Rules for Significant Figures and Rounding

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Rules to identify signficant figures are:

  1. Zeros between non-zero digits are significant.
    1.023 has 4 significant figures, 30405 has five significant figures.
  2. Leading zeros (zeros to the left of non-zero digits) are never significant.
    00509 has three significant figures, 0.00345 has three significant figures (3,4 and 5).
  3. In numbers with decimal point, zeros to the right of the last non-zero digit after the decimal point (trailing zeros) are significant.
    509.0 has four significant figures, 0.0034500 has five significant figures.
  4. In numbers without decimal point, trailing zeros (zeros to the right of the last non-zero digit) may or may not be significant.


Rules for rounding off

  1. If the digit immediately to the right of the last significant digit is 4 or less, the last significant digit is retained and all the digits to the right of it are dropped.
    In 1.023464, digit 3 in bold is the last significant digit. It is rounded off to 1.023. Similarly, in 0.0030435, digit 4 in bold is the last significant digits. This is rounded off to 0.00304.
  2. If the digit immediately to the right of the last significant digits is greater than 5 or is a 5 followed by non-zero digits, the last significant digit is incremented by 1 and all the digit to the right of it are dropped.
    In 13.323564, digit 3 in bold is the last significant digit. It is rounded off to 13.324. Similarly, in 0.00304532, digit 4 in bold is the last significant digits. This is rounded off to 0.00305.
  3. If the digit to be dropped immediately to the right of the last significant digit is 5, then
    a) if last significant digit is odd, 1 is added to it so that it becomes even and 5 to the right of it dropped.
    b) if last significant digit is even, it is left as it is and 5 to the right of it dropped.
    In 1.0235, digit 3 in bold is the last significant digit. It is rounded off to 1.024. Similarly, in 0.003045, digit 4 in bold is the last significant digits. This is rounded off to 0.00304.


Rules for significant figures for quantities calculated from measured quantities

A chain is no stronger than its weakest link.

Precision of a quantity calculated using measured quantities is limited by precision of measured quantities. More precisely, if a quantity a being calculated using two measured quantities - one with low precision and the other with relatively high precision - then the precision of the calculated quantity is limited by that of the quantity measured with lowest precision. Following are the two thumb rules to determine the significant figures for quantities calculated by either adding or subtracting, or by multiplying or dividing quantities measured with varied precision.
  1. Quantities obtained by multiplication and division
    In cases where two or more numbers with different significant figures are multiplied or divided, the resultant must contain significant figures equal to the number with the least significant figures. Consider the case where two measurements obtained using measuring scales with different precision are being multiplied: \[1.023 \times 3.0 = 3.069\] Here, the measurement \(3.0\) has the least number of significant figures — two. Therefore, the product is be rounded to two significant figures also: \[1.023 \times 3.0 = 3.069 \approx 3.1\] On the hand, consider the operation \[\frac{1.023}{3.0} = 0.341.\] The final product is be rounded to two significant figures: \[\frac{1.023}{3.0} = 0.341 \approx 0.34\]

  2. Quantities obtained by addition and subtraction
    In cases where two or more numbers with different significant decimal places (significant digits to the right of the decimal point) are added or subtracted, the resultant must contain no more than the number that has least number of significant digits after the decimal point. Consider the case where two measurements obtained using measuring scales with different precision are being added or subtracted: \[12.025 + 3.23 = 15.255\]\[12.025 - 3.23 = 8.795\] Of the two numbers involved, the measurement \(3.23\) has the largest significant decimal place - hundredths (or the least number of significant digits after the decimal point). Therefore, the resultant must be rounded so that it is stated upto hundredths place (or two significant digits after the decimal point): \[12.025 + 3.23 = 15.255 \approx 15.26\]\[12.025 - 3.23 = 8.795 \approx 8.80\]

Thus, if the numbers used in calculations are stated with proper significant figures, the final result of calculation to significant figures that indicate the degree of precision and reliability of the result.

EXAMPLES

  1. \( 2.345 + 0.2873 + 1.45 = 4.0823 \approx 4.08\)
    because \(1.45\) has two digits after the decimal point.

  2. \( \displaystyle \frac{2.345 \times 0.2873}{2.50} = 0.2694874 \approx 0.2695 \approx 0.270\)
    because \(2.50\) has three significant figures.

  3. \( \displaystyle \frac{(0.125)^{2}}{2.173} - 3.45 = 0.00719052 - 3.45 = 0.00719 -3.45 = -3.4428095 \approx -3.44\)
    In the first term, \(0.125\) has three significant figures, so we limit the result of the first term to three significant figure (\(0.00719\)). Then, when subtracting second term from the first, we limit the final result to two digits after the decimal because \(3.45\) has two digits after the decimal point.

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