Chapter 10: Integers , Multiplication and Division

You already know how to add and subtract integers , including negative ones. Now we extend the toolkit to multiplication and division.

The biggest surprise: when you multiply or divide two negative integers, the result is positive. "Negative times negative is positive" sounds strange the first time. But once you see why (using patterns, the distributive law, or real-world models), it makes perfect sense.

In this chapter you will master the sign rules, learn the properties of integer multiplication (closure, commutative, associative, distributive), and apply them to expressions and real-life problems.

By the end you will be just as comfortable computing $(-7) \times (-8)$ as $7 \times 8$ .

What's inside

Recap of integers , the number line, addition and subtraction.
Multiplying integers , sign rules and computation.
Properties of multiplication , closure, commutativity, distributive law.
Dividing integers , sign rules, special cases, zero.
Word problems and real-life integers , temperature, banking, scores.

Key results

Rule	Statement
Sign in multiplication	Same signs $\Rightarrow$ positive; different signs $\Rightarrow$ negative
Sign in division	Same as multiplication
Multiplication by zero	$a \times 0 = 0$ for any integer $a$
Multiplication by $-1$	$a \times (-1) = -a$
Identity	$a \times 1 = a$
Commutative	$a \times b = b \times a$
Associative	$(a \times b) \times c = a \times (b \times c)$
Distributive	$a \times (b + c) = a \times b + a \times c$

Division by zero is undefined , never write $a \div 0$ .

How to read this chapter

Always think of multiplication as "repeated addition" when in doubt. $(-3) \times 4 = -3 + -3 + -3 + -3 = -12$ . This grounds the abstract rules.

Chapter 10: Integers , Multiplication and Division

What's inside

Key results

How to read this chapter

Sub-topics

Practice quiz