Editing Scientific notation (section)

== Styles ==
=== Normalized notation ===
{{main|Normalized number}}
Any real number can be written in the form {{gaps|''m''|e=''n''}} in many ways: for example, 350 can be written as {{val|3.5E2}} or {{val|35E1}} or {{val|350E0}}.<!-- "Any given real number" includes irrational numbers, which can't be represented exactly. It also includes rational numbers, sometimes requiring recurring decimal representation, e.g., 1/3 = 1.333333... -->

In ''normalized'' scientific notation (called "standard form" in the United Kingdom), the exponent ''n'' is chosen so that the [[absolute value]] of ''m'' remains at least one but less than ten ({{nowrap|1 ≤ {{abs|''m''}} < 10}}). Thus 350 is written as {{val|3.5E2}}. This form allows easy comparison of numbers: numbers with bigger exponents are (due to the normalization) larger than those with smaller exponents, and subtraction of exponents gives an estimate of the number of [[orders of magnitude]] separating the numbers.  It is also the form that is required when using tables of [[common logarithm]]s. In normalized notation, the exponent ''n'' is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as {{val|5E-1}}). The 10 and exponent are often omitted when the exponent is 0. For a series of numbers that are to be added or subtracted (or otherwise compared), it can be convenient to use the same value of ''m'' for all elements of the series.

Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized or differently normalized form, such as [[engineering notation]], is desired. Normalized scientific notation is often called '''[[exponentiation|exponential]] notation''' – although the latter term is more general and also applies when ''m'' is not restricted to the range 1 to 10 (as in engineering notation for instance) and to [[radix|base]]s other than 10 (for example, {{gaps|3.15|base=2|e=20}}).

=== Engineering notation ===
{{Main|Engineering notation}}
Engineering notation (often named "ENG" on scientific calculators) differs from normalized scientific notation in that the exponent ''n'' is restricted to [[multiple (mathematics)|multiples]] of 3. Consequently, the absolute value of ''m'' is in the range 1 ≤ |''m''| < 1000, rather than 1 ≤ |''m''| < 10. Though similar in concept, engineering notation is rarely called scientific notation. Engineering notation allows the numbers to explicitly match their corresponding [[SI prefixes]], which facilitates reading and oral communication. For example, {{val|12.5E-9|u=m}} can be read as "twelve-point-five nanometres" and written as {{val|12.5|u=nm}}, while its scientific notation equivalent {{val|1.25E-8|u=m}} would likely be read out as "one-point-two-five times ten-to-the-negative-eight metres".

=== E notation <span class="anchor" id="Q notation"></span> ===
{| class="wikitable" style="float:right; margin: 0.5em 0 1.3em 1.4em"
!Explicit notation
!E notation
|-
|{{val|2E0}}
|{{codett|2E0}}
|-
|{{val|3E2}}
|{{codett|3E2}}
|-
|{{val|4.321768E3}}
|{{codett|4.321768E3}}
|-
|{{val|-5.3E4}}
|{{codett|-5.3E4}}
|-
|{{val|6.72E9}}
|{{codett|6.72E9}}
|-
|{{val|2E-1}}
|{{codett|2E-1}}
|-
|{{val|9.87E2}}
|{{codett|9.87E2}}
|-
|{{val|7.51E-9}}
|{{codett|7.51E-9}}
|}

[[Calculator]]s and [[computer program]]s typically present very large or small numbers using scientific notation, and some can be configured to uniformly present all numbers that way. Because [[superscript]] exponents like 10<sup>7</sup> can be inconvenient to display or type, the letter "E" or "e" (for "exponent") is often used to represent "times ten raised to the power of", so that the notation {{nowrap|''m''&hairsp;E&hairsp;''n''}} for a decimal significand ''m'' and integer exponent ''n'' means the same as {{nowrap|''m'' × 10<sup>''n''</sup>}}. For example [[Avogadro constant|{{val|6.022E23}}]] is written as {{code|6.022E23}} or {{code|6.022e23}}, and [[Planck length|{{val|1.6E-35}}]] is written as {{code|1.6E-35}} or {{code|1.6e-35}}. While common in computer output, this abbreviated version of scientific notation is discouraged for published documents by some style guides.<ref name="Edwards_2009"/><ref>{{Cite book |url=https://www.worldcat.org/title/ocm62872860 |title=The ACS style guide: effective communication of scientific information |date=2006 |publisher=American Chemical Society; Oxford University Press |isbn=978-0-8412-3999-9 |editor-last=Coghill |editor-first=Anne M. |edition=3rd |location=Washington, DC : Oxford; New York |pages=210 |oclc=ocm62872860 |editor-last2=Garson |editor-first2=Lorrin R. |editor-last3=American Chemical Society}}</ref>

