Editing Scientific notation

{{Short description|Concise notation for large or small numbers}}
{{About|a numeric notation|the musical notation|Scientific pitch notation}}
{{Redirect|E notation|the series of preferred numbers|E series of preferred numbers{{!}}E series|the food additive codes|E number}}
{{Use dmy dates|date=June 2022|cs1-dates=y}}
{{Use list-defined references|date=December 2022}}

'''Scientific notation''' is a way of expressing [[real numbers|numbers]] that are too large or too small to be conveniently written in [[decimal form]], since to do so would require writing out an inconveniently long string of digits. It may be referred to as '''scientific form''' or '''standard index form''', or '''standard form''' in the United Kingdom. This [[base ten]] notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certain [[arithmetic operations]]. On [[scientific calculators]], it is usually known as "SCI" display mode.

{| class="wikitable" style="float:right; margin:5px;"
!Decimal notation
!Scientific notation
|-
|{{val|2}}
|{{val|2E0}}
|-
|{{val|300}}
|{{val|3E2}}
|-
|{{val|4321.768}}
|{{val|4.321768E3}}
|-
|{{val|−53000}}
|{{val|-5.3E4}}
|-
|{{val|6720000000}}
|{{val|6.72E9}}
|-
|{{val|0.2}}
|{{val|2E-1}}
|-
|{{val|987}}
|{{val|9.87E2}}
|- 
|{{val|0.00000000751}}
|{{val|7.51E-9}}
|}
In scientific notation, nonzero numbers are written in the form 
{{block indent | em = 1.5 | text = ''m'' × 10{{sup|''n''}}}}
or ''m'' times ten raised to the power of ''n'', where ''n'' is an [[integer]], and the [[coefficient]] ''m'' is a nonzero [[real number]] (usually between 1 and 10 in absolute value, and nearly always written as a [[decimal|terminating decimal]]). The integer ''n'' is called the [[exponent]] and the real number ''m'' is called the ''[[significand]]'' or ''mantissa''.<ref name="Calio_2017"/> The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the [[fractional part]] of the [[common logarithm]]. <!-- These are actually floating-point terms, not scientific notation. --> If the number is negative then a minus sign precedes ''m'', as in ordinary decimal notation. In [[#Normalized notation|normalized notation]], the exponent is chosen so that the [[absolute value]] (modulus) of the significand ''m'' is at least 1 but less than 10.

[[Decimal floating point]] is a computer arithmetic system closely related to scientific notation.

== History ==
{{See also|Exponentiation#20th century}}
For performing calculations with a [[slide rule]], standard form expression is required. Thus, the use of scientific notation increased as engineers and educators used that tool. See [[Slide rule#History]].

== 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"/>

== Significant figures ==
{{Main|Significant figures}}
A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes [[significant figures#Identifying significant digits|indicated to be significant]]. Leading and trailing zeroes are not significant digits, because they exist only to show the scale of the number. Unfortunately, this leads to ambiguity. The number {{val|1230400}} is usually read to have five significant figures: 1, 2, 3, 0, and 4, the final two zeroes serving only as placeholders and adding no precision. The same number, however, would be used if the last two digits were also measured precisely and found to equal 0 – seven significant figures.

When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the placeholding zeroes are no longer required. Thus {{val|1230400}} would become {{val|1.2304E6}} if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as {{val|1.23040E6}} or {{val|1.230400E6}}. Thus, an additional advantage of scientific notation is that the number of significant figures is unambiguous.

=== Estimated final digits ===
It is customary in scientific measurement to record all the definitely known digits from the measurement and to estimate at least one additional digit if there is any information at all available on its value. The resulting number contains more information than it would without the extra digit, which may be considered a significant digit because it conveys some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together).

Additional information about precision can be conveyed through additional notation. It is often useful to know how exact the final digit or digits are. For instance, the accepted value of the mass of the [[proton]] can properly be expressed as {{val|1.67262192369E-27|(51)|u=kg}}, which is shorthand for {{val|1.67262192369E-27|0.00000000051|u=kg}}. However it is still unclear whether the error ({{val|5.1E-37}} in this case) is the maximum possible error, [[standard error]], or some other [[confidence interval]].

== Use of spaces ==
In normalized scientific notation, in E notation, and in engineering notation, the [[Space (punctuation)|space]] (which in [[typesetting]] may be represented by a normal width space or a [[thin space]]) that is allowed ''only'' before and after "×" or in front of "E" is sometimes omitted, though it is less common to do so before the alphabetical character.<ref name="Samples"/>

