Significant Figures
Chapter 1R
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While not reflecting reality, for this class:
Treat the following as being exact numbers
- Provided constants (physical and mathematical)
- Atomic masses (on your periodic table)
- All Imperial-to-metric conversion factors
When rounding (when the following digit is a 5 or greater), always round up.
A “bar” over a digit is used to indicate the last significant figure in a number.
Read more about Significant Figures on Wikipedia.
Counting Sig. Figs.
Note: Given inexact numbers should contain all certain digits and the first uncertain digit.
Count left-to-right when counting significant figures.
All non-zero digits of a provided number are significant.
\[1\]
\[2.5\]
\[23.9\]
- Zeroes between two other significant digits are significant.
\[203\]
- Zeroes to the right of a non-zero number, and also to the right of a decimal place, are significant
\[123.0\]
- Zeroes that are placeholders are not significant.
\[0.001~3\]
\[13~000\]
Ambiguous (could be 2, 3, 4, or 5). For the purpose of this class, 2 significant figures.
Use of a decimal place at the end would make the trailing zeroes significant.\[13~000.\]
\[1.3\times 10^4\]
\[1.30\times 10^{4}\]
\[1.300\times 10^{4}\]
\[1.300~0\times 10^{4}\]
- Exact numbers (those obtained by counting or simply by their definition such as a minute is defined as being composed of exactly 60 seconds) have an infinite number of significant figures. Fractions can be exact.
Physical constants such as the molar gas constant can also be exact if derived from other exact numbers.
See NIST to determine if a physical constant is exact or inexact (i.e. has a non-zero uncertainty).
\[30~\mathrm{lemons}\]
Some conversion factors are exact while others are inexact. For example, 1 inch is defined as being exactly 2.54 cm. Therefore, both values in the following conversion factor, 1 in = 2.54 cm, are exact. However, 1 gallon (US) is approximately equal to 3.785412 L (inexact). The quantity given for L is inexact and would have 7 significant figures. For this class, treat imperial-to-metric conversions as being exact.
A mathematical or physical constant has significant figures to its known digits. For example, as of March 2024, π is known to 102 trillion digits, each of which are significant.
Constants such as speed of light ( c ), gas constant ( R ), etc. also fall into this category. Note: Using a rounded off physical constant (e.g. 3.00 × 108 m s–1 instead of 299 792 458 m s–1 for speed of light) will limit the number of significant figures for that constant.
See NIST to determine if a physical constant is exact or inexact (i.e. has a non-zero uncertainty). For the purpose of this class, treat provided constants as being exact unless explicitly told otherwise.
\[c = 299~792~458~\mathrm{m~s^{-1}}\]
\[c = 2.998\times 10^{8}~\mathrm{m~s^{-1}}\]
\[R = 8.314~462~618~153~24~\mathrm{J~mol^{-1}~K^{-1}}\]
\[R = 8.314~\mathrm{J~mol^{-1}~K^{-1}}\]
Significant Figures in Calculations
- When adding or subtracting numbers, the result contains no significant digits beyond the place of the last significant digit of any datum.
When adding or subtracting numbers in scientific notation, these numbers should be converted to a common base and exponent before performing the operation.
Examples: Addition and Subtraction
\[\begin{align*} 3.24 + 1.9 + 12.482 &= 17.\bar{6}22 \\[1.5ex] &= 17.6 \\[4ex] 5.421 - 10.138 + 3.41 &= -1.3\bar{0}7 \\[1.5ex] &= -1.31 \\[4ex] 346 - 343.4 &= \bar{2}.6 \\[1.5ex] &= 3 \\[4ex] 25 - 10.1 &= 1\bar{4}.9 \\[1.5ex] &= 15 \\[4ex] 10. + 16.3 &= 2\bar{6}.3 \\[1.5ex] &= 26 \\[4ex] 10.0 + 16.3 &= 26.\bar{3} \\[1.5ex] &= 26.3 \end{align*}\]
Example: Addition and Subtraction using Scientific Notation
In the following example, the common base-10 exponent is chosen to be 3.
\[\begin{align*} 3.42 \times 10^{3} + 4.52 \times 10^{2} &= \\[1.5ex] 3.42 \times 10^{3} + 0.452 \times 10^{3} &= \end{align*}\]
- 3.42 × 103; the last significant figure is in the hundredths place.
- 0.452 × 103; the last significant figure is in the thousandths place.
The final answer should have its last significant figure in the hundredths place.
\[\begin{align*} 3.42 \times 10^{3} + 0.452 \times 10^{3} &= \\[1.5ex] &= 3.8\bar{7}2 \times 10^{3} \\[1.5ex] &= 3.87 \times 10^{3} \\[4ex] \end{align*}\]
Example: Addition and Subtraction using Scientific Notation
In the following example, the common base-10 exponent is chosen to be 2.
\[\begin{align*} 3.42 \times 10^{3} + 4.52 \times 10^{2} &= \\[1.5ex] 34.2 \times 10^{2} + 4.52 \times 10^{2} &= \end{align*}\]
- 34.2 × 102; the last significant figure is in the tenths place.
- 4.52 × 102; the last significant figure is in the hundredths place.
