268 CHAPTER 5 Number Theory and the Real Number System Scientific Notation Often scientific problems deal with very large and very small numbers. For example, the distance from Earth to the sun is about 93,000,000 miles. The wavelength of a yellow color of light is about 0.0000006 meter. Numbers such as 93,000,000 and 0.0000006 are written in a form called decimal notation. We now introduce a more efficient way to write these numbers. Because working with many zeros is difficult, scientists developed a notation that expresses such numbers with exponents. For example, consider the distance from Earth to the sun, 93,000,000 miles. = × = × 93,000,000 9.3 10,000,000 9.3 107 The wavelength of a yellow color of light is about 0.0000006 meter. = × = × − 0.0000006 6.0 0.0000001 6.0 10 7 The numbers × 9.3 107 and × − 6.0 10 7 are written in a form called scientific notation. Each number written in scientific notation is written as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. Some examples of numbers in scientific notation are × × × × − − 3.7 10 , 2.05 10 , 5.6 10 , and 1.00 10 3 3 8 5 The following is a procedure for writing a number in scientific notation. TO CHANGE A NUMBER IN DECIMAL NOTATION TO SCIENTIFIC NOTATION 1. Move the decimal point in the original number to the right or left until you obtain a number greater than or equal to 1 and less than 10. 2. Count the number of places you have moved the decimal point to obtain the number in Step 1. If the decimal point was moved to the left, the count is to be considered positive. If the decimal point was moved to the right, the count is to be considered negative. 3. Multiply the number obtained in Step 1 by 10 raised to the count determined in Step 2. (Note that the count determined in Step 2 is the exponent on the base 10.) PROCEDURE Example 6 Converting from Decimal Notation to Scientific Notation Write each number in scientific notation. a) One drop of blood contains about 5,800,000 blood cells. b) In 2023, about 7,330,000,000 (7.33 billion) people in the world owned a mobile phone. c) In 2023, the population of the United States was about 336,800,000 (336.8 million). d) The diameter of a hydrogen atom nucleus is about 0.0000000000011 millimeter. e) The wavelength of an x-ray is about 0.000000000492 meter. Solution a) = × 5,800,000 5.8 106 b) = × 7,330,000,000 7.33 109 c) = × 336,800,000 3.368 108 d) = × − 0.0000000000011 1.1 10 12 e) = × − 0.000000000492 4.92 10 10 7 Now try Exercise 23 Michaeljung/Shutterstock
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