== Further examples of scientific notation ==
* An [[electron]]'s mass is about {{val|0.000000000000000000000000000000910938356|u=kg}}.<ref name="CODATA2014Full"/> In scientific notation, this is written {{val|9.10938356E-31|u=kg}}.
* The [[Earth]]'s [[mass]] is about {{val|5972400000000000000000000|u=kg}}.<ref name="Luzum_2011"/> In scientific notation, this is written {{val|5.9724E24|u=kg}}.
* The [[Earth#Size and shape|Earth's circumference]] is approximately {{val|40000000|u=m}}.<ref name="Lide_2000"/> In scientific notation, this is {{val|4E7|u=m}}. In engineering notation, this is written {{val|40E6|u=m}}. In [[International System of Units|SI writing style]], this may be written {{val|40|u=Mm}} (''{{val|40|u=megametres}}'').
* An [[inch]] is defined as ''exactly'' {{val|25.4|u=mm}}. Using scientific notation, this value can be uniformly expressed to any desired precision, from the nearest tenth of a [[millimeter]] {{val|2.54E1|u=mm}} to the nearest [[nanometer]] {{val|2.5400000E1|u=mm}}, or beyond.
* [[Hyperinflation]] means that too much money is put into circulation, perhaps by printing banknotes, chasing too few goods. It is sometimes defined as inflation of 50% or more in a single month. In such conditions, money rapidly loses its value. Some countries have had events of inflation of 1 million percent or more in a single month, which usually results in the rapid abandonment of the currency. For example, in November 2008 the monthly inflation rate of the [[Zimbabwean dollar]] reached 79.6 billion percent (470% per day); the approximate value with three significant figures would be {{val|7.96E10}} %,<ref name="Kadzere_2008"/><ref name="BBC"/> or more simply a rate of {{val|7.96E8}}.

== Converting numbers ==
Converting a number in these cases means to either convert the number into scientific notation form, convert it back into decimal form or to change the exponent part of the equation. None of these alter the actual number, only how it's expressed.

=== Decimal to scientific ===
First, move the decimal separator point sufficient places, ''n'', to put the number's value within a desired range, between 1 and 10 for normalized notation. If the decimal was moved to the left, append <code class=nowrap>× 10''<sup>n</sup>''</code>; to the right, <code class=nowrap>× 10''<sup>−n</sup>''</code>. To represent the number {{val|1230400|fmt=commas}} in normalized scientific notation, the decimal separator would be moved 6 digits to the left and <code class=nowrap>× 10<sup>6</sup></code> appended, resulting in {{val|1.2304E6}}. The number {{val|-0.0040321}} would have its decimal separator shifted 3 digits to the right instead of the left and yield {{val|-4.0321E-3}} as a result.

=== Scientific to decimal ===
Converting a number from scientific notation to decimal notation, first remove the <code class=nowrap>× 10''<sup>n</sup>''</code> on the end, then shift the decimal separator ''n'' digits to the right (positive ''n'') or left (negative ''n''). The number {{val|1.2304E6}} would have its decimal separator shifted 6 digits to the right and become {{val|1230400|fmt=commas}}, while {{val|-4.0321E-3}} would have its decimal separator moved 3 digits to the left and be {{val|-0.0040321}}.

=== Exponential ===
Conversion between different scientific notation representations of the same number with different exponential values is achieved by performing opposite operations of multiplication or division by a power of ten on the significand and an subtraction or addition of one on the exponent part. The decimal separator in the significand is shifted ''x'' places to the left (or right) and ''x'' is added to (or subtracted from) the exponent, as shown below.

{{block indent | em = 1.5 | text = {{val|1.234E3}} = {{val|12.34E2}} = {{val|123.4E1}} = 1234}}

== Basic operations ==
<!-- This section is linked from [[Addition]] -->
Given two numbers in scientific notation,
<math display="block">x_0=m_0\times10^{n_0}</math>
and
<math display="block">x_1=m_1\times10^{n_1}</math>

[[Multiplication]] and [[division (mathematics)|division]] are performed using the rules for operation with [[exponentiation]]:
<math display="block">x_0 x_1=m_0 m_1\times10^{n_0+n_1}</math>
and
<math display="block">\frac{x_0}{x_1}=\frac{m_0}{m_1}\times10^{n_0-n_1}</math>

Some examples are:
<math display="block">5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-5+2} = 13.3\times10^{-3} = 1.33\times10^{-2}</math>
and
<math display="block">\frac{2.34\times10^2}{5.67\times10^{-5}}  \approx 0.413\times10^{2-(-5)} = 0.413\times10^{7} = 4.13\times10^6</math>

[[Addition]] and [[subtraction]] require the numbers to be represented using the same exponential part, so that the significand can be simply added or subtracted:
{{block indent | em = 1.5 | text = <math>x_0 = m_0 \times10^{n_0}</math> and <math>x_1 = m_1 \times10^{n_1}</math> with <math>n_0 = n_1</math>}}

Next, add or subtract the significands:
<math display="block">x_0 \pm x_1=(m_0\pm m_1)\times10^{n_0}</math>

An example:
<math display="block">2.34\times10^{-5} + 5.67\times10^{-6} = 2.34\times10^{-5} + 0.567\times10^{-5} = 2.907\times10^{-5}</math>

== Other bases <span class="anchor" id="B notation"></span><span class="anchor" id="H notation"></span><span class="anchor" id="O notation"></span><span class="anchor" id="C notation"></span><span class="anchor" id="P notation"></span> ==
While base ten is normally used for scientific notation, powers of other bases can be used too,<ref name="TI_1974_SR-22"/> base 2 being the next most commonly used one.

