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Significant Figures

Numbers in science have three parts. They have a value or size. They have a unit that gives the number some real-world context. They also have a precision. Remember, precision refers to the level of uncertainty in a measurement. Usually, we measure this in the lab by taking several measurements of the same quantity and then looking at how close together the different measurements are. If they are close together then the results overall are precise. If they vary widely, then the results are not precise. Precision has nothing to do with whether a result is *correct* or accurate.

Precision can be recognized in individual numbers based on the number of digits, or figures, that are included in the number. For example, 4.5 cm is less precise than 4.52 cm, which is in turn less precise than 4.524 cm. The more precise the number, the more significant figures are included in the written result. This can be understood by thinking of the last digit in the measurment as the *estimated digit*. In all measurements you make with non-electronic tools in the lab the final digit is always estimated to 1/10 of the smallest division on the instrument. This makes the minimum uncertainty for that estimated number ± 1. So the 4.5 cm result is ± 0.1. The 4.52 cm result is ± 0.01. And the 4.524 cm result is ± 0.001.

If precision is not clearly stated in the reporting of a result then we must make an assumption about the precision of the number. If you report a result by writing down every digit displayed on your calculator then you are forcing the reader to assume a super-high level of precision. For example, the density of aluminum is reported as 2.7 g/cm^{3} in a reference book. This forces the assumption that the precision is ± 0.1 g/cm^{3}. If you measure some aluminum and calculate the density as 2.69567839 g/cm^{3} then you are forcing the assumption that your precision is ± 0.00000001 g/cm^{3}. This is absurd, though it may be hard to appreciate it.

Often we need to figure out the precision of a number outside of an experimental context where the spread of data points makes it clear. One problem that arises in trying to understand the precision of a number standing on its own is deciding whether zeros in the number are significant. For example, in the number 0.0034 m, are the three leading zeros siginficant figures or just placeholders? The following rules should be memorized and applied consitently in the exercises that follow so that you can learn to identify the significant figures in all numbers used in science.

There are three rules for counting the number of significant figures in a measurement.

- Nonzero integers are always significant.
- Zeros are sometimes significant:
- Leading zeros are never significant figures. The number 0.0042 has only two significant figures.
- Captive zeros are zeros between nonzero integers. These are always significant figures. The number 1.004 has four significant figures.
- Trailing zeros are significant only if they are after a decimal point. The number 4,000 has one significant figure. The number 3.00 × 10
^{8}has three significant figures. - Zeros before the decimal
*may*be significant if and only if a decimal point is placed at the end of the number. 4,000 has one significant figures but 4,000**.**has four significant figures.

- Exact numbers have an infinite number of significant figures. Exact numbers include things like numbers that come from counting: 14 beakers is an exact number. Other exact numbers come about because of definitions: one inch is
*defined*as exactly 2.54 cm.

It is always easiest to tell the number of significant figures if you use scientific notation. In scientific notation you never have trailing zeros or leading zeros. Those zeros only tell you how far the measurement is from the decimal point. In scientific notation this job is taken care of by the exponent. If there are trailing zeros after all other digits in a number written in scientific notation then you may safely conclude that those zeros are meant to be taken as siginficant figures.

Underline the significant zeros:

- 0.00800
- 0.0050
- 1.000
- 1,000,002
- 6,002,300,000
- 0.00893
- 2.070
- 20
- 5.980 × 10
^{8} - 1.405 × 10
^{–17}

Determine the number of significant figures and write it in the blank. Put a box around certain digits and circle the estimated digit.

- 7.890 ________
- 0.003 ________
- 600,000 ________
- 3.0800 ________
- 0.00418 ________
- 91,600 ________
- 0.003005 ________
- 780,000,000 ________
- 2.50 × 10
^{18}________ - 6.893 × 10
^{–15}________

Continue as at left but circle the number with more significant digits.

