Lecture-3a

Lecture-3a - Lecture 3 Order of Operations 1 Some...

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Unformatted text preview: Lecture 3. Order of Operations 1 Some arithmetic expressions appear to allow different interpretations. For example, most people will simplify 6- 3 + 2 as follows: 6- 3 + 2 = (6- 3) + 2 = 3 + 2 = 5 , but a few may read 6- 3 + 2 as 6- (3 + 2), which equals 1. Most people will simplify 6 + 3 × 2 this way: 6 + 3 × 2 = 6 + (3 × 2) = 6 + 6 = 12 , but a few will read 6 + 3 × 2 as (6 + 3) × 2, which is 18. To assure consistency, mathematics users have agreed on a set of rules to govern the order in which operations are to be performed. These rules explicitly sanction specific readings of expressions that might be read in more than one way. The rules for the order of operations I. A string of additions and/or subtractions is to be evaluated from left to right. Each minus-sign operates on the entire expression to its left, but only on the number that is immediately to its right. Thus, 6- 1 + 5- 2- 3 is to be interpreted in the following manner: 6- 1 + 5- 2- 3 . The steps in simplifying the expression are: 6- 1 + 5- 2- 3 = 6- 1 + 5- 2- 3 = 5 + 5- 2- 3 = 10- 2- 3 = 8- 3 = 5 . II. A string of multiplications and/or divisions is also to be evaluated from left to right. Thus, 8 ÷ 4 × 5 ÷ 2 × 3 is to be interpreted as: 8 ÷ 4 × 5 ÷ 2 × 3 . 1 Acknowledgement. These notes were written in 2008 in cooperation with Eric Hsu, San Fran- cisco State University. Dr. Hsu’s web site is http://math.sfsu.edu/hsu 1 III. In a string involving additions and/or subtractions as well as multiplications and/or divisions, the multiplications and/or divisions are to be done before the additions and/or subtractions. Thus, 7 + 2 × 6 ÷ 3 × 5- 9 + 5 × 4 is to be interpreted as: 7 + 2 × 6 ÷ 3 × 5- 9 + 5 × 4 = 7 + 20- 9 + 20 = 38 . IV. Parentheses may be used to override any of the rules above. Expressions inside parentheses are evaluated first, no matter what operations they involve. (1 + 2) × 3 = 3 × 3 = 9 , 5- (3- 2) = 5- 1 = 4 , 5- (5- (3- 2)) = 5- (5- 1) = 5- 4 = 1 . When division is represented using a horizontal bar (as in 3+1 2 ), the bar effectively creates parentheses around the numerator and denominator: 4 + 5 1 + 2 = (4 + 5) ÷ (1 + 3) = 9 / 3 = 3 ....
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This note was uploaded on 11/23/2011 for the course MATH 6302 taught by Professor Madden during the Summer '11 term at LSU.

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Lecture-3a - Lecture 3 Order of Operations 1 Some...

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