For example, in base-2 scientific notation, the number 1001<sub>b</sub> in [[binary numeral system|binary]] (=9<sub>d</sub>) is written as {{nowrap|1.001<sub>b</sub> × 2<sub>d</sub><sup>11<sub>b</sub></sup>}} or {{nowrap|1.001<sub>b</sub> × 10<sub>b</sub><sup>11<sub>b</sub></sup>}} using binary numbers (or shorter {{nowrap|1.001 × 10<sup>11</sup>}} if binary context is obvious).{{citation needed|date=March 2024}} In E notation, this is written as {{nowrap|1.001<sub>b</sub>E11<sub>b</sub>}} (or shorter: 1.001E11) with the letter "E" now standing for "times two (10<sub>b</sub>) to the power" here. In order to better distinguish this base-2 exponent from a base-10 exponent, a base-2 exponent is sometimes also indicated by using the letter "B" instead of "E",<ref name="HP16C-Lib"/> a shorthand notation originally proposed by [[Bruce Alan Martin]] of [[Brookhaven National Laboratory]] in 1968,<ref name="Martin_1968"/> as in {{nowrap|1.001<sub>b</sub>B11<sub>b</sub>}} (or shorter: 1.001B11). For comparison, the same number in [[decimal representation]]: {{nowrap|1.125 × 2<sup>3</sup>}} (using decimal representation), or 1.125B3 (still using decimal representation). Some calculators use a mixed representation for binary floating point numbers, where the exponent is displayed as decimal number even in binary mode, so the above becomes {{nowrap|1.001<sub>b</sub> × 10<sub>b</sub><sup>3<sub>d</sub></sup>}} or shorter 1.001B3.<ref name="HP16C-Lib"/>

This is closely related to the base-2 [[floating-point]] representation commonly used in computer arithmetic, and the usage of IEC [[binary prefixes]] (e.g. 1B10 for 1×2<sup>10</sup> ([[kibi-|kibi]]), 1B20 for 1×2<sup>20</sup> ([[mebi-|mebi]]), 1B30 for 1×2<sup>30</sup> ([[gibi-|gibi]]), 1B40 for 1×2<sup>40</sup> ([[tebi-|tebi]])).

Similar to "B" (or "b"<ref name="HP16C-Add"/>), the letters "H"<ref name="HP16C-Lib"/> (or "h"<ref name="HP16C-Add"/>) and "O"<ref name="HP16C-Lib"/> (or "o",<ref name="HP16C-Add"/> or "C"<ref name="HP16C-Lib"/>) are sometimes also used to indicate ''times 16 or 8 to the power'' as in 1.25 = {{nowrap|1.40<sub>h</sub> × 10<sub>h</sub><sup>0<sub>h</sub></sup>}} = 1.40H0 = 1.40h0, or 98000 = {{nowrap|2.7732<sub>o</sub> × 10<sub>o</sub><sup>5<sub>o</sub></sup>}} = 2.7732o5 = 2.7732C5.<ref name="HP16C-Lib"/>

Another similar convention to denote base-2 exponents is using a letter "P" (or "p", for "power"). In this notation the significand is always meant to be hexadecimal, whereas the exponent is always meant to be decimal.<ref name="Rationale_2003_C"/> This notation can be produced by implementations of the ''[[printf]]'' family of functions following the [[C99]] specification and ([[Single Unix Specification]]) [[IEEE Std 1003.1]] [[POSIX]] standard, when using the ''%a'' or ''%A'' conversion specifiers.<ref name="Rationale_2003_C"/><ref name="printf_2013"/><ref name="Beebe_2017_Hex"/> Starting with [[C++11]], [[C++]] I/O functions could parse and print the P notation as well. Meanwhile, the notation has been fully adopted by the language standard since [[C++17]].<ref name="C++17"/> [[Apple Inc.|Apple]]'s [[Swift (programming language)|Swift]] supports it as well.<ref name="Swift_2017"/> It is also required by the [[IEEE 754-2008]] binary floating-point standard. Example: 1.3DEp42 represents {{nowrap|1.3DE<sub>h</sub> × 2<sup>42</sup>}}.

[[Engineering notation]] can be viewed as a base-1000 scientific notation.

