Significant figures

Digits in light blue are significant figures; those in black are not.

Note that identifying the significant figures in a number requires knowing which digits are reliable (e.g., by knowing the measurement or reporting resolution with which the number is obtained or processed) since only reliable digits can be significant; e.g., 3 and 4 in 0.00234 g are not significant if the measurable smallest weight is 0.001 g.[4]

• Non-zero digits within the given measurement or reporting resolution are significant.
  • 91 has two significant figures (9 and 1) if they are measurement-allowed digits.
  • 123.45 has five significant digits (1, 2, 3, 4 and 5) if they are within the measurement resolution. If the resolution is 0.1, then the last digit 5 is not significant.
• Zeros between two significant non-zero digits are significant (significant trapped zeros).
  • 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 (leading zeros) are not significant.
  • If a length measurement gives 0.052 km, then 0.052 km = 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 (3 and 4) if the resolution is 0.00001.
• Zeros to the right of the last non-zero digit (trailing zeros) in a number with the decimal point are significant if they are within the measurement or reporting resolution.
  • 1.200 has four significant figures (1, 2, 0, and 0) if they are allowed by the measurement resolution.
  • 0.0980 has three significant digits (9, 8, and the last zero) if they are within the measurement resolution.
  • 120.000 consists of six significant figures (1, 2, and the four subsequent zeroes).
• Trailing zeros in an integer may or may not be significant, depending on the measurement or reporting resolution.
  • 45,600 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; 45600 m can be expressed as 45.6 km or as 4.56 × 10⁴ m 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 (exact number), then this number is 4.0000... (with infinite trailing zeros to the right of the decimal point). As a result, 4 does not impact the number of significant figures or digits in the result of calculations with it.
• A mathematical or physical constant has significant figures to its known digits.
  • π, as the ratio of the circumference to the diameter of a circle, is 3.14159265358979323... known to more than 62 trillion digits[5] calculated as of 19 August 2021, and that calculated 'π' approximation has that many significant digits, while in practical applications far fewer are used (and π itself has infinite significant digits, as all irrational numbers do). Often 3.14 is used in numerical calculations, i.e. 3 significant decimal digits, with 7 correct binary digits (while the more accurate 22/7 is also used, even though it also only amounts to the same 3 significant correct decimal digits, it has 10 correct binary digits), which is a good enough approximation for many practical uses. Most calculators, and computer programs, can handle 3.141592653589793, 16 decimal digits, that is commonly used in computers and used by NASA for "JPL's highest accuracy calculations, which are for interplanetary navigation".[6] For "the largest size there is: the visible universe [..] you would need 39 or 40 decimal places."[6]
  • The Planck constant is ${\displaystyle h=6.62607015\times 10^{-34}J\cdot s}$ and is defined as an exact value so that it is more properly defined as ${\displaystyle h=6.62607015(0)\times 10^{-34}J\cdot s}$.[7]

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 (just happens coincidentally to be an exact multiple of a hundred) 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, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten).

• 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.[8]

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 1.30×10³. Likewise 0.0123 can be rewritten as 1.23×10⁻². The part of the representation that contains the significant figures (1.30 or 1.23) is known as the significand or mantissa. The digits in the base and exponent (10³ or 10⁻²) are considered exact numbers so for these digits, significant figures are irrelevant.

• Explicitly state the number of significant figures (the abbreviation s.f. is sometimes used): For example "20 000 to 2 s.f." or "20 000 (2 sf)".

• State the expected variability (precision) explicitly with a plus–minus sign, as in 20 000 ± 1%. This also allows specifying a range of precision in-between powers of ten.

It is recommended for a measurement result to include the measurement uncertainty such as ${\displaystyle x_{best}\pm \sigma _{x}}$, where x_best and σ_x are the best estimate and uncertainty in the measurement respectively.[11] x_best can be the average of measured values and σ_x can be the standard deviation or a multiple of the measurement deviation. The rules to write ${\displaystyle x_{best}\pm \sigma _{x}}$are:[12]

• σ_x has only one or two significant figures as more precise uncertainty has no meaning.
  • 1.79 ± 0.06 (correct), 1.79 ± 0.96 (correct), 1.79 ± 1.96 (incorrect).
• The digit positions of the last significant figures in x_best and σ_x are the same, otherwise the consistency is lost. For example, in 1.79 ± 0.067 (incorrect), it does not make sense to have more accurate uncertainty than the best estimate. 1.79 ± 0.9 (incorrect) also does not make sense since the rounding guideline for addition and subtraction below tells that the edges of the true value range are 2.7 and 0.9, that are less accurate than the best estimate.
  • 1.79 ± 0.06 (correct), 1.79 ± 0.96 (correct), 1.79 ± 0.067 (incorrect), 1.79 ± 0.9 (incorrect).

In chemistry (and may also be for other scientific branches), uncertainty may be implied by the last significant figure if it is not explicitly expressed.[2] The implied uncertainty is ± the half of the minimum scale at the last significant figure position. For example, if the volume of water in a bottle is reported as 3.78 L without mentioning uncertainty, then ± 0.005 L measurement uncertainty may be implied. If 2.97 ± 0.07 kg, so the actual weight is somewhere in 2.90 to 3.04 kg, is measured and it is desired to report it with a single number, then 3.0 kg is the best number to report since its implied uncertainty ± 0.05 kg tells the weight range of 2.95 to 3.05 kg that is close to the measurement range. If 2.97 ± 0.09 kg, then 3.0 kg is still the best since, if 3 kg is reported then its implied uncertainty ± 0.5 tells the range of 2.5 to 3.5 kg that is too wide in comparison with the measurement range.

