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Significant Figures

Measurements and calculations of various kinds are a fundamental part of science. When you make a measurement, it is important to indicate in some way the precision of the measurement--that is, how sure you are of the measurement. Scientists generally use the term significant digits or significant figures to refer to numbers in a measurement or a calculation that are precise.

For example, suppose you are trying to determine the mass of an object. You might first hold the object in your hand and "guesstimate" its mass at about 8 grams. This guess isn't very precise--you are really only sure that its mass is somewhere around 8 grams. Therefore, the measurement 8 g has one significant figure because you are sure only of the ones place. If you used a simple balance to find the mass of the object, you might be able to measure its mass to the nearest tenth of a gram. You could then say with certainty that its mass is 8.4 g. Because your balance gives readings only to the tenths place, the measurement 8.4 g has two significant figures. If you put the object on a more precise balance, you might find the mass of the object to be 8.42 grams. The more precise balance allows you to describe the object's mass to three significant figures.

What if you measure the mass of an object and find that its mass is 0.0032 grams? How many significant figures are in that measurement? The reality is that the leading zeros aren't really significant--you didn't actually measure anything in the tenths place or in the hundredths place. The leading zeros are placeholders and are not counted as significant. Therefore, the measurement 0.0032 g has two significant figures.

Similar rules apply to trailing zeros (zeros that follow a number). For example, the measurement 900 g has only one significant figure.

You can use decimal points to indicate when trailing zeros are significant. For example, suppose you used a balance that is accurate to the nearest gram to measure an apple. If you found that the apple's mass was exactly 300 g, you would not want to give its mass as just 300 g. That measurement--300 g--has only one significant figure. But if your balance is accurate to the nearest gram, you really know that all three digits are significant. You can indicate that all the trailing zeros in a number are significant using a decimal point. Thus, 300 g has one significant figure, but 300. g has three.

One way you can avoid confusing people with unnecessary zeros is to use scientific notation. In scientific notation, the number 0.0032 becomes 3.2 ? 10?3. It is obvious there are only two significant digits in that number. The number 900 becomes 9 ? 102. Again, this notation makes it obvious that there is only one significant figure. The number 900. becomes 9.00 ? 102, making it obvious that there are three significant figures. Scientific notation was invented to help eliminate the use of non-significant digits and to facilitate the writing of very large and very small numbers. It is easier to write that the speed of light is 3.0 ? 108 m/s than to write 300,000,000 m/s.

The next page contains a quick reference for you to use in determining how many significant figures to write in measurements and calculations.

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Rules for Significant Figures

? All nonzero numbers are always significant. o 15.2 ? three significant figures o 8926 ? four significant figures

? Zeros located between nonzero numbers are always significant. Zeros located between a nonzero digit and a decimal place are significant. (Leading zeros are never significant, and trailing zeros are only significant if they follow a decimal.) o 20073 ? five significant figures o 5800. ? four significant figures o 0.000019 ? two significant figures o 36.20 ? four significant figures

? In scientific notation, all written digits are assumed to be significant. o 1.90 ? 104 ? three significant figures o 4.5 ? 10-12 ? two significant figures

? Defined quantities (for example, Avogadro's number), conversion factors (for example, 1000 g/kg), and counted quantities (for example, 7 students) are assumed to be infinitely precise. (This means they are precise to an infinite number of decimal places.)

Using the correct number of significant figures isn't just important when you're reporting your measurements. It's also important that you use the correct number of significant figures when performing calculations. Use the rules below to determine how many significant figures to leave in the result of a calculation.

Rules for Significant Figures in Simple Calculations

? When you multiply or divide, the final answer should contain the same number of significant figures as the measurement with the smallest number of significant figures. o 582.80/39.07 = 14.9168 = 14.92 (582.80 has five significant figures, and 39.07 only has four. Therefore, the answer should be rounded so that it only has four significant figures). o 12 ? 302.1 = 3625.2 = 3600 (12 has two significant figures, while 302.1 has four. Therefore, the answer should contain only two significant figures. Just round the answer after the first two significant figures. Remember that trailing zeros don't count as significant unless they are after the decimal.)

? When you add or subtract, the final answer should contain the same number of significant figures as the measurement with the smallest number of decimal places. o 208.8 + 361.11 = 569.91 = 569.9 (208.8 only has one decimal place, while 361.11 has two decimal places. Therefore, our answers should contain one decimal place, just like 208.8) o 50.834 ? 37 = 13.834 = 14 (50.834 has three decimal places and 37 has none, so the answer should have no decimal places, just like the 37.)

Keeping Track of Significant Figures in Combined Operations

? Denote, with a small line, where the significant figures in each number end. ? If you are using a calculator that will accept multiple operations, keep all digits during the

calculations, and then round up if the last non-significant digit is 5 or greater and round down if the

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last non-significant digit is less than 5. If you are using a calculator that cannot accept multiple operations, write all the digits the calculator produces for each intermediate value. Use those numbers for all intermediate calculations. When you get your final answer, round up if the last nonsignificant digit is 5 or greater and round down if the last non-significant digit is less than 5.

Following are some examples to help you understand these rules. Consider these examples:

Example 0.00341 g 4 test tubes 50.0037 mL 9.200 ? 104 J 100 mm/cm 3.568 ? 5.8

4.186 5.41 g ? 0.348 g 3.38 mL ? 3.01 mL

5 beakers ? 136.9 g/beaker

942.65 g ? 1 kg/1000 g 4.18 ? 58.16 ? (3.38 ? 3.01)

Answer and Explanation This number has three significant figures because leading zeros are never signficant. This number has infinite precision because you must always have a whole number of test tubes. This number has six significant figures because the zeros are between nonzero numbers. This number has four significant figures because trailing zeros after the decimal are significant. This conversion factor has infinite precision.

The factor 5.8 has the fewest significant figures (two), so round the answer to two significant figures. The correct answer is 4.9.

The factor 5.41 has two decimal places; the factor 0.348 has three. Your answer should therefore have two decimal places; the correct answer is 5.06. Both factors have two decimal places, so the answer should also have two decimal places; the correct answer is 0.37. (Note that you actually lose a significant figure here.) The "5 beakers" measurement is a counted item, so it has infinite precision. Your answer should therefore have four significant figures (the same as the measurement of mass per beaker). The correct answer is 684.5 g. The 1000 in the conversion factor is infinitely precise, so your final answer should have five significant figures: 0.94265 kg. Following the rules of algebra, do the subtraction within the parentheses first, which yields 0.37. (The line is placed under the 7 to help you remember where the proper number of significant figures should be.) At this point, the equation becomes 4.18 ? 58.16 ? 0.37. Do the multiplication next, again marking the last significant figure with a line: 58.16 ? 0.37 = 21.5192. Finally, do the last subtraction: 4.18 ? 21.5192 = -17.3392. To determine how many significant figures to include in the final answer, remember that during subtraction you carry the smallest number of decimal places. The number 21.5192 has no significant decimal places, so your answer should not have any either. The correct answer is therefore -17.

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