Significant Figures

Significant figures (or significant digits) are used to express, in an approximate way, the number of figures that are known with some degree of precision. The precision varies with number of significant figures, which are all absolute in value except the right most. High significant figures represent an answer known to a high degree of precision.


Rules for deciding the number of significant figures in a measured quantity

Rule Examples
All nonzero digits are significant.
  1. 1.234 g has 4 significant figures.
  2. 1.2 g has 2 significant figures.
Any zeroes between nonzero digits are significant.
  1. 1.002 kg has 4 significant figures.
  2. 3.07 mL has 3 significant figures.
Leading zeros to the left of the first nonzero digits are not significant; such zeroes indicate the position of the decimal point.
  1. 0.0043 has 2 significant figures
  2. 0.01 has 1 significant figure.
Trailing zeroes that are also to the right of a decimal point in a number are significant.
  1. 0.0230 mL has 3 significant figures.
  2. 0.20g has 2 significant figures.
When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant.

To be explicit about the number of significant figures in this situation, it is a good practice to express the number in terms of scientific notation.
  1. 230 miles may be 2 or 3 significant figures.
  2. 21,400 calories may be 3,4, or 5 significant figures.
For example:
  1. 230 = 23 × 101 = 2 significant figures
  2. 21,400 = 214 × 102 = 3 significant figures
  3. 2,140 = 2.14 × 103 = 3 significant figures
  4. 2.140 × 104 = 4 significant figures

Source: Pharmaceutical Calculations, 2016, by Payal Aggarwal, Jones and Bartlett Learning


Rules for rounding off numbers

Digit to Be Dropped Rule Example
Digit to be dropped > 5 The last retained digit is increased by 1. 39.6 is rounded to 40.
Digit to be dropped < 5 The last retained digit is not changed. 39.4 is rounded to 39.
Digit to be dropped = 5 The last retained digit is increased by 1. 39.5 is rounded to 40.
12.2502 is rounded to 12.3.

Source: Pharmaceutical Calculations, 2016, by Payal Aggarwal, Jones and Bartlett Learning


Rules for performing mathematical operations and reporting the end value with the correct number of significant figures

Addition and subtraction
Report the final value with the same number of decimal places as the measurement with the least number of herokdecimal places. 104.3 (1 decimal place) + 23.643 (3 decimal places)
Calculator answer = 127.943
Correct answer = 127.9 (round to 1 decimal place)
Multiplication and Division
Report the final value with the same number of significant figures as in the component with the last number of significant figures. 3.0 (2 significant figures) × 12.60 (4 significant figures)
Calculator answer = 37.8000
Correct answer = 38 (2 significant figures)

Source: Pharmaceutical Calculations, 2016, by Payal Aggarwal, Jones and Bartlett Learning


Solved Problem: How many significant figures are there in the number 2.673

Approach: All the digits provided are significant, there are no zero present at the end of the number.

Answer: 4

Solved Problem: Round 2.673 to three significant figures.

Approach: There are 4 significant figures in the number, look at the last one, since it is less than 5, the number can be rounded to 2.67

Answer: 2.67