Machinery's Handbook, 31st Edition
POWERS AND ROOTS
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2.5 10 2 ( × ) 4.698 10 3 ( × ) × 3.9 10 –4 × ( × ) 4.3678 10 4 × ( ) 2 10 –3 × ( × ) 1.47 10 –2 ( × ) × ------------------------------------------------------
2.5 4.698 × 3.9 × ( ) 10 2+3 – 4 ( ) 4.3678 2 1.47 × × ( ) 10 4 – 3 – 2 ( ) -------------------------------------- 45.8055 10 × 12.8413 10 –1 × = ------------------- = 3.5670 10 1 –1 ( ) – × = 3.5670 10 2 × 356.70 = = (rounded)
Factorial Notation.— A factorial is a mathematical shortcut denoted by the symbol ! fol lowing a number (for example, 3! is “three factorial”). n ! is found by multiplying together all the positive integers less than or equal to the factorial number n . Zero factorial (0!) is defined as 1. For example: 3! = 1 × 2 × 3 = 6; 4! = 1 × 2 × 3 × 4 = 24; 7! = 1 × 2 × 3 × 4 × 5 × 6 × 7 = 5040; etc. Factorial notation is used in certain areas, including probability and analysis. The following two topics (permutations and combinations) relate to probability and statistics. Permutation.— A permutation is an arrangement of objects of a set into a sequence or order. In mathematics, the number of arrangements of n objects is given by n !. For exam- ple, 4 objects can be arranged 4! ways, that is, 4 × 3 × 2 × 1 = 24 ways. The number of ways r objects can be arranged (that is, ordered) from a set of n is given by the permutation formula P n r n ! n r – ( ) ! = --------- Example: How many ways can the letters X, Y, and Z be arranged? Solution: Three objects ( r = 3) out of a set of 3 ( n = 3) are being arranged. The numbers of possible arrangements for the three letters are 3!/(3 – 3)!= (3 × 2 × 1)/1 = 6. Listing them is not difficult, since there are so few: XYZ, XZY, YXZ, YZX, ZXY, ZYX. Example: There are 10 people participating in a foot race. How many arrangements of first, second, and third place winners are there? Solution: Here r is 3 and n is 10. The number of possible arrangements of winners are: P 10 3 10! 10– 3 ( ) ! ----------- 10! 7! ---- 10 9 8 × × 720 = = = = Combination.— This is the number of ways r objects can be chosen from n in a way that order does not matter. It is expressed as “ n choose r .” There are fewer combinations than permutations of r objects out of n , since it does not matter in what order the three objects are chosen. So in a combination, choosing ABC is the same as choosing ACB or BAC and so on. The formula is C n r n ! n r – ( ) ! r ! = ------------ Example: How many possible sets of 6 numbers can be picked with no regard for order from the numbers 1 to 52? Solution: Here r is 6 and n is 52. So the possible number of combinations is: C 52 6 52! 52– 6 ( ) !6! -------------- = Prime Factorization of Numbers.— Tables of prime numbers and factors of numbers are particularly useful for calculations involving change-gear ratios for compound gearing, dividing heads, gear-generating machines, and mechanical designs having gear trains. Definition: p is a factor of a number n if the division n / p leaves no remainder. Thus, any number n has factors of itself and 1, because n / n = 1 and n /1 = n . Other factors of a number are found as follows: 2 is a factor of any even number. Thus, 28 = 2 × 14, and 210 = 2 × 105. 3 is a factor of any number where the sum of its digits is divisible by 3. Thus, 3 is a factor of 1869, because 1 + 8 + 6 + 9 = 24, and 24 ÷ 3 = 8. 52! 46!6! ------- 52 51 50 49 48 47 × × × × × 1 2 3 4 5 6 × × × × × ---------------------------------- = 20,358,520 = =
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