Section 2: Scientific Notation and Dimensional Analysis Scientists use scientific methods to systematically pose and test solutions to questions and assess the results of the tests. K What I Know W What I Want to Find Out L What I Learned

Essential Questions Why use scientific notation to express numbers? How is dimensional analysis used for unit conversion? Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis

Vocabulary Review New quantitative data scientific notation dimensional analysis conversion factor Scientific Notation and Dimensional Analysis Copyright © McGraw-Hill Education

Scientific Notation Scientific notation can be used to express any number as a number between 1 and 10 (known as the coefficient) multiplied by 10 raised to a power (known as the exponent). Carbon atoms in the Hope Diamond = 4.6 x 1023 4.6 is the coefficient and 23 is the exponent. Count the number of places the decimal point must be moved to give a coefficient between 1 and 10. The number of places moved equals the value of the exponent. The exponent is positive when the decimal moves to the left and negative when the decimal moves to the right. 800 = 8.0 × 102 0.0000343 = 3.43 × 10–5 Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis

Scientific Notation Addition and subtraction Exponents must be the same. Rewrite values to make exponents the same. Example: 2.840 x 1018 + 3.60 x 1017, you must rewrite one of these numbers so their exponents are the same. Remember that moving the decimal to the right or left changes the exponent. 2.840 x 1018 + 0.360 x 1018 Add or subtract coefficients. Example: 2.840 x 1018 + 0.360 x 1017 = 3.2 x 1018 Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis

Scientific Notation Problem Response SOLVE FOR THE UNKNOWN Move the decimal point to give a coefficient between 1 and 10. Count the number of places the decimal point moves, and note the direction. Move the decimal point six places to the left. Move the decimal point eight places to the right. Write the coefficients, and multiply them by 10n where n equals the number of places moved. When the decimal point moves to the left, n is positive; when the decimal point moves to the right, n is negative. Add units to the answers. a. 1.392 × 106 km b. 2.8 × 10-8 g/cm3 Use with Example Problem 2. Problem Write the following data in scientific notation. a. The diameter of the Sun is 1,392,000 km. b. The density of the Sun’s lower atmosphere is 0.000000028 g/cm3 . Response ANALYZE THE PROBLEM You are given two values, one much larger than 1 and the other much smaller than 1. In both cases, the answers will have a coefficient between 1 and 10 multiplied by a power of 10. Scientific Notation and Dimensional Analysis Copyright © McGraw-Hill Education

Scientific Notation EVALUATE THE ANSWER The answers are correctly written as a coefficient between 1 and 10 multiplied by a power of 10. Because the diameter of the Sun is a number greater than 1, its exponent is positive. Because the density of the Sun’s lower atmosphere is a umber less than 1, its exponent is negative. Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis

Scientific Notation Multiplication and division To multiply, multiply the coefficients, then add the exponents. Example: (4.6 x 1023)(2 x 10-23) = 9.2 x 100 To divide, divide the coefficients, then subtract the exponent of the divisor from the exponent of the dividend. Example: (9 x 107) ÷ (3 x 10-3) = 3 x 1010 Note: Any number raised to a power of 0 is equal to 1: thus, 9.2 x 100 is equal to 9.2. Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis

Multiplying and Dividing Numbers in Scientific Notation SOLVE FOR THE UNKNOWN Problem A: State the problem. a. (2 × 103) × (3 × 102) Multiply the coefficients. 2 × 3 = 6 Add the exponents. 3 + 2 = 5 Combine the parts. 6 × 105 Problem B: b. (9 × 108) ÷ (3 × 10-4) Divide the coefficients. 9 ÷ 3 = 3 Use with Example Problem 3. Problem Solve the following problems. a. (2 × 103) × (3 × 102) b. (9 × 108) ÷ (3 × 10-4) Response ANALYZE THE PROBLEM You are given numbers written in scientific notation to multiply and divide. For the multiplication problem, multiply the coefficients and add the exponents. For the division problem, divide the coefficients and subtract the exponent of the divisor from the exponent of the dividend. 9 × 108 3 × 10−4 In this equation, the exponent of the dividend is 8. The exponent of the divisor is −4. Scientific Notation and Dimensional Analysis Copyright © McGraw-Hill Education

Multiplying and Dividing Numbers in Scientific Notation EVALUATE THE ANSWER To test the answers, write out the original data and carry out the arithmetic. For example, Problem a becomes 2000 × 300 = 600,000, which is the same as 6 × 105. SOLVE FOR THE UNKNOWN Problem B continued: Subtract the exponents. 8 − (−4) = 8 + 4 = 12 Combine the parts. 3 × 1012 Scientific Notation and Dimensional Analysis Copyright © McGraw-Hill Education

Dimensional Analysis Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move, or convert, from one unit to another. A conversion factor is a ratio of equivalent values having different units. Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis

Dimensional Analysis Writing conversion factors Conversion factors are derived from equality relationships, such as 1 dozen eggs = 12 eggs. Percentages can also be used as conversion factors. They relate the number of parts of one component to 100 total parts. Using conversion factors A conversion factor must cancel one unit and introduce a new one. Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis

USING CONVERSION FACTORS KNOWN UNKNOWN Length = 6 Egyptian cubits length= ? g 7 palms = 1 cubit 1 palm = 4 fingers 1 finger = 18.75 mm 1 m = 1000 mm Use with Example Problem 4. Problem In ancient Egypt, small distances were measured in Egyptian cubits. An Egyptian cubit was equal to 7 palms, and 1 palm was equal to 4 fingers. If 1 finger was equal to 18.75 mm, convert 6 Egyptian cubits to meters. SOLVE FOR THE UNKNOWN Use dimensional analysis to convert the units in the following order. cubits → palms → fingers → millimeters → meters Multiply by a series of conversion factors that cancels all the units except meter, the desired unit. 6 cubits× 7 palms 1 cubit × 4 fingers 1 palm × 18.75 mm 1 finger × 1 meter 1000 mm = ? m Response ANALYZE THE PROBLEM A length of 6 Egyptian cubits needs to be converted to meters. Scientific Notation and Dimensional Analysis Copyright © McGraw-Hill Education

USING CONVERSION FACTORS SOLVE FOR THE UNKNOWN 6 cubits× 7 palms 1 cubit × 4 fingers 1 palm × 18.75 mm 1 finger × 1 meter 1000 mm = 3.150 m EVALUATE THE ANSWER Each conversion factor is a correct restatement of the original relationship, and all units except for the desired unit, meters, cancel. Scientific Notation and Dimensional Analysis Copyright © McGraw-Hill Education

Review Essential Questions Vocabulary Why use scientific notation to express numbers? How is dimensional analysis used for unit conversion? Vocabulary scientific notation dimensional analysis conversion factor Copyright © McGraw-Hill Education Scientific Notation and Dimensional Analysis