Most popular programming languages – including [[Fortran]], [[C (programming language)|C]]/[[C++]], [[Python (programming language)|Python]], and [[JavaScript]] – use this "E" notation, which comes from Fortran and was present in the first version released for the [[IBM 704]] in 1956.<ref name="Fortran"/> The E&nbsp;notation was already used by the developers of [[SHARE Operating System]] (SOS) for the [[IBM 709]] in 1958.<ref name="DiGri-King_1958"/><!-- Not necessarily the first use, but an early one. --> Later versions of Fortran (at least since [[FORTRAN IV]] as of 1961<!-- Possibly also by some versions of FORTRAN II and III. -->) also use "D" to signify [[double precision]] numbers in scientific notation,<ref name="UH-Manoa"/> and newer Fortran compilers use "Q" to signify [[quadruple precision]].<ref name=FortranQ/> The [[MATLAB]] programming language supports the use of either "E" or "D".

The [[ALGOL|ALGOL 60]] (1960) programming language uses a subscript ten "<sub>10</sub>" character instead of the letter "E", for example: <code class=nowrap>6.022<sub>10</sub>23</code>.<ref name="Naur_1960"/><ref name="Savard_2005"/> This presented a challenge for computer systems which did not provide such a character, so [[ALGOL W]] (1966) replaced the symbol by a single quote, e.g. <code>6.022'+23</code>,<ref name="Bauer-Becker-Graham_1968"/> and some Soviet ALGOL variants allowed the use of the Cyrillic letter "[[Yu (Cyrillic)|ю]]", e.g. {{code|6.022ю+23}}{{Citation needed|date=December 2024}}. Subsequently, the [[ALGOL 68]] programming language provided a choice of characters: {{code|E}}, {{code|e}}, {{code|\}}, {{code|⊥}}, or <code><sub>10</sub></code>.<ref name="Algol_1973"/> The ALGOL "<sub>10</sub>" character was included in the Soviet [[GOST 10859]] text encoding (1964), and was added to [[Unicode]] 5.2 (2009) as {{unichar|23E8|DECIMAL EXPONENT SYMBOL}}.<ref name="Unicode"/>

Some programming languages use other symbols. For instance, [[Simula]] uses {{code|&}} (or {{code|&&}} for [[Double precision|long]]), as in {{code|class=nowrap|6.022&23}}.<ref name="SIMULA_1986"/> [[Mathematica]] supports the shorthand notation {{code|class=nowrap|6.022*^23}} (reserving the letter {{code|E}} for the [[e (mathematical constant)|mathematical constant ''e'']]).

{{anchor|Decapower|D notation}}
[[Image:Avogadro's number in e notation.jpg|thumb|upright=1|A [[Texas Instruments]] [[TI-84 Plus series|TI-84 Plus]] calculator display showing the [[Avogadro constant]] to three significant figures in E notation]]

The first [[pocket calculator]]s supporting scientific notation appeared in 1972.<ref name="TI_1973_SR-10"/> To enter numbers in scientific notation calculators include a button labeled "EXP" or "×10<sup>''x''</sup>", among other variants. The displays of pocket calculators of the 1970s did not display an explicit symbol between significand and exponent; instead, one or more digits were left blank (e.g. <code>6.022 23</code>, as seen in the [[HP-25]]), or a pair of smaller and slightly raised digits were reserved for the exponent (e.g. <code>6.022 <sup>23</sup></code>, as seen in the [[Commodore International|Commodore PR100]]). In 1976, [[Hewlett-Packard]] calculator user Jim Davidson coined the term ''decapower'' for the scientific-notation exponent to distinguish it from "normal" exponents, and suggested the letter "D" as a separator between significand and exponent in typewritten numbers (for example, {{code|6.022D23}}); these gained some currency in the programmable calculator user community.<ref name="Decapower"/> The letters "E" or "D" were used as a scientific-notation separator by [[Sharp Corporation|Sharp]] [[pocket computer]]s released between 1987 and 1995, "E" used for 10-digit numbers and "D" used for 20-digit double-precision numbers.<ref name="Sharp"/> The [[Texas Instruments]] [[TI-83 series|TI-83]] and [[TI-84 Plus series|TI-84]] series of calculators (1996–present) use a [[small caps|small capital]] <code><small>E</small></code> for the separator.<ref name="TI-83"/>

In 1962, Ronald O. Whitaker of Rowco Engineering Co. proposed a power-of-ten system nomenclature where the exponent would be circled, e.g. 6.022&nbsp;×&nbsp;10<sup>3</sup> would be written as "6.022③".<ref name="Whitaker_1962"/>