== See also ==
* [[Positional notation]]
* [[ISO/IEC 80000]] – an international standard which guides the use of physical quantities and units of measurement in science
* [[Suzhou numerals]] – a Chinese numeral system formerly used in commerce, with order of magnitude written below the significand
* [[RKM code]] – a notation to specify resistor and capacitor values, with symbols for powers of 1000

== References ==
{{reflist |refs=
<ref name="Calio_2017">{{cite book |author-last1=Caliò |author-first1=Franca |author-first2=Lazzari |author-last2=Alessandro |title=Elements of Mathematics with Numerical Applications |publisher=Società Editrice Esculapio |pages=30–32 |date=September 2017 |isbn=978-8-89385052-0}}</ref>
<ref name="Edwards_2009">{{cite book |author-last=Edwards |author-first=John |date=2009 |title=Submission Guidelines for Authors: HPS 2010 Midyear Proceedings |publisher=Health Physics Society |place=McLean, VA |page=5 |url=http://hps.org/documents/2010_midyear_author-submission-guidelines.pdf |access-date=2013-03-30 }}</ref>
<ref name="Fortran">However, E notation was not included in the preliminary specification of Fortran, as of 1954. {{pb}}
{{cite book |title=Specifications for: The IBM Mathematical FORmula TRANSlating System, FORTRAN |type=Preliminary report |author-first1=John Warner |author-last1=Backus |author-link1=John Warner Backus |author-first2=Harlan L. |author-last2=Herrick |author-first3=Robert A. |author-last3=Nelson |author-first4=Irving |author-last4=Ziller |display-authors=0 |editor-first=John Warner |editor-last=Backus |editor-link=John Warner Backus |publisher=Programming Research Group, Applied Science Division, [[International Business Machines Corporation]] |place=New York |date=1954-11-10 |url=https://archive.computerhistory.org/resources/text/Fortran/102679231.05.01.acc.pdf |access-date=2022-07-04 }} (29 pages) {{pb}}
{{cite book |title=The FORTRAN Automatic Coding System for the IBM 704 EDPM: Programmer's Reference Manual |publisher=Applied Science Division and Programming Research Department, [[International Business Machines Corporation]] |place=New York |date=1956-10-15 |editor-first=David |editor-last=Sayre |editor-link=David Sayre |author-first1=John Warner |author-last1=Backus |author-link1=John Warner Backus |author-first2=Robert J. |author-last2=Beeber |author-first3=Sheldon F. |author-last3=Best |author-first4=Richard |author-last4=Goldberg |author-link4=Richard Goldberg |author-first5=Harlan L. |author-last5=Herrick |author-first6=Robert A. |author-last6=Hughes |author-first7=Lois B. |author-last7=Mitchell (Haibt) |author-link7=Lois B. Mitchell |author-first8=Robert A. |author-last8=Nelson |author-first9=Roy |author-last9=Nutt |author-link9=Roy Nutt |author-first10=David |author-last10=Sayre |author-link10=David Sayre |author-first11=Peter B. |author-last11=Sheridan |author-first12=Harold |author-last12=Stern |author-first13=Irving |author-last13=Ziller |display-authors=0 |pages=9, 27 |url=http://archive.computerhistory.org/resources/text/Fortran/102649787.05.01.acc.pdf |access-date=2022-07-04 }} (2+51+1 pages)</ref>
<ref name="DiGri-King_1958">{{cite journal |title=The SHARE 709 System: Input-Output Translation |author-first1=Vincent J. |author-last1=DiGri |author-first2=Jane E. |author-last2=King |journal=[[Journal of the ACM]] |volume=6 |issue=2 |date=April 1959 |orig-date=1958-06-11 |doi=10.1145/320964.320969 |s2cid=19660148 |pages=141–144 |quote=It tells the input translator that the field to be converted is a decimal number of the form ~X.XXXXE ± YY where E implies that the value of ~x.xxxx is to be scaled by ten to the ±YY power.|doi-access=free }} (4 pages) (NB. This was presented at the ACM meeting 11–13 June 1958.)</ref>
<ref name="Decapower">
Jim Davidson coined ''decapower'' and recommended the "D" separator in the ''[[65 Notes]]'' newsletter for [[Hewlett-Packard]] [[HP-65]] users, and Richard C. Vanderburgh promoted these in the ''[[52-Notes]]'' newsletter for [[Texas Instruments]] [[SR-52]] users. {{pb}}
{{cite journal |author-first=Jim |author-last=Davidson |editor-first=Richard J. |editor-last=Nelson |journal=[[65 Notes]] |place=Santa Ana, CA |date=January 1976 |volume=3 |number=1 |page=4 |id=V3N1P4 |title=(([title unknown]))}} {{pb}}
