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1.4 Key Math Skills for Chemistry A graph represents the relationship between two variables. Learning Goal Review math concepts used in chemistry: place values, positive and negative numbers, percentages, solving equations, interpreting graphs, and writing numbers in scientific notation. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. A. Identifying Place Values For any number, we can identify the place value for each of the digits in that number. The place values for two numbers are listed below: A premature baby has a mass of 2518 grams. A silver coin has a mass of 6.407 grams. Key Math Skill Identifying Place Values General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. B. Using Positive and Negative Numbers in Calculations A positive number is any number that is greater than zero and has a positive sign (+). Often the positive sign is understood and not written in front of the number. A negative number is any number that is less than zero and is written with a negative sign (−). Key Math Skill Using Positive and Negative Numbers in Calculations General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Multiplication and Division: Positive and Negative Numbers 1. When two positive numbers or two negative numbers are multiplied or divided, the answer is positive (+). 2. When a positive number and a negative number are multiplied or divided, the answer is negative (−). General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Addition: Positive and Negative Numbers 1. When two positive numbers are added, the answer is positive. 2. When two negative numbers are added, the answer is negative. 3. When a positive number and a negative number are added, the smaller number is subtracted from the larger number and the result has the same sign as the larger number. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Subtraction: Positive and Negative Numbers When two numbers are subtracted, change the sign of the number to be subtracted and then follow the rules for addition. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Calculator Operations On your calculator, there are four keys used for basic mathematical operations. The change sign key is used to change the sign of a number. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Calculator Operations—Example General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. C. Calculating Percentages To determine a percentage, divide the parts by the total (whole) and multiply by 100%. If the percentage of red balls is 5, it means there are 5 red balls in every 100 balls. Key Math Skill Calculating Percentages General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. D. Solving Equations Equations can be rearranged to solve for an unknown variable. 1. Place all like items on one side. 2. Isolate the variable you need to solve for. 3. Check your answer. Key Math Skill Solving Equations General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Study Check Solve the following equation for P1. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Solution Solve the following equation for P1. To solve for P1, divide both sides by V1. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. E. Interpreting Graphs A graph • represents the relationship between two variables. • contains two perpendicular axes. • has a horizontal axis, or x axis. • has a vertical axis, or y axis. Key Math Skill Interpreting Graphs General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Interpreting Graphs In the following graph, the • volume of a gas in a balloon is plotted against its temperature. • y axis represents the volume (L) of the gas in the balloon. • x axis represents the temperature (°C) of the gas. • title indicates that the volume of the balloon was measured at different temperatures. • points represent the volume of gas that was measured at a specific temperature. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Interpreting Graphs The line on the graph • indicates a direct relationship: that the volume of a gas in a balloon increases with increasing temperature. • can be used to determine the gas volume at various temperatures. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. F. Scientific Notation People have an average of 1 × 105 hairs on the scalp. Each hair is about 8 × 10−6 m wide. Key Math Skill Writing Numbers in Scientific Notation General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Scientific Notation Numbers written in scientific notation have two parts: Coefficient Power of 10 To write 2400 in the correct scientific notation, • the coefficient is 2.4. • the power of 10 is 3. • write the product of the coefficient multiplied by a power of 10. 2.4 × 103 General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Scientific Notation 2 400 = 2.4 × 1 000 3 places 0.00086 = 4 places = 2.4 × 103 Coefficient × Power of 10 = 8.6 × 10−4 Coefficient × Power of 10 General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Some Positive Powers of 10 General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Some Negative Powers of 10 General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Measurements in Scientific Notation Diameter of chickenpox virus General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake = 0.000 000 3 m = 3 × 10−7 m © 2016 Pearson Education, Inc. Some Measurements Written in Scientific Notation General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Scientific Notation and Calculators Numbers in scientific notation can be entered into a calculator using the EXP or EE key. The calculator display shows a number between 1 and 10 followed by a space and the power of 10. General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Guide to Writing a Number in Scientific Notation General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Study Check Write each of the following in correct scientific notation: A. 64 000 B. 0.021 General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Solution Write each of the following in correct scientific notation: A. 64 000 STEP 1 Move the decimal point to obtain a coefficient that is at least 1 but less than 10. 6.4 STEP 2 Express the number of places moved as a power of 10. 104 STEP 3 Write the product of the coefficient multiplied by the power of 10. 6.4 × 104 General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Solution Write each of the following in correct scientific notation: B. 0.021 STEP 1 Move the decimal point to obtain a coefficient that is at least 1 but less than 10. 2.1 STEP 2 Express the number of places moved as a power of 10. 10−2 STEP 3 Write the product of the coefficient multiplied by the power of 10. 2.1 × 10−2 General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc. Concept Map General, Organic, and Biological Chemistry: Structures of Life, 5/e Karen C. Timberlake © 2016 Pearson Education, Inc.