Significant Figures

When physical quantities are measured, the measured values are known only to within the limits of the experimental uncertainty. The value of this uncertainty can depend on various factors, such as the quality of the apparatus, the skill of the experimenter, and the number of measurements performed.
Suppose that we are asked to measure the area of a computer disk label using a meter stick as a measuring instrument. Let us assume that the accuracy to which we can measure with this stick is $\pm0.1 cm$. If the length of the label is measured to be 5.5 cm, we can claim only that its length lies somewhere between 5.4 cm and 5.6 cm. In this case, we say that the measured value has two significant figures.

Significant figures are each of the digits of a number that are used to express it to the required degree of accuracy, starting from the first non-zero digit.

The precision of a reading can be determined by the number of significant figures (digits) present. Significant digits are those integers (0 to 9) that can be assumed to be accurate for the measurement being made. The result is that all nonzero numbers are considered significant, with zeros being significant in only some cases.

Significant Figures Rules

Specifically, the rules for identifying significant figures when writing or interpreting numbers are as follows:
  • All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).
  • Zeros appearing anywhere between two non-zero digits are significant. Example: 101.1203 has seven significant figures: (1, 0, 1, 1, 2, 0 and 3).
  • Leading zeros are not significant. For example, 0.00052 has two significant figures: (5 and 2).
  • Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: (1, 2, 2, 3, 0 and 0.) The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23 then it might be understood that only two decimal places of precision are available. Stating the result as 12.2300 makes clear that it is precise to four decimal places (in this case, six significant figures).
  • The significance of trailing zeros in a number not containing a decimal point can be ambiguous.
    For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty.
Many conventions exist to address this issue:
  • An overline may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 13$\bar{0}$0 has three significant figures (and hence indicates that the number is precise to the nearest ten).
  • A decimal point may be placed after the number; For example "100." indicates specifically that three significant figures are meant.

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