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9−(−7) = 9 + 7 = 16

−3 −8 = −3 +(−8) = −11

Multiplication

To multiply two integers, multiply the absolute values and then determine the sign. If the integers have the same sign, the product will be positive. If the factors have different signs, the product is negative.

4 × 7 = 28

−4 ×(−7) = 28

4 ×(−7) = −28

−4 × 7 = −28

Division

Just as subtraction is defined as adding the inverse, division is defined as multiplying by the inverse. The multiplicative inverse, or reciprocal, of an integer n is . To form the reciprocal of a fraction, swap the numerator and denominator. The reciprocal of 4 is and the reciprocal of is . You probably remember learning that to divide by a fraction, you should invert the divisor and multiply.

Since division is multiplication in disguise, you follow the same rules for signs when you divide that you follow when multiplying. To divide two integers, divide the absolute values. If the signs are the same, the quotient is positive. If the signs are different, the quotient is negative.

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EXERCISE 1.3

Find the value of each expression.

1. −12 + 14

2. −13 − 4

3. 18 ×(−3)

4. −32 ÷(−8)

5. 6 +(−3)

6. 5 −(−9)

7. 2 × 12

8. 12 ÷(−4)

9. −6 − 2

10. −9 ×(−2)

11. −5 + 7

12. 12 − 5

13. 8 ×(−4)

14. −2 − 8

15. −5 × 8

16. −9 − 3

17. 5 ÷(−5)

18. −4 × 12

19. −4 ×(−4)

20. −45 ÷(−9)

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Order of operations

The order of operations is an established system for determining which operations to perform first when evaluating an expression. The order of operations tells you first to evaluate any expressions in parentheses. Exponents are next in the order, and then, moving from left to right, perform any multiplications or divisions as you meet them. Finally, return to the beginning of the line, and again moving from left to right, perform any additions or subtractions as you encounter them.

The two most common mnemonics to remember the order of operations are PEMDAS and Please Excuse My Dear Aunt Sally. In either case, P stands for parentheses, E for exponents, M and D for multiplication and division, and A and S for addition and subtraction.

A multiplier in front of parentheses means that everything in the parentheses is to be multiplied by that number. If you can simplify the expression in the parentheses and then multiply, that’s great. If not, use the distributive property. Remember that a minus sign in front of the parentheses, as in 13 −(2 + 5), acts as −1. If you simplify in the parentheses first, 13 −(2 + 5) = 13 − 7 = 6, but if you distribute, think of the minus sign as −1.

13 −(2 + 5) = 13 − 1(2 + 5) = 13 − 2 − 5 = 11 − 5 = 6

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EXERCISE 1.4

Find the value of each expression.

1. 18 − 32

2.(18 − 3)2

3. 15 − 8 + 3

4. 15 −(8 + 3)

5. 52 − 3·2 + 4

6.(22 − 7)·2 + 15 ÷ 5 −(4 + 8)

7. 9 − 2 + 4(3 − 5) ÷ 2

8. 9 −(2 + 4)(3 − 5) ÷ 2

9. 8 + 42 − 12 ÷ 3

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Using variables

Variables are letters or other symbols that take the place of a number that is unknown or may assume different values. You used the idea of a variable long before you learned about algebra. When you put a number into the box in 4 + = 6, or knew what the question mark stood for in 3 − ? = 2, or even filled in a blank, you were using the concept of a variable. In algebra, variables are usually letters, and determining what number the variable represents is one of your principal jobs.

When you write the product of a variable and a number, you traditionally write the number first, without a times sign, that is, 2x rather than x. 2. The number is called the coefficient of the variable. A numerical coefficient and a variable(or variables) multiplied together form a term. When you want to add or subtract terms, you can only combine like terms, that is, terms that have the same variable, raised to the same power if powers are involved. When you add or subtract like terms, you add or subtract the coefficients. The variable part doesn’t change.

Translating verbal phrases into variable expressions is akin to translating