The significant figures of a number are digits that carry meaning contributing to its measurement resolution. This includes all digits except:
All leading zeros; Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures); and Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.
Significance arithmetic are approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as propagation of uncertainty. Numbers are often rounded to avoid reporting insignificant figures. For example, it would create false precision to express a measurement as 12.34500 kg (which has seven significant figures) if the scales only measured to the nearest gram and gave a reading of 12.345 kg (which has five significant figures). Numbers can also be rounded merely for simplicity rather than to indicate a given precision of measurement, for example, to make them faster to pronounce in news broadcasts.
1 Identifying significant figures
1.1 Concise rules 1.2 Significant figures rules explained 1.3 Scientific notation
Identifying significant figures Concise rules
All non-zero digits are significant: 1, 2, 3, 4, 5, 6, 7, 8, 9. Zeros between non-zero digits are significant: 102, 2005, 50009. Leading zeros are never significant: 0.02, 001.887, 0.000515. In a number with a decimal point, trailing zeros (those to the right of the last non-zero digit) are significant: 2.02000, 5.400, 57.5400. In a number without a decimal point, trailing zeros may or may not be significant. More information through additional graphical symbols or explicit information on errors is needed to clarify the significance of trailing zeros.
Significant figures rules explained 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, 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, "2000" has two significant figures.
A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.
In the combination of a number and a unit of measurement, the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as 1300 g is ambiguous, while in a mass of 13 hg or 1.3 kg it is not.
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 with a plus-minus sign, as in 20 000 ± 1%, so that significant-figures rules do not apply. This also allows specifying a precision in-between powers of ten (or whatever the base power of the numbering system is).
In most cases, 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, 6996120000000000000♠0.00012 (two
significant figures) becomes 6996120000000000000♠1.2×10−4, and
6997122299999999999♠0.00122300 (six significant figures) becomes
6997122300000000000♠1.22300×10−3. In particular, the potential
ambiguity about the significance of trailing zeros is eliminated. For
example, 7003130000000000000♠1300 to four significant figures is
written as 7003130000000000000♠1.300×103, while
7003130000000000000♠1300 to two significant figures is written as
The part of the representation that contains the significant figures
(as opposed to the base or the exponent) is known as the significand
Identify the significant figures before rounding. These are the n consecutive digits beginning with the first non-zero digit. If the digit immediately to the right of the last significant figure is greater than 5 or is a 5 followed by other non-zero digits, add 1 to the last significant figure. For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant figures should be written 1.25. If the digit immediately to the right of the last significant figure 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 figures:
Round half up (also known as "5/4") rounds up to 1.3. This is the default rounding method implied in many disciplines if not specified. Round half to even, which rounds to the nearest even number, rounds down to 1.2 in this case. The same strategy applied to 1.35 would instead round up to 1.4.
Replace non-significant figures in front of the decimal point by zeros. Drop all the digits after the decimal point to the right of the significant figures (do not replace them with zeros).
In financial calculations, a number is often rounded to a given number
of places (for example, to two places after the decimal separator for
many world currencies).
Precision Rounded to significant figures Rounded to decimal places
6 12.3450 12.345000
5 12.345 12.34500
4 12.35 12.3450
3 12.3 12.345
2 12 12.35
1 10 12.3
0 N/A 12
Another example for 0.012345:
Precision Rounded to significant figures Rounded to decimal places
7 0.01234500 0.0123450
6 0.0123450 0.012345
5 0.012345 0.01235
4 0.01235 0.0123
3 0.0123 0.012
2 0.012 0.01
1 0.01 0.0
0 N/A 0
The representation of a positive number x to a precision of p significant digits has a numerical value that is given by the formula:
round(10−n⋅x)⋅10n, where n = floor(log10 x) + 1 − p.
