Significant Figures
The principle of significant figures is important because the precision of a number should not be increased by mathematical computations. Calculators routinely carry 7 to 14 decimals places on screen and/or in memory even though the accuracy and precision of measurements are only two or three places. It is usually necessary to determine the number of digits that should be kept in a number after mathematical computations. The rules for determining significant digits are different for exact and for approximate numbers.
Because there is no uncertainty with exact numbers, all of the digits are considered to be significant. When rounding answers produced with exact numbers, assume that they have the same number of significant figures as the largest exact number.
Determining significant figures for approximate numbers is more complicated. The first issue is the significance of zeros. A common practice is to consider a zero significant if it is between another number and a decimal point or to the right of the decimal point. For example, 540.2 has four significant figures and 540.0 has four significant figures. A zero is not considered significant if it is to the left of the decimal point. For example 0.325 has three significant figures, but 0.0325 would have four. A zero is not considered significant when it is the last number and there is no decimal point. For example, 540 has two significant figures. An exception to this rule is when the zero is the result of rounding. The number 459.8 rounded to three significant figures is 460. In this case the zero is considered significant. The problem is when the reader doesn't know if the zero is the result of rounding. In this case the significance of the zero can be ambiguous.
For nonzero numbers the number of significant digits for an approximate number depends on the precision of the measuring instrument. If you are given the weight of a steer as 551 lb and know that the scale measured to the nearest 0.1 lb, then the actual weight could be less than 551.1 but more than 550.9 lb and be recorded as 551 lb. The number will have four significant figures. To be correct, if the weight of the steer was actually 551 lb, the number should have been written as 551.0.
When the precision of the measuring instrument is not known, determining significant figures is much more difficult. A common practice is to assume the precision is +/ one half of the smallest unit in the number. For example, a distance of 347 ft would have four significant figures because the measurement could be between 347.4 and 346.6 ft and be recorded as 347 ft.
Problem: You are helping measure the weight of a calf on scales that measure to the nearest 0.5 lb. You are told the calf weighs 102 lb. How many significant figures does the weight have?
Solution: The number of significant figures is four. The scale can read 101.5, 102.0, or 102.5. If the pointer on the scale is between 101.5 and 102.0 but closer to 102.0, then the reading is recorded as 102.0. Similarly, if the pointer is between 102.0 and 102.5 but closer to 102.0, then reading is recorded as 102.0. If the weight is exactly 102 lb, the weight of the calf should be recorded as 102.0 lb. Then the weight has four significant digits. Consider the following situation:
Problem: What is the area (ft2) of a room if the width is 12 ft 3 in and the length is 22 ft 3/16 in?
Solution: The first step is to convert the dimensions to decimal form, 12 ft 3 in = 12.25 ft, and 22 ft 3/16 in = 22.1875 ft. Completing the multiplication gives a value of 271.79687 ft2. How many digits are significant?
Two rules have been developed to help determine the number of significant digits during mathematical computations.
• Adding or subtracting: the answer should be rounded to the number of decimal places in the least precise number.
• Multiplication and division: the answer should be rounded to the number of significant figures in the least accurate number.
For this problem, rule number two applies. The product, 271.79687, is reduced to four significant figures. The correct area for the room is 271.8 ft2.
Problem: You need to know the perimeter of the room to calculate the amount of paint needed to paint the walls. The dimensions of the floor are 12.25 ft by 12.1875 ft.
Solution: In this case the perimeter is the sum of the lengths of the four walls intheroom, P = 12.25 + 12.25 + 22.1875 + 22.1875. For this problem, rule number one applies. The sum, 68.1750, is reduced to four significant figures. The answer is 68.18 ft.
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