Editing Significant figures (section)

== Identifying significant figures ==
{{Refimprove section|date=May 2021}}
=== Rules to identify significant figures in a number ===
[[File:SigFigs.svg|thumb|upright|Digits in light blue are significant figures; those in black are not.]]
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&nbsp;g, then in a measurement given as 0.00234&nbsp;g the &quot;4&quot; is not useful and should be discarded, while the &quot;3&quot; is useful and should often be retained.<ref>Giving a precise definition for the number of correct significant digits is not a straightforward matter: see {{cite book|last=Higham|first=Nicholas|url=http://ftp.demec.ufpr.br/CFD/bibliografia/Higham_2002_Accuracy%20and%20Stability%20of%20Numerical%20Algorithms.pdf|title=Accuracy and Stability of Numerical Algorithms|publisher=SIAM|year=2002|edition=2nd|pages=3–5}}</ref>
* '''Non-zero digits within the given measurement or reporting resolution''' are '''significant'''.
** 91 has two significant figures (9 and&nbsp;1) if they are measurement-allowed digits.
** 123.45 has five significant digits (1, 2, 3, 4 and&nbsp;5) if they are within the measurement resolution. If the resolution is, say, 0.1, then the 5 shows that the true value to 4 sig figs is equally likely to be 123.4 or 123.5.
* '''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&nbsp;0.
* '''Zeros to the left of the first non-zero digit''' ([[leading zero]]s) are '''<u>not</u> significant'''.
** If a length measurement gives {{val|0.052|u=km}}, then {{val|0.052|u=km}} = 52&nbsp;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&nbsp;4) if the resolution is 0.00001. 
* '''Zeros to the right of the last non-zero digit ([[trailing zero]]s) 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&nbsp;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) if, as before, they are within the measurement resolution.
* '''Trailing zeros in an integer''' '''may or may <u>not</u> be significant''', depending on the measurement or reporting resolution.
** {{val|45600}} 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&nbsp;m without information about the reporting or measurement resolution, then it is not clear if the road length is precisely measured as 45600&nbsp;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; {{val|45600|u=m}} can be expressed as {{val|45.6|u=km}} or as {{val|4.56|e=4|u=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 {{val|4.0000|end=...}} (with infinite trailing zeros to the right of the decimal point). As a result, 4&nbsp;does not impact the number of significant figures or digits in the result of calculations with it.
** The [[Planck constant]] is defined as exactly {{nowrap|1=''h'' = {{val|6.62607015|e=-34|u=J.s}}}}.<ref>{{cite web |url=https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf |date=2018-11-16 |work=BIPM |access-date=2018-11-20 |language=en-GB |title=Resolutions of the 26th CGPM |archive-url=https://web.archive.org/web/20181119214326/https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf |archive-date=2018-11-19 |url-status=dead }}</ref>
* '''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 {{val|3.14159|end=...}} are significant. Although many properties of these digits are known &ndash; for example, they do not repeat, because ''π'' is irrational &ndash; not all of the digits are known. As of March 2024, more than 102&nbsp;trillion digits<ref>[https://www.numberworld.org/y-cruncher/records/2024_2_27_pi.txt "y-cruncher validation file"]</ref> have been calculated. A 102 trillion-digit approximation has&nbsp;102 trillion significant digits. In practical applications, far fewer digits are used. The everyday approximation 3.14 has three significant figures and 7&nbsp;correct [[binary number|binary]] digits. The approximation 22/7 has the same three correct decimal digits but has 10&nbsp;correct binary digits. Most calculators and computer programs can handle a 16-digit approximation sufficient for interplanetary navigation calculations.<ref name="NASA/JPL">{{cite web |title=How Many Decimals of Pi Do We Really Need? - Edu News |url=https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ |access-date=2021-10-25 |website=NASA/JPL Edu }}</ref>

=== 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 (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 (symbol)|vinculum]], may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, 13{{overline|0}}0 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 [[underline]]d; for example, "1<span style="text-decoration: underline;">3</span>00" 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.<ref name="Chemistry Significant Figures">{{cite book |last1= Myers |first1= R. Thomas |last2= Oldham |first2= Keith B. |last3= Tocci |first3= Salvatore |title= Chemistry |year= 2000 |publisher= Holt Rinehart Winston |location= Austin, Texas |isbn= 0-03-052002-9 |page= [https://archive.org/details/holtchemistryvis00myer/page/59 59] |url-access= registration |url= https://archive.org/details/holtchemistryvis00myer/page/59}}</ref>

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&nbsp;g is ambiguous, while if stated as 1.30&nbsp;kg it is not. Likewise 0.0123&nbsp;L can be rewritten as 12.3&nbsp;mL.
*Eliminate ambiguous or non-significant zeros by using Scientific Notation: For example, 1300 with three significant figures becomes {{val|1.30|e=3}}. Likewise 0.0123 can be rewritten as {{val|1.23|e=-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 ({{val||e=3}} or {{val||e=-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&nbsp;000 to 2&nbsp;s.f." or "20&nbsp;000 (2&nbsp;sf)".
*State the expected variability (precision) explicitly with a [[plus–minus sign]], as in 20&nbsp;000&nbsp;±&nbsp;1%. This also allows specifying a range of precision in-between powers of ten.