�Before You Begin:
You do not need any particular background to be successful at this tutorial. The material starts very slowly and builds up to more complicated problems.
Significant Figures: Why Bother?
l A number that is associated with a measurement gives information about the measurement itself.
l Specifically, the number of digits in a number implies the limitations of the measuring device.
l Because all instruments have limitations, scientists have developed a method of representing numbers so that they indicate a degree of trustworthiness.
l In common usage, the word ‘uncertainty’ means ‘not known for sure.’
l In physical science, the word ‘uncertainty’ means the lowest decimal place that can be measured (by estimating) for a particular instrument.
l All measurements have uncertainty.
Uncertainty in Measurements
l If we use a ruler to measure length, we read the centimeters directly from the scale. We count the millimeters marks directly. These numbers are ‘certain.’
l We estimate the length to the tenth of a millimeter based on where it falls between marks, halfway is 0.5 mm, for instance.
l The ‘uncertainty’ for that particular measurement (because of the limitations of the ruler) is to the tenth of a millimeter.
l If we use the same ruler to measure an object then report the measurement as 12.546 mm, the 4 and 6 are meaningless.
Significant Figures and Uncertainty
l If we perform mathematical operations with the numbers from a measurement, the result should have the same degree of uncertainty as the original measurement.
l We need to round our numbers so that they don’t imply less uncertainty than our instruments warrant.
l Today I weigh 65 kg. The measurement was made with a scale sensitive to the nearest kilogram.
l If I weigh myself every day for a week and take the average, the result on the calculator display is 64.42857143.
l The number 64.42857143 implies I made measurements with a scale sensitive to the nearest 0.00000001.
l I need to round the result to 64 kilograms to reflect the degree of uncertainty in my measurements.
l Chemists use the term ‘significant figures’ to refer to the certain digits and one estimated digit of a measurement.
l If a chemist reads someone else’s work, s/he will assume that all of the digits were read directly except the last, which was estimated.
l The term ‘significant figures’ is also used loosely to refer to a systematic method of determining at which decimal place to round a number.
Which Digits Are Significant?
l Non-zero digits are always significant.
l All digits (even zeroes) written in scientific notation are significant.
l Zeroes appearing in numbers that are not written in scientific notation may be significant.
l Zeroes between two non-zero digits are significant.
l Zeroes that cannot have been the result of a rounding operation are significant.
l Zeroes that are measured are significant IF you know their history (i.e. if you actually did the measuring).
l Zeroes that serve to “hold” the decimal place are NOT significant (the number of these zeroes changes if you change the metric system prefix).
l Zeroes that replace non-zeroes during a rounding operation are NOT significant.
l Zeroes that might have been the result of a rounding operation are assumed to be non-significant (yes, we assume the worst!).
In a Nutshell
l The whole trick to significant figures is determining if the zeroes were part of the original measurement or if they were the result of rounding.
l Remember that, when you round a number to a decimal place greater than one, you round up or down then replace digits with zeroes.
l If we round 25543 to the thousands place, it becomes 26000, for example.
l For this reason, zeroes to the left of an understood decimal place are suspect.
l A number like 440 has two significant figures because we assume the zero resulted from a rounding operation.
Rounding Right of a Decimal
l Remember that, when you round a number to a decimal place less than one, you drop digits (you never replace them with zeroes).
l If we round 25.543 to the tenths place, it becomes 25.5, for example. (Look, no zeroes!)
l For this reason, zeroes to the right of a non-zero digit and the decimal place are significant.
l A number like 4.40 has three significant figures. This zero is significant because it can’t possibly be the result of a rounding operation.
l When a zero is part of a measurement but it looks like the result of a rounding operation, write the number in scientific notation.
l If you measure 310 mL to the nearest mL, the number should have three significant figures.
l Report the volume as 3.10 x 102 mL so everyone will know the zero is significant.
l How many significant figures does the number 251 have?
l The number 251 has 3 significant figures
l Remember that all non-zero digits are significant
l How many significant figures does the number 2012 have?
l The number 2012 has 4 significant figures
l The zero is significant because it is between two non-zero digits.
l How many significant figures does the number 0.007 have?
