Exercise 1 Lab - Page |1 Astronomy Laboratory Exercise 1...

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P a g e | 1 Astronomy Laboratory Exercise 1 Math Review/Laboratory Methods I. Purpose: To review and practice several mathematical concepts and methods useful in the study of astronomy. II. Review : (Note: in the following explanations and exercise, for the most part, I have used the asterisk (*)to show multiplication. In a couple of cases where I copied a formula from an old document, x or ∙ are used to show multiplication. The division operator is either a slash (/) or a horizontal line.) A. Exponents: Exponents are a mathematical way to show how many times a number (called the base) is multiplied by itself. The exponent is placed in the upper right-hand corner of the base. The exponent can be a whole number, a decimal, a fraction and can be positive or negative. Exponents are also sometimes called “powers”. Examples: 3 3 3 3 3*3*3 3 1 = 3 No written exponent means exponent = 1 B. Algebra of exponents 1 . x a * x b = x (a+b) e.g. 3 4 * 3 -2 = 3 2 2 . ( ) a a b b x x x - = e.g. 4 (4 2) 2 2 3 3 3 9 3 - = = = 3 . x 0 1 also 1 x 0 Proof : ( ) 0 0 0 1 1 1 a a a a x x x x x and x - = = = = = 1 Exponent Means “is defined as”
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P a g e | 2 4 . 1 a a x x - = also 1 a a x x - = Proof: 0 (0 ) 1 a a a a x x x x x - - = = = 5 . ( ) a b ab x x = Example: 4 2 8 (3 ) (3·3·3·3)(3·3·3·3) 3 = = 6 . n a b an bn c cn x y x y z z = C. Scientific Notation: Using the rules and definition of exponents, it is easy to show that: 1 = 10 0 0.1 = 1 10 = 10 -1 10 = 10 1 0.01 = 1 100 = 10 -2 100 = 10 2 0.001= 1 1000 = 10 -3 1000 = 10 3 0.0001= 1 10000 = 10 -4 etc. Note that for numbers ≥ 1, the exponent = the number of zeros after the 1. For numbers < 1, the absolute value of the exponent = (the number of zeros + 1) between the decimal place and the 1. To turn a decimal number into scientific notation , we want to convert the decimal number to look like X.YZ * 10 n Where X, Y, and Z represent integers Example 1: Convert 3,560,000,000 to scientific notation. 2
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P a g e | 3 Note that 3,560,000,000 3.56 1,000,000,000 = But if we divide by 1,000,000,000 we must also multiply by 1,000,000,000 or the value of the number will be changed. But 1,000,000,000 = 10 9 Therefore : 3,560,000,000 = 9 9 3,560,000,000 10 3.56 10 1,000,000,000 x x = Example 2: Convert 0.000000123 to scientific notation. Note that 0.000000123 * 10,000,000 = 1.23 But if we multiply by 10,000,000 we must also divide by 10,000,000 or the value of the number will be changed. But 10,000,000 = 10 7 7 7 7 0.000000123 10,000,000 1.23 0.000000123 1.23 10 10 10 x x - = = = Naturally, we don’t want to go through these laborious processes every time we want to convert from decimal to scientific notation, so: Rule 1: For numbers >1, move the decimal point to the left until it falls just to the right of the 1 st non-zero digit. Then multiple by 10 n where n = the number of places the decimal point was moved. Rule 2: For numbers <1, move the decimal point to the right until it falls just to the right of the 1 st non-zero digit. Then multiple by 10 -n where n = the number of places the decimal point was moved. D. Doing Calculations with numbers written in scientific notation: One reason that scientific notation is so popular with scientists is that using it greatly simplifies doing quick calculations. However to do this, we must abide by a couple of rules.
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This note was uploaded on 04/01/2011 for the course AST 1002 taught by Professor Wilsonkey during the Spring '11 term at St. Johns River Community College.

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Exercise 1 Lab - Page |1 Astronomy Laboratory Exercise 1...

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