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# Scientific Notation

Many of the numbers that are encountered in science are either very large or very small. For example, the distance from the earth to the sun is 93,000,000 miles, and a hydrogen atom weighs 0.0000000000000000000000017 gram. Scientific notation provides a convenient way of writing very large and very small numbers. In scientific notation the distance from the earth to the sun is 9.3 Ã— 107 miles and a hydrogen atom weighs 1.7 Ã— 10-24 gram. In scientific notation the times symbol, , is used to indicate multiplication. Converting a number from scientific notation to standard notation is simply a matter of multiplication.

Example 1

Scientific notation to standard notation

Write each number using standard notation.

a) 7.62 Ã— 105

b) 6.35 Ã— 10-4

Solution

a) Multiplying a number by 105 moves the decimal point five places to the right: b) Multiplying a number by 10-4 or 0.0001 moves the decimal point four places to the left: The procedure for converting a number from scientific notation to standard notation is summarized as follows.

## Strategy for Converting from Scientific Notation to Standard Notation

1. Determine the number of places to move the decimal point by examining the exponent on the 10.

2. Move to the right for a positive exponent and to the left for a negative exponent.

A positive number in scientific notation is written as a product of a number between 1 and 10, and a power of 10. Numbers in scientific notation are written with only one digit to the left of the decimal point. A number larger than 10 is written with a positive power of 10, and a positive number smaller than 1 is written with a negative power of 10. Numbers between 1 and 10 are usually not written in scientific notation. To convert to scientific notation, we reverse the strategy for converting from scientific notation.

## Strategy for Converting from Standard Notation to Scientific Notation

1. Count the number of places (n) that the decimal point must be moved so that it will follow the first nonzero digit of the number.

2. If the original number was larger than 10, use 10n.

3. If the original number was smaller than 1, use 10-n.

Example 2

Standard notation to scientific notation

Convert each number to scientific notation.

a) 934,000,000

b) 0.0000025

Solution

a) In 934,000,000 the decimal point must be moved eight places to the left to get it to follow 9, the first nonzero digit.

934,000,000 = 9.34 Ã— 108 Use 8 because 934,000,000 > 10.

b) The decimal point in 0.0000025 must be moved six places to the right to get the 2 to the left of the decimal point.

0.0000025 = 2.5 Ã— 10-6 Use -6 because 0.0000025 < 1.

We can perform computations with numbers in scientific notation by using the rules of exponents on the powers of 10.

Example 3

Using scientific notation in computations

Evaluate by first converting each number to scientific notation.

Solution   Commutative and associative properties = 0.5 Ã— 105 = 5 Ã— 10-1 Ã— 105 Write 0.5 in scientific notation. = 5 Ã—104