Machinery's Handbook, 31st Edition
Order of Operations 5 As explained in Fractions , the horizontal line in a fraction implies division. The top number (called the numerator ) is divided by the bottom number (called the denomina- tor ). For example, In formulas, the multiplication sign ( × ) may be omitted (when letters—called variables—are multiplied) or replaced by parentheses, which serve the same purpose. A × B = AB , 6 × 4 = (6)(4), 8 × a = 8 a A multiplication dot ( ⋅ ) is also sometimes used. Fractions Rational numbers can be written as common fractions or as decimal fractions . Com- mon fractions are written as a b or a / b , where a (the numerator) and b (the denominator) 50 10 --- 50 10 ÷ 5 = = are integers (but b cannot be 0, since division by zero is not defined). The denominator represents the number of equal parts that a whole quantity is broken into. The numerator is the number of these parts under consideration. For example, 2 5 indicates the whole of something is broken into 5 equal parts, and 2 of these parts are being considered. Any integer is a fraction with a denominator of 1. For example, 6 1 6 = . The implied operation in a fraction is division. Thus, a b a b means ÷. Multiple: A multiple of a number n is the result of multiplying n by positive integer 1, 2, 3, . . . Thus, the multiples of 3 are 3, 6, 9, 12, . . . The least common multiple (LCM) of two or more num bers is the smallest multiple the numbers have in common. In the exam- ple below, the first few multiples of 6 and 20 are shown, with the LCM indicated in bold: 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 , 66, . . . 20: 20, 40, 60 , 80, . . . Thus, 60 is the LCM of 6 and 20. Factor: An integer a is a factor of n if there is no remainder when n is divided by a . That is, if the result of n ÷ a is an integer. For example, 3 is a factor of 12 because 12 ⁄ 3 = 4. The greatest common factor (GCF) of two or more numbers is the largest of their common factors. Thus, the common factors of 12 and 18 are 2, 3, and 6; 6 is the GCF. Unit Fraction: A fraction having the same numerator and denominator is the unit frac- tion, 1 (or “one whole”). For example, 2 ⁄ 2 , 4 ⁄ 4 , 8 ⁄ 8 , 16 ⁄ 16 , 32 ⁄ 32 , and 64 ⁄ 64 all equal 1. Proper Fraction: A fraction whose numerator is less than its denominator. 1 ⁄ 4 , 1 ⁄ 2 , and 47 ⁄ 64 are examples of proper fractions. The value of any proper fraction is less than 1. Improper Fraction: A fraction whose numerator is greater than its denominator. 3 ⁄ 2 , 5 ⁄ 4 , and –17 ⁄ 8 are examples of improper fractions. The absolute value of any improper fraction is greater than 1. Reducible Fraction: A reducible fraction is a common fraction in which numera- tor and denominator have a common factor and so can be reduced to lowest terms by dividing both numerator and denominator by this common factor. For example, in the fraction 12 ⁄ 18 , the numerator and denominator have a GCF of 6. Thus, 12 ⁄ 18 reduces to 2 ⁄ 3 by dividing each part of the fraction by 6. A fraction such as 16 ⁄ 21 cannot be reduced, since 16 and 21 do not have a common factor. Mixed Number: A mixed number is a combination of a whole number and a proper fraction. The implied operation between them is addition. For example, 4 2 9 2 9 4 means . + A mixed number is converted to an improper fraction by multiplying the whole number part with the denominator and adding the numerator to obtain the numerator of the final fraction; the denominator remains the same.
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