{{cite journal |editor-first=Richard C. |editor-last=Vanderburgh |title=Decapower |journal=52-Notes – Newsletter of the SR-52 Users Club |volume=1 |number=6 |page=1 |date=November 1976 |id=V1N6P1 |place=Dayton, OH |url=http://www.claudiolarini.altervista.org/pdf/52NOTES.pdf |access-date=2017-05-28 |quote=''Decapower'' – In the January 1976 issue of [[65 Notes|65-Notes]] (V3N1p4) Jim Davidson ([[HP-65]] Users Club member #547) suggested the term "decapower" as a descriptor for the power-of-ten multiplier used in scientific notation displays. I'm going to begin using it in place of "[[exponent]]" which is technically incorrect, and the letter D to separate the "mantissa" from the decapower for typewritten numbers, as Jim also suggests. For example, {{sic|<code>123|<sup>−45</sup></code>|expected=<code>123×10<sup>−45</sup></code>}} which is displayed in scientific notation as <code>1.23 -43</code> will now be written <code>1.23D-43</code>. Perhaps, as this notation gets more and more usage, the calculator manufacturers will change their keyboard abbreviations. HP's EEX and TI's EE could be changed to ED (for enter decapower).}} [https://www.rskey.org/DOCUMENTS/52NOTES/52v1n6.html<!-- https://web.archive.org/web/20170528122342/https://www.rskey.org/DOCUMENTS/52NOTES/52v1n6.html -->] {{Cite news |title=Decapower |volume=1 |number=6 |page=1 |date=November 1976 |newspaper=52-Notes – Newsletter of the SR-52 Users Club |place=Dayton, OH |url=http://www.rskey.org/DOCUMENTS/52NOTES/52v1n6.html |access-date=2018-05-07 }} (NB. The term ''decapower'' was frequently used in subsequent issues of this newsletter up to at least 1978.)</ref>
<ref name="UH-Manoa">{{cite web |url=http://www.math.hawaii.edu/lab/197/fortran/fort3.htm#double |title=UH Mānoa Mathematics » Fortran lesson 3: Format, Write, etc. |publisher=Math.hawaii.edu |date=2012-02-12 |access-date=2012-03-06 }}</ref>
<ref name="Sharp">
Specifically, models [[Sharp PC-1280|PC-1280]] (1987), [[Sharp PC-1470U|PC-1470U]] (1987), [[Sharp PC-1475|PC-1475]] (1987), [[Sharp PC-1480U|PC-1480U]] (1988), [[Sharp PC-1490U|PC-1490U]] (1990), [[Sharp PC-1490UII|PC-1490UII]] (1991), [[Sharp PC-E500|PC-E500]] (1988), [[Sharp PC-E500S|PC-E500S]] (1995), [[Sharp PC-E550|PC-E550]] (1990), [[Sharp PC-E650|PC-E650]] (1993), and [[Sharp PC-U6000|PC-U6000]] (1993). {{pb}}
{{cite book |title=SHARP Taschencomputer Modell PC-1280 Bedienungsanleitung |language=de |trans-title=SHARP Pocket Computer Model PC-1280 Operation Manual |date=1987 |publisher=[[Sharp Corporation]] |id=7M 0.8-I(TINSG1123ECZZ)(3) |pages=56–60 |url=http://www.instructionsmanuals.com/u2/pdf/calculadoras/Sharp-PC1280-de.pdf |access-date=2017-03-06 }} {{pb}}
{{cite book |title=SHARP Taschencomputer Modell PC-1475 Bedienungsanleitung |language=de |trans-title=SHARP Pocket Computer Model PC-1475 Operation Manual |pages=105–108, 131–134, 370, 375 |date=1987 |publisher=[[Sharp Corporation]] |url=http://www.instructionsmanuals.com/u2/pdf/calculadoras/Sharp-PC1475-de.pdf |access-date=2017-02-25 |url-status=dead |archive-url=https://web.archive.org/web/20170225090805/http://www.instructionsmanuals.com/u2/pdf/calculadoras/Sharp-PC1475-de.pdf |archive-date=2017-02-25}} {{pb}}
{{cite book |title=SHARP Pocket Computer Model PC-E500 Operation Manual |publisher=[[Sharp Corporation]] |date=1989 |id=9G1KS(TINSE1189ECZZ) <!-- |url=http://basic.hopto.org/basic/manual/Sharp%20PC-E500.pdf |access-date=2017-02-24 -->}} {{pb}}
{{cite book |title=SHARP Taschencomputer Modell PC-E500S Bedienungsanleitung |language=de |trans-title=SHARP Pocket Computer Model PC-E500S Operation Manual |publisher=[[Sharp Corporation]] |date=1995 |id=6J3KS(TINSG1223ECZZ) |url=http://vininc.de/Literatur/Sharp%20Bedienungsanleitungen/PC-E500S-DE.pdf |access-date=2017-02-24 |url-status=dead |archive-url=https://web.archive.org/web/20170224235944/http://vininc.de/Literatur/Sharp%20Bedienungsanleitungen/PC-E500S-DE.pdf |archive-date=2017-02-24}} {{pb}}
{{cite book |script-title=ja:電言板5 PC-1490UII PROGRAM LIBRARY |trans-title=Telephone board 5 PC-1490UII program library |publisher=University Co-op |language=ja |date=1991 |volume=5}} {{pb}}
{{cite book |script-title=ja:電言板6 PC-U6000 PROGRAM LIBRARY |trans-title=Telephone board 6 PC-U6000 program library |publisher=University Co-op |language=ja |date=1993 |volume=6}} <!-- https://ja.wikipedia.org/w/index.php?title=%E3%83%8E%E3%83%BC%E3%83%88%3A%E3%83%9D%E3%82%B1%E3%83%83%E3%83%88%E3%82%B3%E3%83%B3%E3%83%94%E3%83%A5%E3%83%BC%E3%82%BF%E3%81%AE%E8%A3%BD%E5%93%81%E4%B8%80%E8%A6%A7&type=revision&diff=64766669&oldid=64653791 --></ref>
