Significant figures

The significant figures (also called significant digits) of a number are those digits that carry meaning contributing to its precision. This includes all digits except:

• leading and trailing zeros where they serve merely as placeholders to indicate the scale of the number.
• spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports.

The concept of significant digits is often used in connection with rounding. Rounding to n significant digits is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant digits (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases.

Computer representations of floating point numbers typically use a form of rounding to significant digits, but with binary numbers.

The term "significant digits" can also refer to a crude form of error representation based around significant-digit rounding; for this use, see Significance arithmetic.

Identifying significant digits

The rules for identifying significant digits when writing or interpreting numbers are as follows:

• All non-zero digits are considered significant. For example, 91 has two significant digits (9 and 1), while 123.45 has five significant digits (1, 2, 3, 4 and 5).
• Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant digits: 1, 0, 1, 1 and 2.
• Leading zeros are not significant. For example, 0.00052 has two significant digits: 5 and 2.
• Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant digits: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant digits (the zeros before the 1 are not significant). In addition, 120.00 has five significant digits. This convention clarifies the accuracy of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.
• 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 accurate 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. Various conventions exist to address this issue:

• A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, $13{\bar {0}}0$ has three significant digits (and hence indicates that the number is accurate to the nearest ten).

• The last significant digit of a number may be underlined; for example, "20000" has two significant digits.

• A decimal point may be placed after the number; for example "100." indicates specifically that three significant digits are meant.

However, these conventions are not universally used, and it is often necessary to determine from context whether such trailing zeros are intended to be significant. If all else fails, the level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply.

Scientific notation

Generally, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant digits) becomes 1.2×10⁻⁴, and 0.000122300 (six significant digits) becomes 1.22300×10⁻⁴. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant digits is written as 1.300×10³, while 1300 to two significant digits is written as 1.3×10³.

Rounding

Main article: Rounding

To round to n significant digits:

• Start with the leftmost non-zero digit (e.g. the "1" in 1200, or the "2" in 0.0256).
• Keep n digits. Replace the rest with zeros.
• Round up by one if appropriate. For example, if rounding 0.039 to 1 significant digit, the result would be 0.04.
• There are several different rules for handling borderline cases — see Rounding for more details.
  • If the first non-significant digit is a 5 followed by other non-zero digits, round up the last significant digit (away from zero). For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant digits should be written 1.25.
  • If the first non-significant digit is a 5 not followed by any other digits or followed only by zeros, rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant digits, Round half up rounds up to 1.3, while Round half to even rounds to the nearest even number 1.2.

Arithmetic

Main article: Significance arithmetic

For multiplication and division, the result should have as many significant digits as the measured number with the smallest number of significant digits.

For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.

Contrary to the common practice, when performing a calculation, do not follow these guidelines for intermediate results— keep as many digits as is practical to avoid rounding errors. You may verify this rule by reading instructions from college/university professors, such as from Cal Tech.

Importance

The last digit is the least significant digit. This is sometimes referred to in product manuals specifying accuracy. For example, an accuracy of +-(0.3%+0.2) for 73.6 volts means there is an uncertainty of 0.42 volts in the nominal value. For the same accuracy, an uncertainty of 0.2 volts would exist for a reading of zero, and an uncertainty of 0.5 for a reading of 100 volts.

See also

• Engineering notation
• Interval arithmetic

References

• http://www.ligo.caltech.edu/~vsanni/ph3/SignificantFiguresAndMeasurements/SignificantFiguresAndMeasurements.pdf

Additional reading

• ASTM E29-06b, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications

External links

• Significant Figures Calculator
• Interactive Significant Figures Resources