Scientific Notation
Scientific Notation Essential Questions How do I write numbers in scientific notation? How do I calculate with scientific notation? When would I use scientific notation?
Vocabulary scientific notation
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.
You can find the product of a number and a power of 10 by moving the decimal point of the number. You may need to write zeros to the right or left of the number in order to move the decimal point.
Additional Example 1: Multiplying by Powers of Ten Since the exponent is a positive 4, move the decimal point 4 places to the right. 14.0 0 0 0 140,000 B. 3.6 10-5 Since the exponent is a negative 5, move the decimal point 5 places to the left. 0 0 0 0 3.6 0.000036
Check It Out! Example 1 Multiply. A. 2.5 105 Since the exponent is a positive 5, move the decimal point 5 places to the right. 2.5 0 0 0 0 250,000 B. 10.2 10-3 Since the exponent is a negative 3, move the decimal point 3 places to the left. 0 10.2 0.0102
Powers of 10 are used when writing numbers in scientific notation Powers of 10 are used when writing numbers in scientific notation. Scientific notation is a way to express numbers that are very large or very small. 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.
Additional Example 2: Writing Numbers in Scientific Notation Write the number in scientific notation. A. 0.00709 Think: The decimal needs to move 3 places to get a number between 1 and 10. 7.09 10-3 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.
Additional Example 2: Writing Numbers in Scientific Notation Write the number in scientific notation. B. 23,000,000,000 Think: The decimal needs to move 10 places to get a number between 1 and 10. 2.3 1010 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.
Think: The number is less than 1, so the exponent will be negative. Check It Out! Example 2 Write the number in scientific notation. A. 0.000811 Think: The decimal needs to move 4 places to get a number between 1 and 10. 8.11 10 -4 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.
Check It Out! Example 2 Write the number in scientific notation. B. 480,000,000 Think: The decimal needs to move 8 places to get a number between 1 and 10. 4.8 108 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.
Additional Example 3: Reading Numbers in Scientific Notation Write the number in standard form. A. 1.35 105 1.35 10 5 Think: Move the decimal right 5 places. 1.35000 135,000
Additional Example 3: Reading Numbers in Scientific Notation Write the number in standard form. B. 2.7 10–3 2.7 10–3 Think: Move the decimal left 3 places. 0002.7 0.0027
Write the number in standard form. Check It Out! Example 3 Write the number in standard form. A. 2.87 109 2.87 10 9 Think: Move the decimal right 9 places. 2.870000000 2,870,000,000
Write the number in standard form. Check It Out! Example 3 Write the number in standard form. B. 1.9 10–5 1.9 10 –5 000001.9 Think: Move the decimal left 5 places. 0.000019
Additional Example 4: 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.
Check It Out! Example 4 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.