Significant Figures

Key Concepts

•   Significant figures, or significant digits, establish the value of a number.

•   Zeros shown merely to locate a decimal point are NOT significant figures

•   Zeros located to the right of another number after a decimal point are significant

•   The last significant figure on the right is the one which is somewhat uncertain

•   An exact number, such as the number of objects counted, can be considered to have an infinite number of zeros after the decimal point, all of which are significant

•   It is impossible to tell how many significant figures are in a large number with zeros to the left of the decimal point without converting the number to scientific notation

•   To find the number of significant figures in a given number:

1. count all the digits starting at the first non-zero digit on the left
2. for a number written in scientific notation count only the digits in the coefficient

•   When adding or subtracting numbers, the number of digits to the right of the decimal point in the result should be the same as the number of digits to the right of the decimal point in the number with the fewest digits to the right of the decimal point

•   When multiplying or dividing numbers, the number of significant figures in the result is the same as the least number of significant figures in any of the multiplied or divided terms

Examples

Finding the Number of Significant Figures in:

(a) 5 mL

    Count all the digits starting at the first non-zero digit on the left.

    1 significant figure

(b) 5.2 g

    Count all the digits starting at the first non-zero digit on the left.

    2 significant figures

(c) 5.0 kg

    Count all the digits starting at the first non-zero digit on the left.

    2 significant figures

(d) 5.000 L

    Count all the digits starting at the first non-zero digit on the left.

    4 significant figures

(e) 0.005 m

    Count all the digits starting at the first non-zero digit on the left.

    1 significant figure

(f) 5 football players

    An exact number, such as the number of objects counted, can be considered to have an infinite number of zeros after the decimal point, all of which are significant.

    infinite number of significant figures

(g) 500 mm

    It is impossible to tell how many significant figures are in a large number with zeros to the left of the decimal point without converting the number to scientific notation.

    unknown number of significant figure

(h) 5.00 x 10³ g

    For a number written in scientific notation count only the digits in the coefficient.

    3 significant figures

Finding the number of Significant Figures in the Result of Calculations:

(a) 12.47g + 7g

    When adding or subtracting numbers, the number of digits to the right of the decimal point in the result should be the same as the number of digits to the right of the decimal point in the number with the fewest digits to the right of the decimal point.

    "7" has no numbers to the right of the decimal point so the final result will also have no numbers to the right of the decimal point

&bnsp   12.47 + 7 = 19 (rounded down to 19 from 19.47 because the number after the decimal point is less than 5)

(b) 32.56mm - 4.9mm

    When adding or subtracting numbers, the number of digits to the right of the decimal point in the result should be the same as the number of digits to the right of the decimal point in the number with the fewest digits to the right of the decimal point.

    "4.1" has one number to the right of the decimal point so the final result will also have one number to the right of the decimal point

&bnsp   32.56 - 4.9 = 27.7 (rounded up to 27.7 from 27.66 because the number to the right of the last significant figure was greater than 5)

(c) 1.473 ÷ 2.6

    When multiplying or dividing numbers, the number of significant figures in the result is the same as the least number of significant figures in any of the multiplied or divided terms.

    1.473 has 4 significant figures, 2.6 has only 2 significant figures, the result will have 2 significant figures.

    1.473 ÷ 2.6 = 0.57 (rounded up to 0.57 from 0.5665 because the number to the right of the last significant figure was greater than 5)

(d) 4.1 x 10³ x 8.635 x 10²

    When multiplying or dividing numbers, the number of significant figures in the result is the same as the least number of significant figures in any of the multiplied or divided terms.

    4.1 x 10³ has 2 significant figures, 8.635 x 10² has 4 significant figures, the result will have 2 significant figures.

    3.5 x 10⁶ (rounded down to 3.5 x 10⁶ from 3.54 x 10⁶ because the number to the right of the last significant figure is less than 5)

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Practice

Scientific notation may be used for large results or if the number of significant digits would be ambiguous otherwise.

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