<ref name="FortranQ">
For instance, DEC FORTRAN 77 (f77), [[Intel Fortran]], Compaq/Digital Visual Fortran, and [[GNU Fortran]] (gfortran) {{pb}}
{{cite book |title=DEC Fortran 77 Manual |chapter=Double Precision, REAL**16 |publisher=[[Digital Equipment Corporation]] |date= |url=https://wwwth.mpp.mpg.de/members/hahn/decfortman.html |access-date=2022-12-21 |quote=Digital Fortran 77 also allows the syntax Qsnnn, if the exponent field is within the T_floating double precision range. […] A REAL*16 constant is a basic real constant or an integer constant followed by a decimal exponent.  A decimal exponent has the form: Qsnn […] s is an optional sign […] nn is a string of decimal digits […] This type of constant is only available on [[DEC Alpha|Alpha system]]s.}} {{pb}}
{{cite book |title=Intel Fortran: Language Reference |date=2005 |orig-date=2003 |publisher=[[Intel Corporation]] |id=253261-003 |pages=3-7–3-8, 3-10 |url=https://www.ehu.eus/sgi/ARCHIVOS/lang_for.pdf |access-date=2022-12-22 }} (858 pages) {{pb}}
{{cite book |title=Compaq Visual Fortran – Language Reference |date=August 2001 |publisher=[[Compaq Computer Corporation]] |place=Houston |url=https://jp.xlsoft.com/documents/intel/cvf/cvf_lref.pdf |access-date=2022-12-22 }} (1441 pages) {{pb}}
{{cite book |title=The GNU Fortran Compiler |chapter=6. Extensions: 6.1 Extensions implemented in GNU Fortran: 6.1.8 Q exponent-letter |date=2014-06-12 |url=https://gcc.gnu.org/onlinedocs/gcc-4.7.4/gfortran/_003ccode_003eQ_003c_002fcode_003e-exponent-letter.html#_003ccode_003eQ_003c_002fcode_003e-exponent-letter |access-date=2022-12-21 }}</ref>
<ref name="Naur_1960">{{cite journal |title=Report on the Algorithmic Language ALGOL 60 |journal=Communications of the ACM |editor-first=Peter |editor-last=Naur |editor-link=Peter Naur |place=Copenhagen |date=1960|volume=3 |issue=5 |pages=299–311 |doi=10.1145/367236.367262 |url=https://dl.acm.org/doi/10.1145/367236.367262}}</ref>
<ref name="Savard_2005">{{cite web |title=Computer Arithmetic |at=The Early Days of Hexadecimal |author-first=John J. G. |author-last=Savard |date=2018 |orig-date=2005 |work=quadibloc |url=http://www.quadibloc.com/comp/cp02.htm |access-date=2018-07-16 }}</ref>
<ref name="Bauer-Becker-Graham_1968">{{cite web |title=ALGOL W – Notes For Introductory Computer Science Courses |author-first1=Henry R. |author-last1=Bauer |author-first2=Sheldon |author-last2=Becker |author-first3=Susan L. |author-last3=Graham |date=January 1968 |publisher=[[Stanford University]], Computer Science Department |url=http://i.stanford.edu/pub/cstr/reports/cs/tr/68/86/CS-TR-68-86.pdf |access-date=2017-04-08 }}</ref>
<ref name="Algol_1973">{{cite journal |title=Revised Report on the Algorithmic Language Algol 68 |journal=Acta Informatica |volume=5 |pages=1–236 |date=September 1973 |issue=1–3 |doi=10.1007/BF00265077 |citeseerx=10.1.1.219.3999 |s2cid=2490556}}</ref>
<ref name="Unicode">{{citation |mode=cs2 |type=Working Group Document |title=Revised proposal to encode the decimal exponent symbol |last=Broukhis |first=Leonid |date=2008-01-22 |website=unicode.org |url=https://www.unicode.org/L2/L2008/08030r-subscript10.pdf |id=L2/08-030R}} {{pb}}
{{cite web |url=https://www.unicode.org/versions/Unicode7.0.0/ |title=The Unicode Standard |edition=v. 7.0.0 |access-date=2018-03-23 }}</ref><ref name="Whitaker_1962">{{cite magazine |title=Numerical Prefixes |author-first=Ronald O. |author-last=Whitaker |date=1962-06-15 |magazine=[[Electronics (magazine)|Electronics]] |department=Crosstalk |page=4 |url=http://www.bitsavers.org/magazines/Electronics/Electronics_V35_N24_19620615.pdf |access-date=2022-12-24 }} (1 page)</ref>
<ref name="TI-83">Also see [[TI calculator character sets]]. {{pb}}
{{cite web |url=http://education.ti.com/downloads/guidebooks/sdk/83p/sdk83pguide.pdf |title=TI-83 Programmer's Guide<!-- exact title to be checked --> |access-date=2010-03-09 }}</ref>
<ref name="SIMULA_1986">{{cite web |title=SIMULA standard as defined by the SIMULA Standards Group – 3.1 Numbers |url=http://prosjekt.ring.hibu.no/simula/Standard/chap_1.htm |access-date=2009-10-06 |date=August 1986 }}</ref>