For negative numbers, the formula can be used on the absolute value; for zero, no transformation is necessary. Note that the result may need to be written with one of the above conventions explained in the section "Identifying significant figures" to indicate the actual number of significant digits if the result includes for example trailing significant zeros. Arithmetic Main article: Significance arithmetic As there are rules for determining the number of significant figures in directly measured quantities, there are rules for determining the number of significant figures in quantities calculated from these measured quantities. Only measured quantities figure into the determination of the number of significant figures in calculated quantities. Exact mathematical quantities like the π in the formula for the area of a circle with radius r, πr2 has no effect on the number of significant figures in the final calculated area. Similarly the ½ in the formula for the kinetic energy of a mass m with velocity v, ½mv2, has no bearing on the number of significant figures in the final calculated kinetic energy. The constants π and ½ are considered to have an infinite number of significant figures. For quantities created from measured quantities by multiplication and division, the calculated result should have as many significant figures as the measured number with the least number of significant figures. For example,
1.234 × 2.0 = 2.468… ≈ 2.5,
with only two significant figures. The first factor has four significant figures and the second has two significant figures. The factor with the least number of significant figures is the second one with only two, so the final calculated result should also have a total of two significant figures. For quantities created from measured quantities by addition and subtraction, the last significant decimal place (hundreds, tens, ones, tenths, and so forth) in the calculated result should be the same as the leftmost or largest decimal place of the last significant figure out of all the measured quantities in the terms of the sum. For example,
100.0 + 1.234 = 101.234… ≈ 101.2
with the last significant figure in the tenths place. The first term
has its last significant figure in the tenths place and the second
term has its last significant figure in the thousandths place. The
leftmost of the decimal places of the last significant figure out of
all the terms of the sum is the tenths place from the first term, so
the calculated result should also have its last significant figure in
the tenths place.
The rules for calculating significant figures for multiplication and
division are opposite to the rules for addition and subtraction. For
multiplication and division, only the total number of significant
figures in each of the factors matter; the decimal place of the last
significant figure in each factor is irrelevant. For addition and
subtraction, only the decimal place of the last significant figure in
each of the terms matters; the total number of significant figures in
each term is irrelevant.
In a base 10 logarithm of a normalized number, the result should be
rounded to the number of significant figures in the normalized number.
For example, log10(3.000×104) = log10(104) + log10(3.000) ≈ 4 +
0.47712125472, should be rounded to 4.4771.
When taking antilogarithms, the resulting number should have as many
significant figures as the mantissa in the logarithm.
When performing a calculation, do not follow these guidelines for
intermediate results; keep as many digits as is practical (at least 1
more than implied by the precision of the final result) until the end
of calculation to avoid cumulative rounding errors.
When using a ruler, initially use the smallest mark as the first
estimated digit. For example, if a ruler's smallest mark is cm, and
4.5 cm is read, it is 4.5 (±0.1 cm) or 4.4 –
It is possible that the overall length of a ruler may not be accurate
to the degree of the smallest mark and the marks may be imperfectly
spaced within each unit. However assuming a normal good quality ruler,
it should be possible to estimate tenths between the nearest two marks
to achieve an extra decimal place of accuracy. Failing to do this adds
the error in reading the ruler to any error in the calibration of the
Main article: Estimation
When estimating the proportion of individuals carrying some particular
characteristic in a population, from a random sample of that
population, the number of significant figures should not exceed the
maximum precision allowed by that sample size. The correct number of
significant figures is given by the order of magnitude of sample size.
This can be found by taking the base 10 logarithm of sample size and
rounding to the nearest integer.
For example, in a poll of 120 randomly chosen viewers of a regularly
visited web page we find that 10 people disagree with a proposition on
that web page. The order of magnitude of our sample size is Log10(120)
= 2.0791812460..., which rounds to 2. Our estimated proportion of
people who disagree with the proposition is therefore 0.083, or 8.3%,
with 2 significant figures. This is because in different samples of
120 people from this population, our estimate would vary in units of
1/120, and any additional figures would misrepresent the size of our
sample by giving spurious precision. To interpret our estimate of the
number of viewers who disagree with the proposition we should then
calculate some measure of our confidence in this estimate.
Relationship to accuracy and precision in measurement
Main article: Accuracy and precision
Traditionally, in various technical fields, "accuracy" refers to the
closeness of a given measurement to its true value; "precision" refers
to the stability of that measurement when repeated many times. Hoping
to reflect the way the term "accuracy" is actually used in the
scientific community, there is a more recent standard, ISO 5725, which
keeps the same definition of precision but defines the term "trueness"
as the closeness of a given measurement to its true value and uses the
term "accuracy" as the combination of trueness and precision. (See the
Accuracy and precision
Accuracy and precision
^ Chemistry in the Community; Kendall-Hunt:Dubuque, IA 1988
^ Giving a precise definition for the number of correct significant
digits is surprisingly subtle, see Higham, Nicholas (2002). Accuracy
and Stability of Numerical Algorithms (PDF) (2nd ed.). SIAM.
^ Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000).
Chemistry. Austin, Texas: Holt Rinehart Winston. p. 59.
^ Engelbrecht, Nancy; et al. (1990). "
Significant Figures Vid