Machinery's Handbook, 31st Edition
12 POWERS AND ROOTS Using logarithms can greatly facilitate the process of raising a number to a power or ex- tracting its root. As shown in Logarithms on page 36, this is especially true if the power is not an integer. For example, the square root of 137.1 can only be found with a degree of accuracy through logarithms, a scientific calculator, or Taylor series polynomials. Scientific Notation.— Calculations involving both large and small magnitude numbers are facilitated by scientific notation . In this system, a number is expressed by two factors: (1) an integer from 1 to 9, possibly followed by a decimal, and (2) a power of 10. Large numbers in standard form are converted to scientific notation as shown in the following examples: 50,000 = 5 × 10 4 273.15 = 2.7315 × 10 2 In the example, 50,000 becomes 5 × 10 4 because the positive exponent on 10 is the num- ber of places to the right that the decimal point moves so that the first factor falls between 1 and 10. Numbers less than 1 are converted to scientific notation as shown in the following examples: 0.840 = 8.40 × 10 –1 0.0000001 = 1 × 10 –7 The negative exponent shows the number of places to the left that the decimal point moves, so that the first factor falls between 1 and 10. Science and engineering quantities—which are often quite large or small—lend them- selves to representation in scientific notation. For instance, Avogadro’s number , which is the number of particles in one mole of a substance, is 6.024 × 10 23 . The metric (SI) pressure unit of 1 pascal (Pa) is equivalent to 0.00000986923 atmosphere (atm) or 0.0001450377 pound/square inch (psi). In scientific notation, these figures are 9.86923 × 10 –6 atm and 1.450377 × 10 –4 psi, respectively. Engineering notation is a version of scientific notation in which the exponent of 10 is always a multiple of 3. (See MEASURING UNITS on page 2827 for a table of this system.) Multiplication in Scientific Notation: The procedure is as follows: 1) Multiply the first factors of the numbers to obtain the first factor of the product. 2) Add the exponents of the factors of 10 to obtain the product’s factor of 10. Thus: 4.31 10 –2 ( × ) 9.01 10 ( × ) × 4.31 9.01 ( × ) 10 –2 +1 × 38.8331 10 –1 × = = 5.98 10 4 ( × ) 4.37 10 3 ( × ) × 5.98 4.37 ( × ) 10 4+3 × 26.1326 10 7 × = = 3) Write the final in conventional scientific notation, as explained in the previous sec - tion. So, for the two examples: 38.8331 × 10 –1 = 3.88331 × 10 0 = 3.88331, because 10 0 = 1, and 26.1326 × 10 7 = 2.61326 × 10 8 . When multiplying several numbers written in this notation, the procedure is the same. Thus, (4.02 × 10 –3 ) × (3.987 × 10) × (4.863 × 10 5 ) = (4.02 × 3.987 × 4.863) × 10 (–3+1+5) = 77.94 × 10 3 = 7.79 × 10 4 , rounding off the first factor to two decimal places. Division in Scientific Notation: The procedure is as follows: 1) Divide the first factor of the dividend (the first number) by the first factor of the divisor (the second number) to get the first factor of the quotient. 2) Subtract the exponents of the factors of 10 to obtain the product’s factor of 10: 4.31 10 –2 ( × ) 9.0125 10 × ( ) ÷ = 4.31 9.0125 ( ÷ ) 10 –2 1 ( ) × 0.4782 10 –3 × 4.782 10 –4 × = = – It can be seen that this system of notation is helpful where several numbers of different magnitudes are to be multiplied and divided. Example: Find the solution of 250 4698 × 0.00039 × 43678 0.002 × 0.0147 × ----------------------------- Solution: Changing all these numbers to powers of 10 notation and performing the oper ations indicated:
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