This chapter completes arithmetic on integers by developing multiplication and division of positive and negative numbers, after a recap of addition and subtraction using the number line and token models. Using a "bag of tokens" model, you see that placing tokens means a positive multiplier and removing them means a negative multiplier, leading to the sign rules: the product (or quotient) of two numbers with the same sign is positive, and with opposite signs is negative. Patterns in multiplication tables confirm these rules, first stated historically by Brahmagupta using "fortune" (positive) and "debt" (negative). Multiplication of integers is commutative \((a\times b = b\times a)\), associative \((a\times(b\times c)=(a\times b)\times c)\), and distributive over addition \((a\times(b+c)=a\times b + a\times c)\). The sign of a product of many integers is positive when there is an even number of negative factors and negative when odd. Real contexts — exam scores with negative marking, a mine elevator, temperature drops — show integers at work.
Recap of integers, number line and tokensMultiplication of integers and sign rulesDivision of integersCommutative, associative and distributive propertiesInteger word problems
Key terms
Integer
A whole number that may be positive, negative or zero.
Additive inverse
The number that adds to a given number to give zero; the inverse of \(a\) is \(-a\).
Sign rule for multiplication
Same signs give a positive product; opposite signs give a negative product.
Commutative property
For any integers, \(a \times b = b \times a\).
Associative property
For any integers, \(a \times (b \times c) = (a \times b) \times c\).
Distributive property
For any integers, \(a \times (b + c) = a \times b + a \times c\).
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The product of two negative integers is:
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