Editing Significant figures (section)

=== 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>