Rules for Working with Significant Digits

1. Leading zeros are never significant.
   Imbedded zeros are always significant.
   Trailing zeros are significant only if the decimal point is specified.
   Hint: Change the number to scientific notation. It is easier to see.
2. Addition or Subtraction:
   The last digit retained is set by the first doubtful digit.
3. Multiplication or Division:
   The answer contains no more significant digits than the least accurately known number.

EXAMPLES

Example	Significant Digits	Notes
0.00682	3	Leading zeros are not significant.
1.072	4	Imbedded zeros are always significant.
300	1	Trailing zeros are significant only if the decimal point is specified.
300.	3

OPERATION EXAMPLES

Addition		Even though your calculator gives you the answer 8.0372, you must round off to 8.04. Your answer must only contain 1 doubtful number. Note that the doubtful digits are underlined.
Subtraction		Subtraction is interesting when concerned with significant digits. Even though both numbers involved in the subtraction have 5 significant digits, the answer only has 3 significant digits when rounded correctly. Remember, the answer must only have 1 doubtful digit.
Multiplication		The answer must be rounded off to 2 significant digits since 1.6 only has 2 significant digits.
Division		The answer must be rounded off to 3 significant digits since 45.2 has only 3 significant digits.

Notes on Rounding

You can trust Excel to round properly but you should understand what it is doing.

• When rounding off numbers to a certain number of significant digits, do so to the nearest value.
  Examples:
  • Rounding 2.3467 to 3 significant digits: 2.35
  • Rounding 1.612 to 2 significant digits: 1.6
• What happens if there is a 5? There is an arbitrary rule:
  • If the number before the 5 is odd, round up.
  • If the number before the 5 is even, let it be.
  The justification for this is that in the course of a series of many calculations, any rounding errors will be averaged out.
  Examples:
  • Rounding 2.35 to 2 significant digits: 2.4
  • Rounding 2.45 to 2 significant digits: 2.4
  • Rounding 2.451 to 2 significant digits: 2.5

(Based on material from Department of Chemistry, Texas A&M University math review)