Significant figures


Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement, if the number of digits exceeds what the measurement instrument can resolve, only the digits that are determined by the resolution are dependable and therefore considered significant.
For instance, if a length measurement yields 114.8 millimetres, using a ruler with the smallest interval between marks at 1 mm, the first three digits are certain and constitute significant figures. Further, digits that are uncertain yet meaningful are also included in the significant figures. In this example, the last digit is likewise considered significant despite its uncertainty. Therefore, this measurement contains four significant figures.
Another example involves a volume measurement of 2.98 litres with an uncertainty of ± 0.05 L. The actual volume falls between 2.93 L and 3.03 L. Even if certain digits are not completely known, they are still significant if they are meaningful, as they indicate the actual volume within an acceptable range of uncertainty. In this case, the actual volume might be 2.94 L or possibly 3.02 L, so all three digits are considered significant. Thus, there are three significant figures in this example.
The following types of digits are not considered significant:
  • Leading zeros. For instance, 013 kg has two significant figures—1 and 3—while the leading zero is insignificant since it does not impact the mass indication; 013 kg is equivalent to 13 kg, rendering the zero unnecessary. Similarly, in the case of 0.056 m, there are two insignificant leading zeros since 0.056 m is the same as 56 mm, thus the leading zeros do not contribute to the length indication.
  • Trailing zeros when they serve as placeholders. In the measurement 1500 m, when the measurement resolution is 100 m, the trailing zeros are insignificant as they simply stand for the tens and ones places. In this instance, 1500 m indicates the length is approximately 1500 m rather than an exact value of 1500 m.
  • Spurious digits that arise from calculations resulting in a higher precision than the original data or a measurement reported with greater precision than the instrument's resolution.
A zero after a decimal is significant, and care should be used when appending such a decimal of zero. Thus, in the case of 1.0, there are two significant figures, whereas 1 has one significant figure.
Among a number's significant digits, the most significant digit is the one with the greatest exponent value, while the least significant digit is the one with the lowest exponent value. For example, in the number "123" the "1" is the most significant digit, representing hundreds, while the "3" is the least significant digit, representing ones.
To avoid conveying a misleading level of precision, numbers are often rounded. For instance, it would create false precision to present a measurement as 12345.25 g when the measuring instrument only provides accuracy to the nearest gram. In this case, the significant figures are the first five digits from the leftmost digit, and the number should be rounded to these significant figures, resulting in 12345 g as the accurate value. The rounding error approximates the numerical resolution or precision. Numbers can also be rounded for simplicity, not necessarily to indicate measurement precision, such as for the sake of expediency in news broadcasts.
Significance arithmetic encompasses a set of approximate rules for preserving significance through calculations. More advanced scientific rules are known as the propagation of uncertainty.
Radix 10 is assumed in the following.

Identifying significant figures

Rules to identify significant figures in a number

Identifying the significant figures in a number requires knowing which digits are meaningful, which requires knowing the resolution with which the number is measured, obtained, or processed. For example, if the measurable smallest mass is 0.001 g, then in a measurement given as 0.00234 g the "4" is not useful and should be discarded, while the "3" is useful and should often be retained.
  • Non-zero digits within the given measurement or reporting resolution are significant.
  • * 91 has two significant figures if they are measurement-allowed digits.
  • * 123.45 has five significant digits if they are within the measurement resolution. If the resolution is, say, 0.1, then the 5 shows that the true value to 4 significant figures is equally likely to be 123.4 or 123.5.
  • Zeros between two significant non-zero digits are significant .
  • * 101.12003 consists of eight significant figures if the resolution is to 0.00001.
  • * 125.340006 has seven significant figures if the resolution is to 0.0001: 1, 2, 5, 3, 4, 0, and 0.
  • Zeros to the left of the first non-zero digit are not significant.
  • * If a length measurement gives, then = 52 m so 5 and 2 are only significant; the leading zeros appear or disappear, depending on which unit is used, so they are not necessary to indicate the measurement scale.
  • * 0.00034 has 2 significant figures if the resolution is 0.00001.
  • Zeros to the right of the last non-zero digit in a number with the decimal point are significant if they are within the measurement or reporting resolution.
  • * 1.200 has four significant figures if they are allowed by the measurement resolution.
  • * 0.0980 has three significant digits if they are within the measurement resolution.
  • * 120.000 consists of six significant figures if, as before, they are within the measurement resolution.
  • Trailing zeros in an integer 'may or may not be significant, depending on the measurement or reporting resolution.
  • * has 3, 4 or 5 significant figures depending on how the last zeros are used. For example, if the length of a road is reported as 45600 m without information about the reporting or measurement resolution, then it is not clear if the road length is precisely measured as 45600 m or if it is a rough estimate. If it is the rough estimation, then only the first three non-zero digits are significant since the trailing zeros are neither reliable nor necessary; can be expressed as or as in scientific notation, and neither expression requires the trailing zeros.
  • An exact number has an infinite number of significant figures.
  • * If the number of apples in a bag is 4, then this number is . As a result, 4 does not impact the number of significant figures or digits in the result of calculations with it.
  • * The Planck constant is defined as exactly.
  • A mathematical or physical constant has significant figures to its known digits.'
  • * π'' is a specific real number with several equivalent definitions. All of the digits in its exact decimal expansion are significant. Although many properties of these digits are known - for example, they do not repeat, because π is irrational - not all of the digits are known. As of March 2024, more than 102 trillion digits have been calculated. A 102 trillion-digit approximation has 102 trillion significant digits. In practical applications, far fewer digits are used. The everyday approximation 3.14 has three significant figures and 7 correct binary digits. The approximation 22/7 has the same three correct decimal digits but has 10 correct binary digits. Most calculators and computer programs can handle a 16-digit approximation sufficient for interplanetary navigation calculations.

    Ways to denote significant figures in an integer with trailing zeros

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 the number 1300 is precise to the nearest unit or if it is only shown to the nearest hundreds due to rounding or uncertainty. Many conventions exist to address this issue. However, these are not universally used and would only be effective if the reader is familiar with the convention:
  • An overline, sometimes also called an overbar, or less accurately, a vinculum, may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 130 has three significant figures.
  • Less often, using a closely related convention, the last significant figure of a number may be underlined; for example, "1300" has two significant figures.
  • A decimal point may be placed after the number; for example "1300." indicates specifically that trailing zeros are meant to be significant.
As the conventions above are not in general use, the following more widely recognized options are available for indicating the significance of number with trailing zeros:
  • Eliminate ambiguous or non-significant zeros by changing the unit prefix in a number with a unit of measurement. For example, the precision of measurement specified as 1300 g is ambiguous, while if stated as 1.30 kg it is not. Likewise 0.0123 L can be rewritten as 12.3 mL.
  • Eliminate ambiguous or non-significant zeros by using Scientific Notation: For example, 1300 with three significant figures becomes. Likewise 0.0123 can be rewritten as. The part of the representation that contains the significant figures is known as the significand or mantissa. The digits in the base and exponent are considered exact numbers so for these digits, significant figures are irrelevant.
  • Explicitly state the number of significant figures : For example "20 000 to 2 s.f." or "20 000 ".
  • State the expected variability explicitly with a plus–minus sign, as in 20 000 ± 1%. This also allows specifying a range of precision in-between powers of ten.