Physics Level II Name _____________________________
Up to this time in your studies probably all numbers and all measurements have been treated as exact numbers. An exact number is a number that has been determined as a result of counting, such as
24 students are enrolled in this class, or by some definition, such as 1 hr. = 60 min. or 1 in. = 2.54 cm, or a conversion definition agreed to by the world governments' bureaus of standards. Generally, the treatment of the addition, subtraction, multiplication, and division of exact numbers is the emphasis or main content of elementary mathematics.
However, nearly all data of a technical nature involve approximate numbers ; that is, they have been determined as a result of some measurement process—some direct, as with a ruler, and some indirect, as with a surveying transit. Before studying how to perform the calculations with approximate numbers (measurements), we first must determine the "correctness" of an approximate number. First, we realize that no measurement can be found exactly. The length of the cover of this book can be found using many instruments. The better the measuring device used, the better the measurement.
A measurement may be expressed in terms of its accuracy or its precision. The accuracy of a measurement refers to the number of digits, called significant digits, which indicates the number of units that we are reasonably sure of having counted when making a measurement. The greater the number of significant digits given in a measurement, the better the accuracy, and vice versa.
Learn these rules, paying special attention to the rules involving zeros.
1. All non-zero digits are significant.
3.23 has 3 significant digits
27 has 2 significant digits
2. All zeros between non-zero digits are significant.
408 has 3 significant digits
7,006 has 4 significant digits
3. Initial zeros are never significant since they only mark the position of the decimal point.
0.04 has 1 significant digit
0.000592 has 3 significant digits
4. Final zeros are not significant since they also mark the position of the decimal point. However, note how one would make final zeros significant if they turn out to be:
420 has 2 significant digits
5,000 has 1 significant digit
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5,000 has 2 significant digits (note position of bar)
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5,000 has 3 significant digits (note position of bar) page 1

5. All zeros, which appear to the right of a decimal point AND appear to the right of a non-zero digit are significant.
38.00 has 4 significant digits
107.0 has 4 significant digits
9.0 has 2 significant digits
6. When numbers are expressed in scientific notation, the number of significant digits is the number of significant digits in the base number.
5.3 x 10 9 has 2 significant digits
2.00 x 10 -5 has 3 significant digits
4 x 10 8 has 1 significant digit
Addition and Subtraction Rule:
To add or subtract measurements:
1. Make certain that all the measurements are expressed in the same units. If they are not, convert them all to the same units.
2. Add or subtract.
3. Round the result to the same precision as the least precise measurement.
Example: 216.6
5124.
13.05 mi mi mi
+ 0.837 mi
5354.487 mi ——> 5354 mi
5354.487 should be rounded to 5354 because the least precise number only was measured to the decimal and not beyond. Using 5354.487 presumes that there was seven significant digit accuracy but in this case there was not.
Multiplying and Dividing Rule:
1. Multiply or divide by the measurements as given.
2. Round the result to the same number of significant digits as the measurement with the least number of significant digits.
Example: 230 x 6.34 = 1458.2
Round the answer to 1,500. This may seem surprising, but the instrument which measured 230 was not very precise. What if it had measured 230.00?
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