Significant Figures

Significant figures, or significant digits, are used in science as a way to account for the uncertainty in numbers obtained through measurement.  All measurements have some level of uncertainty.  How much uncertainty depends on how the measurement was made - the equipment used and the skill of the person making the measurement.

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General Rules

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1.  All non-zero digits are significant.

52.76 and 2367 both have four significant figures.

2.  Captive zeros (zeros that are between non-zero digits) are significant.

502 and 2.07 both have three significant figures.

3.  Leading zeros (to locate the decimal point) are not significant.

0.0527 and 0.205 both have three significant figures.

4.  Trailing zeros (located to the right of significant figures by the other three rules) are only significant if a decimal is present.

5.60, 520. and 5760 all have three significant figures.

 

 

 

Examples

• 30525 has five significant figures, four non-zero digits, and one zero between non-zero digits. 
• 0.236 has three significant figures, one non-significant trailing zero that does not count towards the number of significant figures, and three significant non-zero digits. 
• 56.00 has four significant figures, two non-zero digits, and two trailing zeros to the left of the decimal. 
• 320. has three significant figures because of the decimal at the end of this whole number shows the trailing zero is significant
• 2500 has two significant figures. Two non-zero digits that are significant and two non-significant trailing zeros. Since there is no decimal at the end of those trailing zeros they become non-significant. 
• Numbers that are definitions such as 1 km have infinite significant figures. 
•  5.02 x 10⁴  has three significant digits, because of 5.02, 10⁴ is non-significant based on the above rules. 
• 55.62 has four significant figures because all the non-zero numbers are significant

 

Exceptions

Exact numbers, those that are counted or defined rather than measured, do not have uncertainty.  Because they are exact, they are considered to have infinite significant figures. 

 

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Math with Significant Figures

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Rounding Rules

Identify the first digit to be dropped (immediately to the right of the last significant figure).  If it is:

• less than 5 (0, 1, 2, 3, or 4), then leave the last significant figure unchanged (round down)

example:  rounding 52.51 to three significant figures becomes 52.5

• greater than 5 (6, 7, 8, or 9), then increase the last significant figure by 1 (round up)

example:  rounding 52.57 to three significant figures becomes 52.6

• equal to 5:
  • if any non-zero digits follow the 5, then round up
  • if the 5 is the last digit available or is followed only by zeros, then round up or down to make the last significant figure even.

examples:  rounding to three significant figures

52.55 becomes 52.6 (round up to make last sig. fig. even)

52.65 becomes 52.6 (round down to make last sig. fig. even)

52.6500 becomes 52.6 (round down to make last sig. fig. even)

52.6501 becomes 52.7 (round up because there is a non-zero digit following the 5 to be dropped)

 

These rules are designed to reduce bias and rounding error.  Rounding error can be further reduced by carrying extra digits through your calculations and rounding only at the last step.  Keep track of the required number of significant figures at each calculation step (according to the rules below) so you know how many significant figures you need in your final value.  For greater explanation of the reasoning around these rounding rules (and a practice quiz!), check out the University of Calgary Chemistry Textbook.

 

Addition and Subtraction

• Count the number of significant digits after the decimal
• Add or subtract the numbers
• The round the final answer to the least number of places in the decimal part of any number

Example I

2.550 + 3.5001 = 6.0501 (We are not done yet!)

• 2.550 has three digits after the decimal
• 3.5001 has four digits after the decimal
• 2.550 has the least number of digits after the decimal

Therefore the answer should have three digits after the decimal. The fourth number is one. Since it's less than five we should round down the final answer to 6.050 

Example II

52.36095 - 32.232 = 20.12895

• 52.36095 has five digits after the decimal
• 32.232 has three digits after the decimal
• 32.232 has the least number of digits after the decimal

Therefore the answer should have three digits after the decimal. The fourth number is nine. Since it's greater than five we should round up the final answer to 20.129 

 

Multiplication and Division

The significant figures in the result, for both multiplication and division, are determined by the least number of significant figures in any number. 

Example I

An object has a mass of 35.5324 g and a volume of 15.0 cm³, then find the density.

15.0 cm³ has the least number of significant figures (three significant figures), therefore the answer has three significant figures.