Significant Figures Knowledge/Understanding: •

How and why measurements are rounded.

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How rounding and significant figures relate to precision and uncertainty.

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When significant figures do not apply.

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Determine which digits in a number are significant.

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Express the uncertainty of a measurement based on its significant digits.

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Round a number to the desired number of significant digits based on its precision.

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Perform mathematical calculations and round the answer to the correct number of significant figures.

Skills:

significant figures (also called significant digits): the digits whose value is known precisely; the digits that were measured. insignificant figures (also called insignificant digits): the digits that are part of the number, but whose values are not known; the digits that were rounded off or not measured.

Ideally, scientists would always write the value and the uncertainty for all numbers, but this is often impractical. A common shortcut is to express the precision of a measurement through rounding. E.g., the number 125 000 is clearly measured (or rounded off) to the nearest 1 000. If we were to write this number including its uncertainty, we would write 125 000 ± 1 000. The values of the hundreds, tens, and ones digits are not known, because they are beyond the precision of the measurement. If we were able to measure more precisely, the actual number would be between 124 000 and 126 000, and it would round off to 125 000. I.e., the number would most likely lie between 124 500 and 125 499.. For the number 125 000 we would say that the first three digits are significant (the exact value is known), and the last three are insignificant (the exact value is unknown). For any measurement that does not have an explicitly stated uncertainty value, assume the uncertainty is ±1 in the last significant digit.

Identifying the Significant Figures in a Number The first significant digit is where the “measured” part of the number begins—it’s always the first digit that is not zero. The last significant digit is the last “measured” digit—the last digit whose true value is known or accurately estimated (usually ±1). •

If the number doesn’t have a decimal point, the last significant digit will be the last digit that is not zero.

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If the number does have a decimal point, the last significant digit will be the last digit shown.

If the number is in scientific notation, the above rules tell us (correctly) that all of the digits before the “times” sign are significant. In the following numbers, the significant figures have been underlined:

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13 000

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0.0275

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0.0150

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6 804.305 00

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6.0 × 10 3 400.

23 (note the decimal point at the end)

When Not to Use Significant Figures Significant figure rules only apply in situations where the numbers you are working with have a limited precision. This is usually the case when the numbers represent measurements. Exact numbers have infinite precision, and therefore have an infinite number of significant figures. Some examples of exact numbers are: •

Pure numbers, such as the ones you encounter in math class.

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Anything you can count. (E.g., there are 24 people in the room. That means exactly 24 people, not 24.0 ± 0.1 people.)

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Exact mathematical relationships, such as most whole-number 2 exponents in formulas. (E.g., the area of a circle is πr . This is an exact formula, so the numbers π and “2” are exact.)

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Exact definitions, such as 1 km = 1000 m.

What to Do When the Number as Written Doesn’t Show the Correct Number of Significant Figures The best answer is always to state the uncertainty explicitly, such as 13 000 ± 10. However, you also have the option of either placing a line over the last significant digit, or expressing the number in scientific notation. For example, both of the following have four significant digits, and both are equivalent to writing 13 000 ± 10.• 4 1.300 × 10

Where to Round (Math with Significant Figures) Whenever you do math with significant figures, the answer can’t be more precise than the numbers that it came from. There are two simple rules to make sure this doesn’t happen:

Addition & Subtraction: Line up the numbers in a column. Any column that has an uncertain digit—a zero from rounding—is an uncertain column. (Uncertain digits are shown as question marks in the right column below.) You need to round off your answer to get rid of all of the uncertain columns. For example: problem: 23000 0.0075 +1650          24650.0075 meaning:      23???.???? 0.0075 +      165?.????               24???.???? Because we can’t know which digits go in the hundreds, tens, ones, and decimal places of all of the addends, the exact values of those digits must therefore be unknown in the sum.

This means we need to round off the answer of 24 650.0075 to the nearest 1 000, which gives a final answer of 25 000 (which means 25 000 ± 1 000). A silly (but correct) example of addition with significant digits is: 100 + 37 = 100

Multiplication, Division, Etc. For multiplication, division, and just about everything else (except for addition and subtraction, which we have already discussed), round your answer off to the same number of significant digits as the number that has the fewest. For example, consider the problem 34.52 × 1.4 = 48.328 The number 1.4 has the fewest significant digits (2). Remember that 1.4 really means 1.4 ± 0.1. This means the actual value, if we had more precision, could be anything between 1.3 and 1.5. This means the actual answer could be anywhere from: 34.52 × 1.3 = 44.876 to: 34.52 × 1.5 = 51.780 This means the actual answer must be 48.328 ± 3.452 If the ones digit is approximate, then everything beyond it must be unknown and we should report the number as 48 ± 3 In this problem, notice that the least significant term in the problem (1.4) had only 2 significant digits, and the correctly-rounded answer (48) also has 2 significant digits. This is where the rule comes from. A silly (but correct) example of multiplication with significant digits is: 147 × 1 = 100

Keeping a Guard Digit For complicated problems, it may help to keep track of the last significant digit in each step by putting a line over it (even if it’s not a zero). For multi-step operations, keep at least one extra significant digit (called a “guard digit”) until you’re finished, so round-off errors don’t accumulate. Once you have your final answer, round it to the correct number of significant digits. Don’t forget to use the correct order of operations (PEMDAS)! Here’s an example:

Note that in the above example, we kept all of the digits until the end. This is to avoid introducing small rounding errors at each step, which can add up to enough to change the final answer. Notice how keeping one guard digit gives the same answer, but rounding off the numbers at each step gives the wrong answer: With Guard Digit Without Guard Digit  

When All Else Fails Remember that significant figures are always an approximation, and their purpose is to make sure someone looking at your numbers won’t think they are more precise than they actually are. If you’re not sure how many significant figures you need, you can usually make an educated guess based on how significant the original numbers were. With most lab equipment, it is only possible to make measurements to three digits of precision (two digits from the markings, plus one estimated digit). This means that most measurements will have 3 significant digits. Any time you’re not sure how many significant digits you need, 3 is usually a pretty good guess.