College Algebra 9th txtbk

A-14 APPENDIX B SPECIAL TOPICS
B-1 Scientific Notation and Significant Digits
Z Significant Digits
Z Rounding Convention
Z Significant Digits
Most calculations involving problems in the real world deal with numbers that are only
approximate. It therefore seems reasonable to assume that a final answer should not be any
more accurate than the least accurate number used in the calculation. This is an important
point, because calculators tend to give the impression that greater accuracy is achieved than
is warranted.
Suppose we want to compute the length of the diagonal of a rectangular field from
measurements of its sides of 237.8 meters and 61.3 meters. Using the Pythagorean theorem
and a calculator, we find
d 2237.8 2 61.3 2
245.573 878 . . .
d
237.8 meters
61.3 meters
The calculator answer suggests an accuracy that is not justified. What accuracy is justified?
To answer this question, we introduce the idea of significant digits.
Whenever we write a measurement such as 61.3 meters, we assume that the measurement
is accurate to the last digit written. So the measurement 61.3 meters indicates that the measurement
was made to the nearest tenth of a meter. That is, the actual width is between 61.25
meters and 61.35 meters. In general, the digits in a number that indicate the accuracy of the
number are called significant digits. If all the digits in a number are nonzero, then they are
all significant. So the measurement 61.3 meters has three significant digits, and the measurement
237.8 meters has four significant digits.
What are the significant digits in the number 7,800? The accuracy of this number is
not clear. It could represent a measurement with any of the following accuracies:
Between 7,750 and 7,850
Between 7,795 and 7,805
Between 7,799.5 and 7,800.5
Correct to the hundreds place
Correct to the tens place
Correct to the units place
To give a precise definition of significant digits that resolves this ambiguity, we use scientific
notation.
Z DEFINITION 1 Significant Digits
If a number x is written in scientific notation as
x a 10 n
1 a 10, n an integer
then the number of significant digits in x is the number of digits in a.