l The number 0.007 has 1 significant figures
l In this case, the zeroes show the location of the decimal place, so they are not significant.
l How many significant figures does the number 0.1104 have?
l The number 0.1104 has 4 significant figures
l The first zero is not significant because it ‘holds’ the decimal place. The second zero is significant because it lies between two non-zero digits.
l How many significant figures does the number 75,500 have?
l The number 75,500 has 3 significant figures
l The zeroes are not significant because they might be the result of a rounding operation.
l How many significant figures does the number 0.10 have?
l The number 0.10 has 2 significant figures
l The first zero is not significant because it merely ‘holds’ the decimal place. The second zero is significant because it could not have been the result of a rounding operation.
Mathematics and Significant Figures
l If you perform mathematics using numbers that are measurements, the result must reflect the same level of uncertainty as the original measurements.
l The final number must be rounded to a decimal place that gives this information.
Addition/Subtraction and Significant Figures
l When two numbers are added (or subtracted) the result should not appear to have greater certainty than the original measurements.
l To determine where to round the result of addition/subtraction operations, find the lowest significant decimal place for each measurement.
l The measurement with the highest uncertainty is the one with its lowest significant decimal place in the highest decimal place.
l Round the result of addition/subtraction to the same decimal place as the measurement with the highest uncertainty.
Addition Example-Left of Decimal
l Add the numbers 414, 7750, and 2500.
l The result is 10664 before rounding.
l The uncertain decimal places are the ones (414), the tens (7750), and the hundreds (2500).
l The hundreds place is the highest uncertain decimal place.
l The answer rounds to the hundreds place to give 10700.
Addition Example-Right of Decimal
l Add the numbers 0.224, 1.175, and 0.44
l The result is 1.6374 before rounding.
l The uncertain decimal places are the thousandths (0.224 and 1.175) and the hundredths (0.44).
l The hundredths place is the highest uncertain decimal place.
l The answer rounds to the hundredths place to give 1.64
Addition Rule in a Nutshell
l Scan the numbers added/subtracted.
l Round the answer to the same decimal place as the number with the greatest level of uncertainty. This will be the lowest decimal place at which all of the numbers have a significant digit.
l Sometimes it helps to write the numbers in columns so that the decimals ‘line up.’
Addition Example 3
l Add the numbers 1.0556, 7.113, and 0.81.
l This sum rounds to 8.98 because the hundredths place has the highest uncertainty.
Addition Example 4
l Using these atomic weights find the molar mass of zinc oxide: Zn = 65.39 g/mol and O = 15.9994 g/mol/
l Zinc oxide is ZnO, so the molar mass is 65.39 + 15.9994 = 81.39 g/mol.
l We round at the hundredths place because of the limitations on the atomic weight of zinc.
l What is 298 Kelvin in degrees Celcius?
l temperature = 298 – 273.15 = 25 °C
l This rounds to the nearest whole degree because the 298 is to the nearest whole degree.
Multiplication/Division and Significant Figures
l When two numbers are multiplied (or divided) the result should not have less uncertainty than the original measurements.
l Round the result of multiplication/division so that the answer has the same number of significant figures as the measurement with fewest significant figures.
l Multiply the numbers 65.39 and 6.0221421x1023.
l The answer is 3.937878719x1025 before rounding.
l These numbers have four and eight significant figures.
l Round the answer to four significant figures (because four is fewer than eight) to get 3.938x1025.
Multiplication Example 2
l Faraday’s Constant, F, is equal to Avogadro’s Number, N, multiplied by the charge of the electron, e. Use these values for N and e to calculate Faraday’s Constant: N = 6.0221x1023 1/mol and e = 1.602177x10-19 C
l F = 9.6485x104 C/mol
l These values have five and seven significant figures, so the answer is rounded to five significant figures.
l How many kilometers is 9.3 miles?
l Use the conversion factor 1 km = 0.62137 mi.
l 9.3 mi ÷ 0.62137 km/mi = 14.96692792 which rounds to 15 km.
l The numbers used in the problem have two and five significant figures.
l The answer is rounded to two significant figures because two is fewer than five.
l When performing more complicated calculations, determine where to round each step of the way using the appropriate rule.
l It is usually better to round your final answer rather than round at each intermediate step.
