Editing Significant figures (section)

== Arithmetic ==

As there are rules to determine the significant figures in directly ''measured'' quantities, there are also guidelines (not rules) to determine the significant figures in quantities ''calculated'' from these ''measured'' quantities.

Significant figures in ''measured'' quantities are most important in the determination of significant figures in ''calculated quantities'' with them. A mathematical or physical constant (e.g., {{math|π}} in the formula for the [[area of a disk|area of a circle]] with radius {{math|''r''}} as {{math|π''r''<sup>2</sup>}}) has no effect on the determination of the significant figures in the result of a calculation with it if its known digits are equal to or more than the significant figures in the measured quantities used in the calculation. An exact number such as {{math|{{sfrac|1|2}}}} in the formula for the [[kinetic energy]] of a mass {{math|''m''}} with velocity {{math|''v''}} as {{math|{{sfrac|1|2}}''mv''<sup>2</sup>}} has no bearing on the significant figures in the calculated kinetic energy since its number of significant figures is infinite (0.500000...).

The guidelines described below are intended to avoid a calculation result more precise than the measured quantities, but it does not ensure the resulted implied uncertainty close enough to the measured uncertainties. This problem can be seen in unit conversion. If the guidelines give the implied uncertainty too far from the measured ones, then it may be needed to decide significant digits that give comparable uncertainty.

=== Multiplication and division ===
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.<ref>{{cite web|url=http://chemistry.bd.psu.edu/jircitano/sigfigs.html|title=Significant Figure Rules|publisher=Penn State University}}</ref> For example, 
:* 1.234 × 2 = {{overline|2}}.468 ≈ 2
:* 1.234 × 2.0 = 2.{{overline|4}}68 ≈ 2.5
:* 0.01234 × 2 = 0.0{{overline|2}}468 ≈ 0.02
:* 0.012345678 / 0.00234 = 5.2{{overline|7}}59 ≈ 5.28
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.

==== Exception ====
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&nbsp;inch has the implied uncertainty of ±&nbsp;0.5&nbsp;inch = ±&nbsp;1.27&nbsp;cm. If it is converted to the centimeter scale and the rounding guideline for multiplication and division is followed, then {{overline|2}}0.32&nbsp;cm ≈ 20&nbsp;cm with the implied uncertainty of ±&nbsp;5&nbsp;cm. If this implied uncertainty is considered as too overestimated, then more proper significant digits in the unit conversion result may be 2{{overline|0}}.32&nbsp;cm ≈ 20.&nbsp;cm with the implied uncertainty of ±&nbsp;0.5&nbsp;cm.

Another exception of applying the above rounding guideline is to multiply a number by an integer, such as 1.234 ×&nbsp;9. If the above guideline is followed, then the result is rounded as 1.234 × 9.000.... = 11.1{{overline|0}}6 ≈ 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.<ref>{{Cite web|date=2017-06-16|title=Uncertainty in Measurement- Significant Figures|url=https://chem.libretexts.org/@go/page/83744|website=Chemistry - LibreTexts}}</ref> As a result, the final answer is 1.234 + 1.234 + … + 1.234 = 11.10{{overline|6}} = 11.106 (one significant digit increase).

=== Addition and subtraction of significant figures ===
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 = {{overline|3}}.234 ≈ 3
:* 1.234 + 2.0 = 3.{{overline|2}}34 ≈ 3.2
:* 0.01234 + 2 = {{overline|2}}.01234 ≈ 2
:* 12000 + 77 = 1{{overline|2}}077 ≈ 12000
with the last significant figures in the ''ones'' place, ''tenths'' place, ''ones'' place, and ''thousands'' 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.{{citation needed|date=July 2020}}  However, greater accuracy will often be obtained if some non-significant digits are maintained in intermediate results which are used in subsequent calculations.{{citation needed|date=July 2020}}

=== Logarithm and antilogarithm ===
The [[base of a logarithm|base]]-10 [[logarithm]] of a [[normalized number]] (i.e., ''a'' × 10<sup>''b''</sup> with 1 ≤ ''a'' < 10 and ''b'' as an integer), is rounded such that its decimal part (called [[Mantissa (logarithm)|mantissa]]) has as many significant figures as the significant figures in the normalized number.

* log<sub>10</sub>(3.000 × 10<sup>4</sup>) = log<sub>10</sub>(10<sup>4</sup>) + log<sub>10</sub>(3.000) = 4.000000... (exact number so infinite significant digits) + 0.477{{overline|1}}212547... = 4.477{{overline|1}}212547 ≈ 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<sup>4.4771</sup> = 299{{overline|9}}8.5318119... = 30000 = 3.000 × 10<sup>4</sup>.

=== Transcendental functions ===
If a [[transcendental function]] <math>f(x)</math> (e.g., the [[exponential function]], the [[logarithm]], and the [[Trigonometric function|trigonometric functions]]) is differentiable at its domain element 'x', then its number of significant figures (denoted as "significant figures of <math>f(x)</math>") is approximately related with the number of significant figures in ''x'' (denoted as "significant figures of ''x''") by the formula

<math> {\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 ) </math>,

where <math> \left\vert{\frac{df(x)}{dx} \frac{x}{f(x)}}\right\vert </math> is the [[condition number]].

=== Round only on the final calculation result ===
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.<ref>{{cite web |url= http://www.ligo.caltech.edu/~vsanni/ph3/SignificantFiguresAndMeasurements/SignificantFiguresAndMeasurements.pdf |archive-url= https://web.archive.org/web/20130618184216/http://www.ligo.caltech.edu/~vsanni/ph3/SignificantFiguresAndMeasurements/SignificantFiguresAndMeasurements.pdf |archive-date= June 18, 2013 |title= Measurements and Significant Figures (Draft) |first= Virgínio |last= de Oliveira Sannibale |year= 2001 |work= Freshman Physics Laboratory |publisher= California Institute of Technology, Physics Mathematics And Astronomy Division }}</ref>

* (2.3494 + 1.345) × 1.2 = 3.69{{overline|4}}4 × 1.2 = 4.{{overline|4}}3328  ≈ 4.4.
* (2.3494 × 1.345) + 1.2 = 3.15{{overline|9}}943 + 1.2 = 4.{{overline|3}}59943  ≈ 4.4.