Lesson 2 Scientific Notation Scientific notation is a

Lesson 2 Scientific Notation

Ú Scientific notation is a special way of writing numbers. It is used in many different fields of science to show very large and very small numbers. Ú Here are three numbers and their equivalents in scientific notation 230, 000 = 2. 3 x 105 0. 00000105 = 1. 05 x 10 -6 4, 761, 000 = 4. 761 x 109

Ú A number in scientific notation has two factors: § A number greater than or equal to 1 and less than 10. § A power of 10 (10 with an exponent). The exponent can be either positive or negative. Large numbers get positive exponents; very small numbers get negative exponents.

Example 1 Ú Saturn is about 875, 000 miles from the sun. How would this distance be represented in scientific notation? Ú Strategy: Write the number, using the two factors for scientific notation.

Ú Step 1: Write the first factor by using the three non-zero digits (875) 0 f 875, 000. Place a decimal point after the first digit. The first factor becomes 8. 75. Remember, this factor is a number greater than or equal to 1 but less than 10.

Ú Step 2: What number times 8. 75 equals 875, 000? The number is 100, 000 or 108, so 108 is the second factor. A quick way to find the exponent of 10: Count the number of places after the first digit in 875, 000. There are 8 places. So, the exponent of 10 is 8.

Ú Step 3: Write the two factors together: 875, 000 = 8. 75 x 108

Solution Ú The distance of Saturn from the sun is 8. 75 x 108 miles. Note: The exponent 8 in the answer represents 8 places to the left from the original decimal point (875, 000). 8. 75, 000. count 8 places to the left

Finding the Exponent for the Second Factor Ú Count the number of places from the original decimal point to the decimal point of the first factor. Ú Rule 1: If you count to the left from the original decimal point, the exponent is positive. Ú Rule 2: If you count to the right from the original decimal point, the exponent is negative.

Example 2 Ú Write 0. 00017 in scientific notation. Ú Step 1: Use the two non-zero digits of 0. 00017 to write the first factor. The two non-zero digits are 1 and 7. The first factor is a number 1 but < 10. In this case 1. 7 is the first factor.

Ú Step 2: Use the rule to find the power of 10 for the second factor. . 0001. 7 count 4 places to the right The power of 10 is -4. (Rule 2 states that if you count to the right, the exponent is negative. ) Ø Step 3: Write the two factors together.

Solution Ú The number 0. 00017 expressed in scientific notation is 1. 7 x 10 -4.

Ú You often see (in newspapers and books) a large number abbreviated using a mix of numbers and words. Instead of the number 4, 500, 000, you see 4. 5 million. This form is similar to scientific notation, since it starts with a number between 0 and 10.

Example 3 Ú For the number 9, 600, 000 write the number in mixed number-word form. Ú Strategy: Use scientific notation.

Ú Step 1: What is the beginning number in scientific notation for 9, 600, 000? The number 9, 600, 000 in scientific notation is 9. 6 x 106. The beginning number is 9. 6 Ú Step 2: What is the value of 106? 106 = 1, 000 or 1 million.

Solution Ú 9, 600, 000 = 9. 6 million Ú You can convert the answers to traditional multiplication problems into scientific notation.

Example 4 Ú Express the product of 400 x 5000 in scientific notation. Ú Strategy: Find the product, and then change it into scientific notation.

Ú Step 1: Multiply 400 x 5000 2, 000 Ø Step 2: What is 2, 000 in scientific notation? 2 x 106

Solution Ú 2, 000 = 2 x 106