Mixed Operations Example 1
l A student measures the mass and volume of a liquid to determine its density. The volume is 25.00 mL. An empty beaker has a mass of 74.881 g, and the beaker with liquid is 99.437 g. What is the density?
l The mass of the liquid is 24.556 g (by the addition rule).
l The density is 24.556/25.00 = 0.9822 g/mL (the answer is rounded to four significant figures by the multiplication rule applied after the addition rule).
l Note that the volume has four significant figures—zeroes right of a decimal and non-zero digits are significant.
Mixed Operations Example 2
l Use these atomic weights to find the percent chlorine by mass in sodium hypochlorite: Na = 22.989770 g/mol, Cl = 35.4527 g/mol, and O = 15.9994 g/mol.
l Sodium hypochlorite is NaClO so the molar mass is 74.441870 g/mol. This rounds to 74.442 g/mol by the addition rule.
l % Cl = (22.989770 g/mol ÷ 74.442 g/mol)100 = 30.88279466 % which rounds to 30.883%.
l We round to five significant figures by the multiplication/division rule (five is fewer than eight).
l Note that the 100 in this problem did not affect where we round the answer. This is an ‘exact’ number.
l Exact numbers are ones that are the result of a definition rather than a measurement.
l These have no uncertainty because no estimation was ever involved.
l There are 360 degrees in a circle, for example. This is an exact number, because it is how we define a degree rather than a measurement that varies from one circle to another.
Warning: Exact Versus Whole
l ‘Exact’ number does not mean the same as ‘whole’ number!
l Exact numbers have values that never vary because they are part of a definition.
l Even though it is not a whole number, the conversion faction for inches to centimeters is exact. A U.S. federal law re-defined the inch as 2.54 centimeters in 1959.
l Many exact numbers are whole numbers (like the metric system prefixes), but not all whole numbers are exact.
Warning: Exact Versus Exactly
l We use the word ‘exactly’ in common speech to express the idea of ‘no more, and no less.’ This is not the same as an exact number.
l Your lab partner might say “it was exactly 5 cm long” meaning the certain digit of the measurement was 5 and the uncertain digit was estimated to be zero.
l This number should be reported as 5.0 cm. It is a measurement rather than a definition. It is not an exact number, even if it did measure exactly 5!
Exact Numbers and Significant Figures
l Ignore any exact numbers used in a calculation when deciding where to round the result.
Exact Number Example
l A length of time is 3347 seconds to the nearest second. How many minutes is this?
l We use the conversion factor 60 sec = 1 min
l 3347 sec ÷ 60 sec/min = 55.78333333 which rounds to 55.78 minutes.
l This answer has 4 significant figures—just like the original measurement.
l The conversion factor we used is an exact number and does not limit the number of significant figures.
l When taking the average of a series of measurements, follow the addition rule. The final result should have the same significant decimal places as the measurements.
l The number of trials (the denominator of the average calculation) is similar to an exact number and should not limit the number of significant figures.
l A student performs a series of titrations to determine the concentration of a sodium hydroxide solution. Her results are 0.1038 M, 0.09994 M, and 0.1017 M. What is the average?
l The average is 0.1018 M (NOT 0.1 M).
l We round the result of the addition to the fourth decimal place by the addition rule.
Universal Constants and Irrational Numbers
l Irrational numbers like pi or constants like the speed of light should not unnecessarily limit the uncertainty.
l Use values for constants that have at least as many decimal places or significant figures as the measurements themselves.
Example with an Irrational Number
l Some equations use this form of Planck’s Constant: ħ = h/2π. If h = 6.62618 x 10-34 Js, what is ħ?
l ħ = 6.62618 x 10-34 Js ÷ [2(3.141592654)] = 1.05459 x 10-34 Js
l Note that, if you got lazy and used 3.14 as your value for pi, you would have to round ħ to three significant figures!
l Note also that the 2 in the formula is an exact number (it is part of the definition of ħ), so it doesn’t limit the significant figures.
© Copyright 2005, Kelley Whitley, ChemProfessor. All rights reserved.
This site was last updated 09/07/05