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Scientific Notation Section 7-2 Goals Goal • To write numbers in scientific notation and standard form. • To compare and order numbers using scientific notation. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems. Vocabulary • Scientific Notation Powers of 10 The table shows relationships between several powers of 10. • Each time you divide by 10, the exponent in the power decreases by 1 and the decimal point in the value moves one place to the left. • Each time you multiply by 10, the exponent in the power increases by 1 and the decimal point in the value moves one place to the right. Powers of 10 • You can find the product of a number and a power of 10 by moving the decimal point of the number. – If the exponent is positive, move the decimal point to the right. – If the exponent is negative, move the decimal point to the left. • You may need to write zeros to the right or left of the number in order to move the decimal point. Example: Multiplying by Powers of 10 Multiply. A. 14 104 14.0 0 0 0 Since the exponent is a positive 4, move the decimal point 4 places to the right. 140,000 B. 3.6 10-5 0 0 0 0 3.6 0.000036 Since the exponent is a negative 5, move the decimal point 5 places to the left. Your Turn: Multiply. A. 2.5 105 2.5 0 0 0 0 Since the exponent is a positive 5, move the decimal point 5 places to the right. 250,000 B. 10.2 10-3 0 10.2 0.0102 Since the exponent is a negative 3, move the decimal point 3 places to the left. Definition • Scientific Notation - is a way to express numbers that are very large or very small. – Powers of 10 are used when writing numbers in scientific notation. – Numbers written in scientific notation are expressed as 2 factors. • One factor is a number greater than or equal to 1. • The other factor is a power of 10. – Example: • 1.43 ⨯ 1012 • 5.8 ⨯ 10-9 Scientific Notation The first part is a number that is greater than or equal to 1 and less than 10. The second part is a power of 10. Why Use Scientific Notation? • For very large and very small numbers, these numbers can be converted into scientific notation to express them in a more concise form. • Numbers expressed in scientific notation can be used in a computation with far greater ease. Example: Recognizing Scientific Notation Is the number written in scientific notation? Explain. 1. 53 ⨯ 104 No, 53 is not less than 10 2. 3.42 ⨯ 10-7 Yes 3. 0.35 ⨯ 102 No, 0.35 is not greater than or equal to 1 4. 9.6 ⨯ 100 No, 100 is not in power of 10 form Your Turn: Is the number written in scientific notation? Explain. 1. 8.15 ⨯ 10-6 Yes 2. 12.9 ⨯ 108 No, 12.9 is greater than 10 3. 1.003 ⨯ 107 Yes 4. 0.0045 ⨯ No, 0.0045 is not greater than or equal to 1 10-32 Procedure: Writing Numbers in Scientific Notation 1. Place the decimal point so that there is one nonzero digit to the left of the decimal point. 2. Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10. 3. If the original number was less than 1, then the exponent is negative. If the original number was greater than 1, then the exponent is positive. Example: Writing Numbers in Scientific Notation Write the number in scientific notation. A. 0.00709 7.09 10-3 Think: The decimal needs to move 3 places to get a number between 1 and 10. Think: The number is less than 1, so the exponent will be negative. So 0.00709 written in scientific notation is 7.09 10–3. Example: Writing Numbers in Scientific Notation Write the number in scientific notation. B. 23,000,000,000 2.3 1010 Think: The decimal needs to move 10 places to get a number between 1 and 10. Think: The number is greater than 1, so the exponent will be positive. So 23,000,000,000 written in scientific notation is 2.3 1010. Your Turn: Write the number in scientific notation. Think: The decimal needs to move 4 places to get a number between 1 and 10. A. 0.000811 -4 8.11 10 Think: The number is less than 1, so the exponent will be negative. So 0.000811 written in scientific notation is 8.11 10–4. Your Turn: Write the number in scientific notation. B. 480,000,000 4.8 108 Think: The decimal needs to move 8 places to get a number between 1 and 10. Think: The number is greater than 1, so the exponent will be positive. So 480,000,000 written in scientific notation is 4.8 108. Reading Math Standard form refers to the usual way that numbers are written—not in scientific notation. Procedure: Writing Numbers in Standard Form 1. Simply move the decimal point to the right for positive exponent 10. 2. Move the decimal point to the left for negative exponent 10. (Use zeros to fill in places.) Example: Writing a Number in Standard Form Write the number in standard form. A. 1.35 105 1.35 10 5 1.35000 135,000 Think: Move the decimal right 5 places. Example: Writing a Number in Standard Form Write the number in standard form. B. 2.7 10–3 2.7 10–3 0002.7 0.0027 Think: Move the decimal left 3 places. Your Turn: Write the number in standard form. A. 2.87 109 2.87 10 9 2.870000000 2,870,000,000 Think: Move the decimal right 9 places. Your Turn: Write the number in standard form. B. 1.9 10–5 1.9 10 –5 000001.9 0.000019 Think: Move the decimal left 5 places. Example: Comparing Numbers in Scientific Notation A certain cell has a diameter of approximately 4.11 10-5 meters. A second cell has a diameter of 1.5 10-5 meters. Which cell has a greater diameter? 4.11 10-5 1.5 10-5 Compare the exponents. 4.11 > 1.5 Compare the values between 1 and 10. Notice that 4.11 10-5 > 1.5 10-5. The first cell has a greater diameter. Your Turn: A star has a diameter of approximately 5.11 103 kilometers. A second star has a diameter of 5 104 kilometers. Which star has a greater diameter? 5.11 103 5 104 Compare the exponents. Notice that 3 < 4. So 5.11 103 < 5 104 The second star has a greater diameter. Example: Ordering Numbers in Scientific Notation Order the list of numbers from least to greatest. Step 1 List the numbers in order by powers of 10. Step 2 Order the numbers that have the same power of 10 Your Turn: Order the list of numbers from least to greatest. Step 1 List the numbers in order by powers of 10. 2 10-12, 4 10-3, 5.2 10-3, 3 1014, 4.5 1014, 4.5 1030 Step 2 Order the numbers that have the same power of 10 Joke Time • Did you hear about the red ship and the blue ship that collided? • Both crews were marooned! • What did one shark say to the other while eating a clownfish? • This tastes funny! • What did the cobbler say when a cat wandered into his shop? • Shoe! Assignment • 7-2 Exercises Pg. 450 - 452: #12 – 48 even