When adding these two numbers together, the last significant figure in the tenths place while the base-10 exponent is 2. Translating the result into a number with an base-10 exponent of 3 puts the last significant figure in the hundredths place.
\[\begin{align*} 34.2 \times 10^{2} + 4.52 \times 10^{2} &= \\[1.5ex] &= 38.\bar{7}2 \times 10^{2} \\[1.5ex] &= 3.8\bar{7}2 \times 10{3} \\[1.5ex] &= 3.87 \times 10^{3} \end{align*}\]
For numbers that clearly have an ambiguous number of significant figures, assume the zeroes to be insignificant (for the purpose of this class).
\[\begin{align*} 210~000 + 61~435 &= 2\bar{7}1~435 \\[1.5ex] &= 270~000 \end{align*}\]
Here, 210 000 has an ambiguous number of significant figures. The last non-zero digit is located in the ten thousands place. The result should be rounded to the ten thousands place.
Examples: Resolving Ambiguity
\[\begin{align*} 23~100 + 32 &= 23~\bar{13}32 \\[1.5ex] &= 23~100 \\[4ex] 890 + 12 &= 9\bar{0}2 \\[1.5ex] &= 900 \\[4ex] 312 + 300~000 &= \bar{3}00~312 \\[1.5ex] &= 300~000 \\[4ex] 10 + 16.3 &= \bar{2}6.3 \\[1.5ex] &= 30 \end{align*}\]
- In multiplication or division, the number of significant figures in the answer is determined by the value with the fewest significant figures.
Examples: Multiplication and Division
\[\begin{align*} 3.24 \times 812.3 &= 2~6\bar{3}1.852 \\[1.5ex] &= 2.63\times 10^{3}\\[4ex] 1.502 \left ( \dfrac{4.90}{2.11} \right ) &= 1.502 \left ( 2.3\bar{2}2\right ) \\[1.5ex] &= 3.4\bar{8}80 \\[1.5ex] &= 3.49 \\[4ex] \left ( 346 - 343.4 \right ) / 8.4 &= \left ( \bar{2}.6 \right ) / 8.4 \\[1.5ex] &= 0.\bar{3}095 \\[1.5ex] &= 0.3 \end{align*}\]
- As per the textbook, when a number is rounded off, the last digit retained is increased by one (rounded up) only if the following digit is 5 or greater.
NOTE: The round-half-to-even rule is followed by NIST, ANSI, ASTM, etc. where a number only gets rounded up on a 5 if the resulting number was an even number.
Examples: Rounding (per the textbook)
Each number below is rounded to 3 significant figures using the rule provided by the textbook.
\[\begin{align*} 12.\bar{6}96 &\rightarrow 12.7 \\[1.5ex] 18.\bar{3}49 &\rightarrow 18.3 \\[1.5ex] 14.\bar{9}99 &\rightarrow 15.0 \\[1.5ex] 14.\bar{3}5 &\rightarrow 14.4 \\[1.5ex] 1.1\bar{2}51 &\rightarrow 1.13 \end{align*}\]
Examples: Rounding (round-half-to-even rule)
The following examples round the given numbers to 3 significant figures using the round-half-to-even rule.
\[\begin{align*} 14.\bar{3}5 &\rightarrow 14.4 \\[1.5ex] 1.1\bar{2}51 &\rightarrow 1.12 \end{align*}\]
- A rounded value should be obtained in one step by direct rounding of the most precise value available and not in two or more successive roundings.
Example: Round in one step
89 490 rounded to the nearest 1 000 is at once 89 000; it would be incorrect to round first to the nearest 100, giving 89 500 and then to the nearest 1 000, giving 90 000.
- In a multi-step calculation, only round the final value. Determine the number of significant figures in the final result by considering each step in the calculation. In intermediate steps, write the number to the proper number of significant figures and keep at least one additional digit.
Examples: Rounding in multi-step calculations
\[\begin{align*} (2.349~4 + 1.345) \times 1.2 &= 3.69\bar{4}~4 \times 1.2 \\[1.5ex] &= 4.\bar{4}3 \\[1.5ex] &= 4.4 \\[4ex] (2.349~4 \times 1.345) + 1.2 &= 3.15\bar{9}~9 + 1.2 \\[1.5ex] &= 4.\bar{3}5 \\[1.5ex] &= 4.4 \end{align*}\]
- Digits in logarithms, ln(x) or log10(x), are significant through the n-th place after the decimal when x has n significant figures. A quantity resulting from a logarithmic transformation is dimensionless.
Examples: Logarithms
\[\ln(3.46~\mathrm{kPa}) = 1.241\]
3.46 has 3 significant figures. The result should have three places after the decimal. Note that the resulting number is dimensionless (has no unit).
\[\log(3.000\times 10^4) = 4.477~1\]
3.000 × 104 has 4 significant figures. The result should have 4 places after the decimal.
- Significant digits as a result of exponentials and antilogarithms, ex or 10x, is equal to the place of the last significant digit in x after the decimal. A number resulting from an antilog transformation will have dimensions if the original logarithmic value was derived from a quantity with units.
Examples: Exponentials and antilogarithms
\[\begin{align*} e^{1.241} &= 3.4\bar{5}9~07 \\[1.5ex] &= 3.46 \end{align*}\]
Since 1.241 has 3 decimal places, the final answer should have 3 significant figures.
\[\begin{align*} 10^{4.4771} = 29~9\bar{9}8.531~811~9\ldots \\[1.5ex] &= 30~000 \\[1.5ex] &= 3.000\times 10^{4} \end{align*}\]
Since 4.477 1 has 4 decimal places, the final answer should have 4 significant figures.