Significant Figures

What are Significant Figures?

Significant figures are the number of known digits that can be provided reliably.  The number of significant figures can provide a means of understanding the level of precision for a given result.  The level of precision is determined by the starting values provided, ie known values, at the beginning of a problem or calculation. We determine the number of significant figures by following some basic rules.

Basic Rules for Significant Figures

Here are some rules to help you figure out how many significant figures you should use in your results.  The number of significant figures will also help in determining where you should round your final results.

• Non-zero digits are significant
  • 1234 has four non-zero digits.  Therefore the number of significant figures is 4.
    
• Leading zeros are not significant
  • 0.00001234 has four significant figures because the 5 leading zeros are not significant.
• Any zeroes between significant digits are significant
  • 101 has 3 significant figures because the zero is sandwiched between non-zero digits, just as 100000001 has 9 significant figures.
• Trailing zeros after the decimal point are significant
  • 1.00 and 10.0 both have 3 significant figures.
• Trailing zeros may or may not be significant to the left of the decimal point
  
  • 370 is considered to be 2 significant figures, while 370.0 would become 4 significant figures

Significant Figures for Addition and Subtraction

The result during addition or subtraction should contain no more than the least  accurate number used in the calculation.

321.012 + 9.8 = 330.812.  This becomes 330.8, since 9.8 is the least precise.

Significant Figures for Multiplication and Division

Results associated with multiplication or division should have only as many digits as the number with the least number of significant digits. 

Say we have 7.6543 and we divide that by 0.012.  .6543 has 5 significant figures, while 0.012 has 2.  The result would only be 2 digits.  So 637.8583 would simply be 640.  In this case the trailing zero isn’t significant.

When doing a several computational steps, it is typical to keep one extra digit throughout the process until the very end when the final result is then rounded appropriately to meet the appropriate significant figures.

Exercises

There following problems come from Chapter 1 exercises from the textbook Physics for Scientists and Engineers(3rd Edition) by Giancoli (Amazon link here).  Although I am using a Physics textbook for this exercise, the principles are still mathematical.

#2 How many significant figures do each of the following numbers have:

1. 2142
   • 4 sig figs.  All the digits are non-zero numbers.
2. 81.60
   • 4 sig figs. All but the last are non-zero.  Since the trailing zero is after the decimal place, it is counted as well.
3. 7.63
   • 3 sig figs.  All the digits are non-zero numbers.
4. 0.03
   • 1 sig fig.  Leading zeros are not significant. 
5. 0.0086
   • 2 sig figs.  Leading zeros are not significant.
6. 3236
   • 4 sig figs.  All the digits are non-zero numbers.
7. 8700
   • 2 sig figs.  The two trailing zeros in this case are not significant.