- 40 m 40. m
- 540 cm 5.40 m
- 2704 mg 2740 g
- 8230 L 8203 L
- 456.0 g 456,000 mg
- 420 cm 402 m
- 9,975 L 9,970 mL
- 412,001 μm 7,000 μm
- 512 mg 512.0 mg
- 19 L 1,900. mL

When performing calculations it is important not to exaggerate the number of significant figures in the results of your calculations. To avoid doing so you may follow some simple rules that do a good job of keeping results reasonable with respect to the precision they imply. There are different rules that must be applied for multiplication/division and for addition/subtraction.

Rules for multiplication and division involving significant figures:

- The
*least*number of significant figures in any number in your calculation determines the number of significant figures in the answer. - Pure numbers (such as exact conversion factors and defined quantities) do not affect the number of significant figures.

- 140 × 61
- 67,001 ÷ 435
- 0.025 ÷ 0.0056
- 42 × 7.432
- 1,000,000 × 567
- 9.03 ÷ 7.2

- 14.50 ÷ 7.8321
- 13.000 × 42
- 89.0 ÷ 7.567
- 0.003 × 1.045
- 2,000. ÷ 100.
- 49.9 × 57.001

- Determine the number with the smallest number of significant figures to the
*right*of the decimal point when all numbers are written with the same power of ten when using scientific notation. The digits to the left of the decimal point do not contribute to determing the number of significant digits in the answer. - Add and subtract all numbers using all available digits.
- Round the answer so that the decimal portion has the same number of digits as that in the number determined in step one.

Example:

5.67 + 27.893 + 4.3

‘4.3’ has the fewest number of digits after the decimal point

5.67 + 27.893 + 4.3

‘4.3’ has the fewest number of digits after the decimal point

5.67 27.893 + 4.3 —————— 37.863 37.9 is the correct answer

- 60.7 cm – 29.01 cm
- 9.0 m + 6.01 m
- 12.2 cm + 42 cm
- 3.7 km – 2.012 km
- Find the average: 5.01 cm 4.9 cm 5.09 cm
- Find the average: 74.1 m 75.2 m 73.90 m
- Find the average: 16 mL 16.0 mL 15.9 mL
- Find the average: 19.0 g 18.99 g 19.03 g

- 15.25 in + 92.0 in
- 19.8 cm
^{3}+ 12 cm^{3} - 59 m
^{2}– 4.05 m^{2} - 67.9 in
^{2}+ 12 in^{2} - 12.0030 m
^{3}+ 4.3 m^{3} - 4,780 km + 24.3 km
- 9.871 × 10
^{3}m – 4.9 × 10^{2}m - 4.57 × 10
^{1}L – 2.56 × 10^{–1}L

- An archaeologist finds that the fossil human ancestor she discovered is about 2,000,000 years old. How old will the fossil be in one year?
- A house has two bedrooms. One bedroom is 2.7 m × 2.34 m and the other is 2.98 m × 3.32 m. What is the total area of the two bedrooms?
- A tennis ball has a radius of 3.33 cm. Find the
volume in cm
^{3}(V_{sphere}= 4/3πr^{3}). - You are hoping to attract foreign buyers to buy
apartments that you own in New York City. Since nearly all countries in the
world use meters instead of feet you must list the floor space in the
apartments in both ft
^{2}and m^{2}. Your apartments have the following areas: 1,500 square feet, 1,620 square feet, and 1,789 square feet. What are their areas in square meters? (The conversion is (1 m)^{2}= (3.2808 ft)^{2}or 1 m^{2}= 10.764 ft^{2}) - An important tool in chemical identification is the
measurement of density. Density is defined as mass ÷ volume. What is
the density of a substance, in g/cm
^{3}, that has a mass of 14.76 g and a volume of 1.65 cm^{3}? - You find that the mass of 50 sheets of paper is 227 g. What is the mass of a single piece of paper?
- The diameter of a circle is 2.1 m. What is its area?
(A
_{circle}= πr^{2}). - The dimensions of a small box are 14.2 cm x 12.5 cm x 2.76 cm. Sixty-four of these boxes fit into a packing crate. What is the volume of the packing crate?
- In a machine shop the shop foreman tells you that it will cost you $500 to make a cube of aluminum 2.123 cm on a side but only $35 to make one that is 2.1 cm on a side. What is the difference between the two cubes that justifes the difference in price?