<ref name="Samples">Samples of usage of terminology and variants: {{pb}}
{{cite web |title=A Computer Program For The Design And Static Analysis Of Single-Point Sub-Surface Mooring Systems: NOYFB |series=WHOI Document Collection |author-first=Donald A. |author-last=Moller |date=June 1976 |type=Technica Report |publisher=Woods Hole Oceanographic Institution |id=WHOI-76-59 |place=Woods Hole, MA |url=http://darchive.mblwhoilibrary.org/bitstream/1912/665/1/WHOI-76-59.pdf |access-date=2015-08-19}} {{pb}}
{{cite web |url=http://www.brookscole.com/physics_d/templates/student_resources/003026961X_serway/review/expnot.html |archive-url=https://web.archive.org/web/20071019061437/http://brookscole.com/physics_d/templates/student_resources/003026961X_serway/review/expnot.html |archive-date=2007-10-19 |title=Cengage – the Leading Provider of Higher Education Course Materials}} {{pb}}
{{cite web |url=http://www.brynmawr.edu/nsf/tutorial/ss/ssnot.html |access-date=2007-04-07 |title=Bryn Mawr College: Survival Skills for Problem Solving – Scientific Notation }} {{pb}}
{{cite web |url=http://www.lasalle.edu/~smithsc/Astronomy/Units/sci_notation.html |access-date=2007-04-07 |title=Scientific Notation}} {{pb}}
[https://web.archive.org/web/20061029195358/http://www.gnsphysics.com/mathreview.pdf] {{pb}}
{{cite web |url=http://www.ttinet.com/doc/language_v44_003.html#heading_3.2.4.2 |archive-url=https://web.archive.org/web/20150503005623/http://www.ttinet.com/doc/language_v44_003.html |archive-date=2015-05-03 |title=INTOUCH 4GL a Guide to the INTOUCH Language}}
</ref>
<ref name="CODATA2014Full">{{cite journal |author-last1=Mohr |author-first1=Peter J. |author-last2=Newell |author-first2=David B. |author-last3=Taylor |author-first3=Barry N. |date=July–September 2016 |title=CODATA recommended values of the fundamental physical constants: 2014 |url=http://ws680.nist.gov/publication/get_pdf.cfm?pub_id=920687 |journal=[[Reviews of Modern Physics]] |volume=88 |issue=3 |page=035009 |doi=10.1103/RevModPhys.88.035009 |arxiv=1507.07956 |bibcode=2016RvMP...88c5009M |citeseerx=10.1.1.150.1225 |s2cid=1115862}}</ref>
<ref name="Luzum_2011">{{cite journal |author-last1=Luzum |first1=Brian |author-last2=Capitaine |author-first2=Nicole |author-last3=Fienga |author-first3=Agnès |author-last4=Folkner |author-first4=William |author-last5=Fukushima |author-first5=Toshio |author-last6=Hilton |author-first6=James |author-last7=Hohenkerk |author-first7=Catherine |author-last8=Krasinsky |author-first8=George |author-last9=Petit |author-first9=Gérard |author-last10=Pitjeva |author-first10=Elena |author-last11=Soffel |author-first11=Michael |author-last12=Wallace |author-first12=Patrick |title=The IAU 2009 system of astronomical constants: The report of the IAU working group on numerical standards for Fundamental Astronomy |journal=Celestial Mechanics and Dynamical Astronomy |volume=110 |issue=4 |date=August 2011 |pages=293–304 |bibcode=2011CeMDA.110..293L |doi=10.1007/s10569-011-9352-4 |doi-access=free}}</ref>
<ref name="Lide_2000">{{cite book |author=Various |editor-first=David R. |editor-last=Lide |date=2000 |title=Handbook of Chemistry and Physics |edition=81st |publisher=CRC |isbn=978-0-8493-0481-1}}</ref>
<ref name="Kadzere_2008">{{cite web |url=http://allafrica.com/stories/200810090256.html |title=Zimbabwe: Inflation Soars to 231 Million Percent |publisher=[[The Herald (Zimbabwe)|The Herald]] |place=Harare, Zimbabwe |author-first=Martin |author-last=Kadzere |date=2008-10-09 |access-date=2008-10-10 }}</ref>
<ref name="BBC">{{cite news |url=http://news.bbc.co.uk/1/hi/world/africa/7660569.stm |title=Zimbabwe inflation hits new high |date=9 October 2008 |archive-url=https://web.archive.org/web/20090514123525/http://news.bbc.co.uk/1/hi/world/africa/7660569.stm |archive-date=2009-05-14 |publisher=[[BBC News]] |access-date=2009-10-09}}</ref>
<ref name="TI_1974_SR-22">{{cite book |title=electronic hexadecimal calculator/converter SR-22 |publisher=[[Texas Instruments Incorporated]] |date=1974 |page=7 |edition=Revision A |id=1304-389 Rev A |url=http://www.datamath.net/Manuals/SR-22_US.pdf |access-date=2017-03-20 }} (NB. This calculator supports floating point numbers in scientific notation in bases 8, 10 and 16.)</ref>
