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Scientific Notation Definition a x 10n ; 1 < a < 10 and n is an integer Scientific notation is used as shorthand for extremely large numbers or extremely small numbers. You will need to be able to change a number that is in scientific notation into standard notation (the way it is normally written). You will also need to be able to change a number that is in standard notation into a number written in scientific notation. The steps for both of these processes are explained below. Scientific Notation Æ Standard Notation 1. Rewrite the decimal part. 2. find the absolute value of the exponent of the 10 3. Move the decimal that many places (left if exponent is negative and right if exponent is positive) 4. Fill in empty loops with zeros. 5. Put commas and/or decimal wherever necessary. 1. rewrite decimal part Examples 3.4 x 107 = 3.4 3.4 2. exponent of 10 is 7 3. move decimal 7 units to right . 4. Fill in empty loops with zeros 3 4 0 0 0 0 0 0. 5. Put commas and/or decimal 3.4 x 107 = 34,000,000 5.06 x 10-8 = 1. rewrite decimal part 2. exponent of 10 is -8 3. move decimal 8 units to left 4. Fill in empty loops with zeros 5. Put commas and/or decimal 5.06 . . 5.06 00000005 06 5.06 x 10-8 = 0.0000000506 © LaurusSoft, Inc. Standard Notation Æ Scientific Notation 1. Locate the decimal or insert one at the end of the number. 2. Move the decimal left or right towards the numbers that are not zeros using loops. 3. Stop when you get to a number between 1 and 10. 4. Count the loops to determine exponent. a. If you started with a number less than 1 (0.000…) or if you moved the decimal to the right your exponent is negative. b. If you started with a number greater than 10 or if you moved the decimal to the left, your exponent is positive. 5. Rewrite the number with the number between 1 and 10 (ignoring all of the 0’s at the end) times 10 to the power that you determined. Examples 437,000,000 = 1. Insert the decimal 2. Move the decimal left 3. Stop between 1 and 10 4. Count the loops 437,000,000. 4 . 3 7 0 0 0 0 0 0. b. you moved decimal to the left, 8 places to the left your exponent is positive. b. Rewrite the number 437,000,000 = 4.37 x 108 0.00000063 = 0 .00000063 1. Locate the decimal 2. Move the decimal right 3. Stop between 1 and 10 4. Count the loops 0 a. you moved decimal to the right, your exponent is negative. 5. Rewrite the number .0 0 0 0 0 0 6.3 7 places to the right 0.00000063 = 6.3 x 10-7 Tip Remember that the numbers that are greater than 1 (the large numbers) are going to have a positive exponent. The numbers that are between 0 and 1 (the decimals) will have the negative exponents. To figure out the exponent you could also count how many digits are between where the decimal started off and where it ended up. 35,000 = 35,000. = 3.5000. = 3.5 x 10 4 digits between decimals © LaurusSoft, Inc. 4 Sample Problem Involving Scientific Notation If the length of a rectangular plot of land is 6.2 x 107 ft and the width of the plot is 2.0 x 105 ft, find the area of this plot of land. Answer: The area of a rectangle is length times width. Using the commutative property of multiplication and properties of exponents, the area can be calculated as follows: (6.2 x 107) x (2.0 x 105) = 6.2 x 2.0 x 107 x 105 = 12.4 x 1012 However 12.4 x 1012 is not in scientific notation since 12.4 is not less than 10. So, rewrite 12.4 as 1.24 x 101. Now, rewrite the expression as (1.24 x 101) x 1012 = 1.24 x 1013 So, the area of the rectangular plot is 1.24 x 1013 square feet. © LaurusSoft, Inc.