Appendix B Mathematics Review

# Appendix B Mathematics Review - Appendix B Mathematics...

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A.14 These appendices in mathematics are intended as a brief review of operations and methods. Early in this course, you should be totally familiar with basic algebraic tech- niques, analytic geometry, and trigonometry. The appendices on differential and inte- gral calculus are more detailed and are intended for those students who have difﬁculty applying calculus concepts to physical situations. B.1 Scientiﬁc Notation Many quantities that scientists deal with often have very large or very small values. For example, the speed of light is about 300 000 000 m/s, and the ink required to make the dot over an i in this textbook has a mass of about 0.000 000 001 kg. Obviously, it is very cumbersome to read, write, and keep track of numbers such as these. We avoid this problem by using a method dealing with powers of the number 10: and so on. The number of zeros corresponds to the power to which 10 is raised, called the exponent of 10. For example, the speed of light, 300 000 000 m/s, can be ex- pressed as 3 ± 10 8 m/s. In this method, some representative numbers smaller than unity are In these cases, the number of places the decimal point is to the left of the digit 1 equals the value of the (negative) exponent. Numbers expressed as some power of 10 multiplied by another number between 1 and 10 are said to be in scientiﬁc nota- tion. For example, the scientiﬁc notation for 5 943 000 000 is 5.943 ± 10 9 and that for 0.000 083 2 is 8.32 ± 10 ² 5 . When numbers expressed in scientiﬁc notation are being multiplied, the following general rule is very useful: 10 n ± 10 m ³ 10 n ´ m 10 ² 5 ³ 1 10 ± 10 ± 10 ± 10 ± 10 ³ 0.000 01 10 ² 4 ³ 1 10 ± 10 ± 10 ± 10 ³ 0.000 1 10 ² 3 ³ 1 10 ± 10 ± 10 ³ 0.001 10 ² 2 ³ 1 10 ± 10 ³ 0.01 10 ² 1 ³ 1 10 ³ 0.1 10 0 ³ 1 10 1 ³ 10 10 2 ³ 10 ± 10 ³ 100 10 3 ³ 10 ± 10 ± 10 ³ 1000 10 4 ³ 10 ± 10 ± 10 ± 10 ³ 10 000 10 5 ³ 10 ± 10 ± 10 ± 10 ± 10 ³ 100 000 Appendix B Mathematics Review

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where n and m can be any numbers (not necessarily integers). For example, 10 2 ± 10 5 ² 10 7 . The rule also applies if one of the exponents is negative: 10 3 ± 10 ³ 8 ² 10 ³ 5 . When dividing numbers expressed in scienti c notation, note that Exercises 1. 86 400 ² 8.64 ± 10 4 2. 9 816 762.5 ² 3. 0.000 000 039 8 ² 3.98 4. (4 ± 10 8 ) (9 ± 10 9 ) ² 5. (3 ± 10 7 ) (6 ± 10 ³ 12 ) ² 6. 7. B.2 Algebra Some Basic Rules as x , y , and z called the unknowns. If we wish to solve for x In general, if x ´ a ² b x ² 45 ± x 5 ² (5) ² 9 ± 5 (3 ± 10 6 )(8 ± 10 ³ 2 ) (2 ± 10 17 )(6 ± 10 5 ) ² 75 ± 10 ³ 11 5 ± 10 ³ 3 ² 1.5 ± 10 10 n 10 m ² 10 n ± 10 ³ m ² 10 n ³ m SECTION B.2 Algebra A.15
In all cases, whatever operation is performed on the left side of the equality must also be per- formed on the right side . The following rules for multiplying, dividing, adding, and subtracting fractions should be recalled, where a , b , and c are three numbers: A.16 Appendix B Mathematics Review Rule Example Multiplying Dividing Adding 2 3 ± 4 5 ² (2)(5) ± (4)(3) (3)(5) ²± 2 15 a b ³ c d ² ad ³ bc bd 2/3 4/5 ² ² 10 12 ( a / b ) ( c / d ) ² bc ± 2 3 4 5 ² ² 8 ± a b c d ² ² ac bd Exercises In the following exercises, solve for x : Answers 1.

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Appendix B Mathematics Review - Appendix B Mathematics...

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