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Transcript
Appendix B Exponential Notation 1. Writing Large and Small Numbers in Exponential Notation Many of the numbers used in science are very large or very small, and are not conveniently written in the sort of notation used for normal-sized numbers. For example, the number of atoms of carbon present in 12.0 g of carbon is 602,000,000,000,000,000,000,000 Clearly, in a number of this magnitude, most of the zeros indicated are merely place-holders, used to locate the decimal point in the correct place. With very large and very small numbers, it is usually most convenient to express the number in scientific or exponential notation. For example, the number of atoms of carbon could be written in exponential notation as 6.02 x 1023 This expression indicates that 6.02 is to be multiplied by 10 twenty-three times. To see how to convert a number to exponential notation, consider the following example. Example: Write 125000 in exponential notatioa The standard format for exponential notation uses a factor between 1 and 10 that gives all the appropriate significant digits of the number, multiplied by a power of ten that locates the decimal point and indicates the order of magnitude of the number. The number 125000 can be thought of as 1.25 x 100000. 100000 is equivalent to 10 x 10 x 10 x 10 x 10 = 105. 125000 in exponential notation is then 1.25 x 105. A simple method for converting a large number to exponential notation is as follows. Move the decimal point of the given number to the position after the first significant digit of the number. This gives the multiplying factor for the number when it has been written in exponential notation. Count the number of places the decimal point has been moved from its original position (at the end of the number) to its new position (after the first significant 576 Copyright © Houghton Mifflin Co. All rights reserved. Appendix B Exponential Notation 577 digit). The number of places the decimal point has been moved represents the exponent of the power of ten of the number when expressed in scientific notation. Example: Write the following in exponential notation: 1723 Move decimal point to position after the first significant digit: 1.723 Decimal point has been moved three spaces to the left from the original 1723 to givel.723. Exponent will be 3: 1723 = 1.723 x 103 Example: Write the following in exponential notation: 7,230,000 Move decimal point to position after the first significant digit: 7.230000 Decimal point has been moved six spaces to the left from its original position. Exponent will be 6: 7,230,000 = 7.23 x 106 Basically the same methods are used when attempting to express very small numbers in exponential notation For example, consider this number: 0.0000359 This number is equivalent to 3.59 x 0.00001, and since 0.00001 is equivalent to 10~5: 0.0000359 = 3.59 x 10~5 A simple method for converting a small number to exponential notation is as follows. Move the decimal point of the given number to the position after the first significant digit of the number. This gives the multiplying factor for the number when it has been written in exponential notation Count the number of places the decimal point has been moved from its original position (at the beginning of the number) to its new position (after the first significant digit). The number of places the decimal point has been moved represents the exponent of the power of ten of the number when expressed in scientific notatioa Because the number is small, the exponent is negative. Example: Write the following in exponential notation: 0.0000000072 Move decimal point to position after the first significant digit: 7.2 Decimal point has been moved nine spaces from its original position at the beginning of the number to after the first significant digit. Exponent will thus be-9: 0.0000000072 = 7.2 x 10"9 Copyright © Houghton Mifflin Co. All rights reserved. 578 Appendix B Exponential Notation Example: Write the following in exponential notation. 0.00498 Move decimal point to position after first significant digit: 4.98 Decimal point has been moved three spaces from its original location. Exponent will be -3: 0.00498 = 4.98 x 10"3 2. Arithmetic with Exponential Numbers Oftentimes in chemistry problems, it becomes necessary to perform arithmetical operations with numbers written in exponential notation. Such arithmetic is basically no different from arithmetic with normal numbers, but certain methods for handling the exponential portion of the numbers are necessary. A. Addition and Subtraction of Exponential Numbers Numbers written in exponential notation can be added together or subtracted only if they have the same power of ten. Consider this problem: 1.233 x 103 + 2.67 x 102 + 4.8 x 101 It is not possible to just add up the coefficients of these numbers. This problem, if written in normal (nonexponential) notation would be 1233 267 + 48 1548 Before numbers written in exponential notation can be added, they must all be converted to the same po> power of tea For example, the numbers could all be written so as to have 10 as their power. , 1233 = 1.233 x 103 2.67 x 102 = 0.267 x 103 4.8 x 101 = 0.048 x 103 and the sum as (1.233 + 0.267 + 0.048) x 103 = 1.548 x 103 = 1548. B. Multiplication and Division of Exponential Numbers According to the commutative law of mathematics, the expression (A x B)(C x D) could be written validly as (A x C)(B x D). When this is applied to numbers written in exponential notation, it becomes very straightforward to find the product of two such numbers. Copyright © Houghton Mifflin Co. All rights reserved. Appendix B Exponential Notation 579 Example: Evaluate: (4.2 x 106X1.5 x 104) (4.2 x 106)(1.5 x 104) = (4.2 x 1.5X106 x 104) (4.2 x 1.5) = 6.3 (10 6 xloVlO (6+4) =10 10 (4.2 x 106)(1.5 x 104) = 6.3 x 1010 When dividing numbers written in exponential notation, a similar application of the commutative law of mathematics is made: (E x F)/(G x H) = (E/G) x (F/H) Example: Evaluate: (4.8 x 109)/(1.5 x 107) (4.8 x 109)/(1.5 x 107) = (4.8/1.5) x (109/107) (4.8/1.5) = 3.2 (109/107) = lO^-7* = 102 (4.8 x 109)/(1.5 x 107) = 3.2 x 102 Copyright © Houghton Mifflin Co. All rights reserved.