<ref name="Martin_1968">{{cite journal |title=Letters to the editor: On binary notation |author-first=Bruce Alan |author-last=Martin |journal=[[Communications of the ACM]] |volume=11 |issue=10 |date=October 1968 |page=658 |doi=10.1145/364096.364107 |s2cid=28248410|doi-access=free }}</ref>
<ref name="HP16C-Lib">{{cite book |title=HP16C Emulator Library for the HP48S/SX |author-first1=Jake |author-last1=Schwartz |author-first2=Rick |author-last2=Grevelle |date=2003-10-20 |orig-date=April 1993 |edition=1 |version=1.20 |url=http://www.pahhc.org/mul8r.htm |access-date=2015-08-15 }} (NB. This library also works on the [[HP 48G]]/[[HP 48GX|GX]]/[[HP 48G+|G+]]. Beyond the feature set of the [[HP-16C]], this package also supports calculations for binary, octal, and hexadecimal [[floating-point number]]s in scientific notation in addition to the usual decimal floating-point numbers.)</ref>
<ref name="HP16C-Add">{{cite book |title=HP16C Emulator Library for the HP48 – Addendum to the Operator's Manual |author-first1=Jake |author-last1=Schwartz |author-first2=Rick |author-last2=Grevelle |date=2003-10-21 |edition=1 |version=1.20 |url=http://www.pahhc.org/mul8r.htm |access-date=2015-08-15 }}</ref>
<ref name="Rationale_2003_C">{{cite web |title=Rationale for International Standard – Programming Languages – C |version=5.10 |date=April 2003 |pages=52, 153–154, 159 |url=http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf |access-date=2010-10-17 }}</ref><ref name="printf_2013">{{cite web |title=dprintf, fprintf, printf, snprintf, sprintf – print formatted output |work=The Open Group Base Specifications |edition=Issue 7, IEEE Std 1003.1, 2013 |date=2013 |orig-date=2001 |author=The IEEE and The Open Group |url=http://pubs.opengroup.org/onlinepubs/9699919799/functions/printf.html |access-date=2016-06-21 }}</ref>
<ref name="Beebe_2017_Hex">{{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook – Programming Using the MathCW Portable Software Library |date=2017-08-22 |place=Salt Lake City |publisher=Springer |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |s2cid=30244721}}</ref>
<ref name="C++17">{{cite web |work=cppreference.com |title=floating point literal |url=http://en.cppreference.com/w/cpp/language/floating_literal |access-date=2017-03-11 |quote=The hexadecimal floating-point literals were not part of C++ until C++17, although they can be parsed and printed by the I/O functions since C++11: both C++ I/O streams when std::hexfloat is enabled and the C I/O streams: std::printf, std::scanf, etc. See std::strtof for the format description.}}</ref>
<ref name="Swift_2017">{{cite web |title=The Swift Programming Language (Swift 3.0.1) |at=Lexical Structure |work=Guides and Sample Code: Developer: Language Reference |publisher=[[Apple Inc.|Apple Corporation]] |url=https://developer.apple.com/library/content/documentation/Swift/Conceptual/Swift_Programming_Language/LexicalStructure.html |access-date=2017-03-11 }}</ref>
<ref name="TI_1973_SR-10">Such as the TI SR-10. {{pb}} {{cite book |title=Texas Instruments electronic slide rule calculator SR-10 |date=1973 |publisher=[[Texas Instruments Incorporated]] |place=Dallas |id=1304-739-266 |url=https://www.sliderulemuseum.com/Calculators/SR-10_US.pdf |access-date=2023-01-01 }} (1+1+45+1 pages) (NB. Although this manual is dated 1973, presumably version 1 of this calculator was introduced in November 1972 according to other sources.)</ref>
}}

== External links ==
{{Wiktionary|scientific notation}}
* [http://www.miniwebtool.com/decimal-to-scientific-notation-converter/ Decimal to Scientific Notation Converter]
* [http://www.miniwebtool.com/scientific-notation-to-decimal-converter/ Scientific Notation to Decimal Converter]
* [http://www.math.toronto.edu/mathnet/plain/questionCorner/scinot.html Scientific Notation in Everyday Life]
* [http://science.widener.edu/svb/tutorial/scinot.html An exercise in converting to and from scientific notation]
* [http://www.scientificnotationconverter.com Scientific Notation Converter]
* [http://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_4.html ''Scientific Notation''] chapter from [http://www.ibiblio.org/kuphaldt/electricCircuits/DC/index.html ''Lessons In Electric Circuits Vol 1 DC''] free ebook and [http://www.ibiblio.org/kuphaldt/electricCircuits/ ''Lessons In Electric Circuits''] series.

{{DEFAULTSORT:Scientific Notation}}
[[Category:Mathematical notation]]
[[Category:Measurement]]
[[Category:Numeral systems]]