Machinery's Handbook, 31st Edition
4 INTEGERS above, at, or below sea level. Angles can be negative, too, as explained in TRIGONOME- TRY . Calculators facilitate computation that involves integers (signed numbers). Knowing the rules of integer operations prevents errors that might occur when a calculator is used. Absolute Value: A number’s absolute value, sometimes called its magnitude , is the num- ber’s distance from zero on the number line. Whether a number is positive or negative, its absolute value is positive. For example, the absolute value of both 5 and – 5 is 5. The absolute value of n is notated | n |; thus, |5| = 5 and |– 5| = 5. Absolute value helps explain the rules of signed number addition and subtraction. Real Number Line: The real number line is generally shown with only the integers marked off (though all numbers are included). A number line is useful for conveying how signed numbers are added or subtracted. Operations on Signed Numbers: The following rules of operations apply to rational and irrational numbers as well. For simplicity, only integers are given as examples. Addition and Subtraction: Adding a negative number is equivalent to subtracting its absolute value. When a larger number is subtracted from a smaller number, the result is negative. The rules for adding and subtracting integers are illustrated with an example using four values: 7, 11, –7, and –11. The following examples illustrate the rules: Examples, Addition Examples, Subtraction 7 + 11 = 18 7 – 11 = –4 7 + (–11) = 7 – 11 = –4 7 – (–11) = 7 + 11 = 18 (–7) + 11 = 11 + (–7) = 11 – 7 = 4 (–7) – (–11) = (–7) + 11 = 11 + (–7) = 11 – 7 = 4 (–7) + (–11) = –18 –7 – 11 = –18 Multiplication and Division: Multiplication or division of numbers with the same sign results in a positive answer. Opposite signed numbers result in negative answers when multiplied or divided. The following examples illustrate the rules: Examples, Multiplication Examples, Division 5 × 2 = 10 12 ÷ 3 = 4 5 × (–2) = –10 (–12) ÷ 3 = –4 (–5) × 2 = –10 (12) ÷ (–3) = –4 (–5) × (–2) = 10 (–12) ÷ (–3) = 4 Order of Operations.—Mathematical operations are performed on numbers in a par- ticular order, commonly referred to as PEMDAS, which stands for “ P arentheses, E xpo- nents, M ultiplication, D ivision, A ddition, S ubtraction.” First, when there are no parenthe- ses or other grouping symbols, multiplication and division are done before addition and subtraction. Then, proceeding from left to right, the addition and subtraction are done in the order they appear. For example: 100 – 26 + 7 × 2 – 100 ÷ 4 = 100 – 26 + 14 – 25 = 74 + 14 – 25 = 88 – 25 = 63 Parentheses ( ) and brackets [ ]—called grouping symbols —indicate if addition and subtraction are to occur before multiplication and division. The operations are performed from the innermost to the outermost grouping symbols. For example: [6 × (15 – 7)] ÷ 2 = [6 × 8] ÷ 2 = 48 ÷ 2 = 24 Exponents are a multiplication operation, but unless parentheses or brackets are present, exponents are applied before multiplication. For example: 4 × 9 2 = 4 × 81 = 324 Also, when parentheses are present next to a multiplication, the × can be omitted: 5(8 – 3) = 5(5) = 25
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