Exponential numbers significant digits

The number of significant figures is the number of digits shown when a quantity is expressed in exponential notation. 

To be sure that you know how many significant digits there are in such a number, you can report the number in standard exponential notation. The population of New York would be 8 X 10h people, and the bank account would be 8.000000 x 106 dollars. All digits in standard exponential form arc significant. 

Another way to determine the number of significant figures in a number is to express it in scientific (exponential) notation. The number of digits shown is the number of significant figures. For example, 2.305 X 10 5 would contain four significant figures. 

When we add or subtract numbers in exponential notation, the exponents must be the same. (This rule is related to the rule that requires numbers being added or subtracted to have their decimal points aligned.) The answer is then the sum or difference of the coefficients times the same exponential part as in each number being added or subtracted. (The calculator does this operation automatically, but we must know what is happening in order to report the proper number of significant digits [see Section 2.4].)... 

Do exponential calculations on the calculator to get the proper magnitude, then redo them mentally to get the proper number of significant digits. 

In Problem 2.73, how can we report the proper number of significant digits in the results without using exponential notation ... 

The magnitudes of the answers are the same, just as the maguitudes of the origiual values are the same. The numbers of millimeters all look the same, but since we know where the values came from, we know how many significant digits each contains. We can solve the problem by using standard exponential form (a) 7.3 x 10 mm, (b) 7.30 x 10 mm,... 

EXAMPLE 11 Perform the following arithmetic operations with exponential numbers, giving the answers in scientific notation and with the proper number of significant digits ... 

X 103 3 6. The number of significant digits in a number written in exponential notation is equal to the number of significant digits in the decimal portion... 

All the digits in the coefficient of a number in standard exponential form are significant. 

All digits in the coefficient of a properly reported value in scientific notation are significant, because the exponential part of the number gives the magnitude. The electronic calculator will do the arithmetic with numbers in scientific notation, but we still have to know how the process works because the calculator does not consider sigrrificant digits. See Section 1.3 for a discussion of calculator processing of numbers in exponential form. 

In scientific notation the digit term indicates the number of significant figures in the number. The exponential term merely locates the decimal point and does not represent significant figures. 

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