Significant Figures

A tutorial adapted from www.highschoolchem.com

What are significant figures?What are significant figures?

Significant figures are a way of expressing Significant figures are a way of expressing precision in measurement.  In a precision in measurement.  In a measurement in lab, the digits which you measurement in lab, the digits which you can read for certain, plus one uncertain can read for certain, plus one uncertain digit, are significant. digit, are significant.

For instance, on the graduated For instance, on the graduated cylinder shown to the left, you will cylinder shown to the left, you will notice that the solution is somewhere notice that the solution is somewhere between 25 mL and 30 mL.  The first between 25 mL and 30 mL.  The first digit is certain.  We know it has to be digit is certain.  We know it has to be 2.  When reading a measurement, 2.  When reading a measurement, always read  the certain plus one always read  the certain plus one uncertain.  Take a good guess at the uncertain.  Take a good guess at the uncertain digit.  Probably 8.  We would uncertain digit.  Probably 8.  We would report the volume of liquid in this report the volume of liquid in this graduated cylinder to be 28 mL.  graduated cylinder to be 28 mL.  These would be the significant digits.  These would be the significant digits.  You wouldn't want to report any You wouldn't want to report any more.  It would be foolish to try to get more.  It would be foolish to try to get any more digits in this answer.  We any more digits in this answer.  We just can't be sure! just can't be sure!

If the graduated cylinder had If the graduated cylinder had markings every mL instead of every 5, markings every mL instead of every 5, we could get even more specific.  You we could get even more specific.  You would certainly know the first two would certainly know the first two digits of the measurement.  The digits of the measurement.  The uncertain digit would be whatever you uncertain digit would be whatever you "guess" to be the fraction of liquid to "guess" to be the fraction of liquid to be between the two markings. be between the two markings.

Determining Significant Figures Determining Significant Figures in a Measurementin a Measurement

There are a few basic rules to remember when counting There are a few basic rules to remember when counting the number of significant figures in a measurement.the number of significant figures in a measurement.

All non-zero numbers ARE significant.All non-zero numbers ARE significant.   The number 33.2 contains THREE significant figures because all The number 33.2 contains THREE significant figures because all

of the digits present are non-zero. of the digits present are non-zero. Zeros between two significant digits ARE significant.Zeros between two significant digits ARE significant.   

2051 has FOUR significant figures. Since the zero is between a 2051 has FOUR significant figures. Since the zero is between a 2 and a 5, it's significant.2 and a 5, it's significant.

Leading zeros are NEVER significant.Leading zeros are NEVER significant.   They're nothing more than "place holders".  For instance, 0.54 They're nothing more than "place holders".  For instance, 0.54

has only TWO significant figures because the zero is leading has only TWO significant figures because the zero is leading and a place holder.  0.0032 also has TWO significant figures. and a place holder.  0.0032 also has TWO significant figures. 

Trailing zeros to the right of a decimal ARE Trailing zeros to the right of a decimal ARE significant.significant.    There are FOUR significant digits in 92.00 because the zeros are There are FOUR significant digits in 92.00 because the zeros are

trailing to the right of  the decimal.  Remember, they must be trailing to the right of  the decimal.  Remember, they must be there because the person measuring this value must have been there because the person measuring this value must have been able to read these numbers from the apparatus. able to read these numbers from the apparatus.

Trailing zeros in a whole number with the decimal Trailing zeros in a whole number with the decimal shown ARE significant.shown ARE significant.   Placing a decimal at the end of a number is usually not Placing a decimal at the end of a number is usually not

done.  By convention, however, this decimal will indicate a done.  By convention, however, this decimal will indicate a significant zero.  For instance, 540. indicates that the significant zero.  For instance, 540. indicates that the (trailing) zero IS significant.  There are a total of THREE (trailing) zero IS significant.  There are a total of THREE significant digits in this number. significant digits in this number.

Trailing zeros in a whole number with no decimal Trailing zeros in a whole number with no decimal shown are NOT significant.shown are NOT significant.    Writing just 540 indicates that the zero is NOT significant Writing just 540 indicates that the zero is NOT significant

and there are only TWO significant figures in this value. and there are only TWO significant figures in this value. Exact numbers have an INFINITE number of Exact numbers have an INFINITE number of

significant figures.significant figures.   Numbers that are definitions or exact have an infinite Numbers that are definitions or exact have an infinite

number of significant figures.  For example1 meter = 1000 number of significant figures.  For example1 meter = 1000 millimeters.  1 meter equals 1000.0000000... millimeters millimeters.  1 meter equals 1000.0000000... millimeters as 1.0000000...meters equals 1000 millimeters.  Both are as 1.0000000...meters equals 1000 millimeters.  Both are definitions and therefore have infinite significant figures. definitions and therefore have infinite significant figures.

PracticePractice

13.06 mL 13.06 mL 0.0450 g 0.0450 g 1.20 kg 1.20 kg 10 lbs 10 lbs 10.0 seconds 10.0 seconds 1.820 L 1.820 L 5902.05 mg 5902.05 mg 1010.2060 g 1010.2060 g

Using significant figures in Using significant figures in mathematical calculationsmathematical calculations

The expression "a chain is only as strong The expression "a chain is only as strong as its weakest link" explains why as its weakest link" explains why significant figures need to be considered significant figures need to be considered when calculating.  Remember, significant when calculating.  Remember, significant figures represent the accuracy of a figures represent the accuracy of a measurement.  When manipulating these measurement.  When manipulating these measurements by adding, subtracting, measurements by adding, subtracting, multiplying, or dividing, your final answer multiplying, or dividing, your final answer cannot be more accurate than the cannot be more accurate than the numbers you started with.  Your answer numbers you started with.  Your answer can only be as accurate as the can only be as accurate as the measurements you start with. measurements you start with.

Multiplication and DivisionMultiplication and Division When multiplying or dividing, count how When multiplying or dividing, count how

many significant figures are in each many significant figures are in each measurement.  Your final answer should measurement.  Your final answer should contain the same number of significant contain the same number of significant figures as which ever starting value has figures as which ever starting value has the least.  the least. 

For example, when you take 2.045 cm X For example, when you take 2.045 cm X 1.3 cm, your two measurements have 4 1.3 cm, your two measurements have 4 significant figures and 2 significant significant figures and 2 significant figures.  Since the LEAST number of figures.  Since the LEAST number of significant figures is 2, your final answer significant figures is 2, your final answer can only have 2 significant figures. can only have 2 significant figures.

Addition and SubtractionAddition and Subtraction

When adding or subtracting, you must When adding or subtracting, you must count decimal places instead of significant count decimal places instead of significant figures.  Your final answer should contain figures.  Your final answer should contain the same number of decimal places as the same number of decimal places as which ever starting value has the least. which ever starting value has the least.

For example, 1.994 + 16.3 = 18.294 which For example, 1.994 + 16.3 = 18.294 which rounds to 18.3 using 1 decimal placerounds to 18.3 using 1 decimal place

PracticePractice

15.04 / 3.1 15.04 / 3.1 188.20 + 92.334 + 1.0008 188.20 + 92.334 + 1.0008 345.04 g - 227.1 g 345.04 g - 227.1 g 2.11 X 0.0006 2.11 X 0.0006 0.891 X 2000.891 X 200.. X 13.8 X 13.8