If there is a need to write the implied uncertainty of a number, then it can be written as ${\displaystyle x\pm \sigma _{x}}$with stating it as the implied uncertainty (to prevent readers from recognizing it as the measurement uncertainty), where x and σ_x are the number with an extra zero digit (to follow the rules to write uncertainty above) and the implied uncertainty of it respectively. For example, 6 kg with the implied uncertainty ± 0.5 kg can be stated as 6.0 ± 0.5 kg.

For quantities created from measured quantities via multiplication and division, the calculated result should have as many significant figures as the least number of significant figures among the measured quantities used in the calculation.[13] For example,

• 1.234 × 2 = 2.468 ≈ 2
• 1.234 × 2.0 = 2.468 ≈ 2.5
• 0.01234 × 2 = 0.02468 ≈ 0.02

with one, two, and one significant figures respectively. (2 here is assumed not an exact number.) For the first example, the first multiplication factor has four significant figures and the second has one significant figure. The factor with the fewest or least significant figures is the second one with only one, so the final calculated result should also have one significant figure.

For unit conversion, the implied uncertainty of the result can be unsatisfactorily higher than that in the previous unit if this rounding guideline is followed; For example, 8 inch has the implied uncertainty of ± 0.5 inch = ± 1.27 cm. If it is converted to the centimeter scale and the rounding guideline for multiplication and division is followed, then 20.32 cm ≈ 20 cm with the implied uncertainty of ± 5 cm. If this implied uncertainty is considered as too overestimated, then more proper significant digits in the unit conversion result may be 20.32 cm ≈ 20. cm with the implied uncertainty of ± 0.5 cm.

Another exception of applying the above rounding guideline is to multiply a number by an integer, such as 1.234 × 9. If the above guideline is followed, then the result is rounded as 1.234 × 9.000.... = 11.106 ≈ 11.11. However, this multiplication is essentially adding 1.234 to itself 9 times such as 1.234 + 1.234 + … + 1.234 so the rounding guideline for addition and subtraction described below is more proper rounding approach.[14] As a result, the final answer is 1.234 + 1.234 + … + 1.234 = 11.106 = 11.106 (one significant digit increase).

For quantities created from measured quantities via addition and subtraction, the last significant figure position (e.g., hundreds, tens, ones, tenths, hundredths, and so forth) in the calculated result should be the same as the leftmost or largest digit position among the last significant figures of the measured quantities in the calculation. For example,

• 1.234 + 2 = 3.234 ≈ 3
• 1.234 + 2.0 = 3.234 ≈ 3.2
• 0.01234 + 2 = 2.01234 ≈ 2

with the last significant figures in the ones place, tenths place, and ones place respectively. (2 here is assumed not an exact number.) For the first example, the first term has its last significant figure in the thousandths place and the second term has its last significant figure in the ones place. The leftmost or largest digit position among the last significant figures of these terms is the ones place, so the calculated result should also have its last significant figure in the ones place.

The rule to calculate significant figures for multiplication and division are not the same as the rule for addition and subtraction. For multiplication and division, only the total number of significant figures in each of the factors in the calculation matters; the digit position of the last significant figure in each factor is irrelevant. For addition and subtraction, only the digit position of the last significant figure in each of the terms in the calculation matters; the total number of significant figures in each term is irrelevant. However, greater accuracy will often be obtained if some non-significant digits are maintained in intermediate results which are used in subsequent calculations.

The base-10 logarithm of a normalized number (i.e., a × 10^b with 1 ≤ a < 10 and b as an integer), is rounded such that its decimal part (called mantissa) has as many significant figures as the significant figures in the normalized number.

• log₁₀(3.000 × 10⁴) = log₁₀(10⁴) + log₁₀(3.000) = 4.000000... (exact number so infinite significant digits) + 0.4771212547... = 4.4771212547 ≈ 4.4771.

When taking the antilogarithm of a normalized number, the result is rounded to have as many significant figures as the significant figures in the decimal part of the number to be antiloged.

• 10^4.4771 = 29998.5318119... = 30000 = 3.000 × 10⁴.

If a transcendental function ${\displaystyle f(x)}$ (e.g., the exponential function, the logarithm, and the trigonometric functions) is differentiable at its domain element x, then its number of significant figures (denoted as "significant figures of ${\displaystyle f(x)}$") is approximately related with the number of significant figures in x (denoted as "significant figures of x") by the formula

${\displaystyle {\rm {(significant~figures~of~f(x))}}\approx {\rm {(significant~figures~of~x)}}-\log _{10}\left(\left\vert {{\frac {df(x)}{dx}}{\frac {x}{f(x)}}}\right\vert \right)}$,

where ${\displaystyle \left\vert {{\frac {df(x)}{dx}}{\frac {x}{f(x)}}}\right\vert }$ is the condition number. See the significance arithmetic article to find its derivation.

When performing multiple stage calculations, do not round intermediate stage calculation results; keep as many digits as is practical (at least one more digit than the rounding rule allows per stage) until the end of all the calculations to avoid cumulative rounding errors while tracking or recording the significant figures in each intermediate result. Then, round the final result, for example, to the fewest number of significant figures (for multiplication or division) or leftmost last significant digit position (for addition or subtraction) among the inputs in the final calculation.[15]

• (2.3494 + 1.345) × 1.2 = 3.6944 × 1.2 = 4.43328 ≈ 4.4.
• (2.3494 × 1.345) + 1.2 = 3.159943 + 1.2 = 4.359943 ≈ 4.4.