Significant Figures: Meaning and Counting Rules

Units And Measurements • Physics

Concept Summary

Every measurement involves errors, so the result must be reported in a way that indicates its precision. A reported result normally includes all the digits known reliably plus the first uncertain digit; together these are called significant digits or significant figures. For example, a pendulum period of 1.62 s has three significant figures because 1 and 6 are certain while 2 is uncertain. Reporting more digits than the significant ones is superfluous and misleading because it suggests a precision that does not exist. Significant figures depend on the least count of the measuring instrument, and a change of units does not change their number: 2.308 cm, 0.02308 m, 23.08 mm and 23080 µm all have four. The rules are as follows. All non-zero digits are significant. Zeros between two non-zero digits are significant. For a number less than 1, zeros to the right of the decimal point but left of the first non-zero digit are not significant. Trailing zeros in a number without a decimal point are not significant, whereas trailing zeros in a number with a decimal point are significant. To remove ambiguity, report measurements in scientific notation \(a\times10^{b}\); all digits in \(a\) are significant. Exact numbers, like the 2 in \(s = 2\pi r\), have infinite significant figures. The exponent \(b\) of 10 after rounding \(a\) is the order of magnitude.

Key Notes
  • Significant figures = all reliably known digits + the first uncertain digit.
  • All non-zero digits are significant (e.g. 287.5 has 4).
  • Zeros between two non-zero digits are always significant (e.g. 2.308 has 4).
  • Leading zeros of a number less than 1 are not significant (0.002308 has 4).
  • Trailing zeros: not significant without a decimal point (12300 has 3); significant with a decimal point (3.500 and 0.06900 have 4).
  • Changing units never changes the number of significant figures: 4.700 m = 470.0 cm = \(4.700\times10^{3}\) mm = 0.004700 km.
  • In scientific notation \(a\times10^{b}\), every digit in \(a\) is significant and the power of 10 is irrelevant.
  • Exact numbers (such as 2 in \(r = d/2\), or \(n\) in \(T = t/n\)) have infinite significant figures.
  • Order of magnitude: round \(a\) to 1 if \(a\le5\) and to 10 if \(5
Important Equations
\[ \text{Scientific notation: } N = a \times 10^{b},\quad 1 \le a < 10 \] \[ 4.700\ \mathrm{m} = 4.700\times10^{2}\ \mathrm{cm} = 4.700\times10^{3}\ \mathrm{mm} = 4.700\times10^{-3}\ \mathrm{km} \]

Order of magnitude: diameter of Earth \(1.28\times10^{7}\ \mathrm{m} \sim 10^{7}\ \mathrm{m}\); diameter of hydrogen atom \(1.06\times10^{-10}\ \mathrm{m} \sim 10^{-10}\ \mathrm{m}\).

\[ \text{Difference in order of magnitude} = 7 - (-10) = 17 \]
Flow Chart
Take the number
Mind Map
  • Significant Figures
    • Definition: certain digits + first uncertain digit
    • Depends on least count of instrument
    • Rules
      • Non-zero digits: significant
      • Zeros between non-zeros: significant
      • Leading zeros: not significant
      • Trailing zeros: depends on decimal point
    • Scientific notation
      • \(a\times10^{b}\)
      • Removes ambiguity
    • Exact numbers: infinite significant figures
    • Order of magnitude: power of 10 nearest to the number
Graphic Organiser
NumberSignificant FiguresReason
287.5 cm4All non-zero digits
2.308 cm / 0.02308 m4Zero between non-zeros counts; leading zeros do not
12300 cm3Trailing zeros without decimal point
3.5004Trailing zeros with decimal point
0.069004Leading zeros ignored; trailing zeros counted
\(4.700\times10^{2}\)4All digits in base number are significant