integer $a$ divided by $1$ gives the integer $a$. Any non-zero integer $a$ divided by itself gives $1$.

Division by Zero: Dividing any integer by zero is undefined ($a \div 0$ is not possible). However, zero divided by any non-zero integer is always zero ($0 \div a = 0$). Visually, you cannot share a quantity into zero groups, but you can share 'nothing' among any number of people.