Scientific notation and standard form were developed to express large or small numbers in a more convenient form. They both use powers of 10 to replace the nonsignificant digits of a large or small number. Standard form uses a whole digit, then a decimal point, and two significant figures. It was primarily used with slide rules because three significant digits was the practical limit. Scientific notation uses the same format. The difference is that in scientific notation the number of significant figures is not limited to three. For example, if the number 1,000,000 has one significant figure, it is expressed as 1.0 x 106. This number is read as "one point zero times ten to the sixth power." The number 43,560,000 expressed in scientific notation with four significant figures would be 4.356 x 107. To use scientific notation effectively you must understand the powers of numbers and how they can be manipulated during mathematical computations.
First a review of the powers of 10:
There are several helpful rules to use when working with powers of 10:
When a number 10 and its exponent are moved from the denominator of a fraction to the numerator, or from the numerator to the denominator, the sign of the exponent is changed. Thus:
When two or more numbers in standard form are multiplied together, the powers of 10 can be added. Thus:
It is important to note that different methods are used to express scientific notation. One method replaces the 10 and exponent with an "E" and the power of 10.
Another method reduces the confusion of using an "x" to represent both multiplication and powers of 10 by replacing the "x" for multiplication with a •.
4.0 • 103 x 5.0 • 105 = 20 • 103+5 = 20 • 108
Other methods are also used.
When two or more numbers in standard form are divided, the powers of 10 are subtracted. Thus:
When two or more numbers in standard form are added or subtracted, the numbers must be converted to the same power of 10, and the power of 10 is not affected during the addition or subtraction. Thus:
4 • 104 - 3 • 103 becomes 40 • 103 - 3 • 103 = 37 • 103
4 • 104 + 3 • 103 becomes 40 • 103 + 3 • 103 = 43 • 103 reduced to 4.3 • 103
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