Engineering notation: Difference between revisions

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{{use list-defined references|date=December 2021}}

'''Engineering notation''' or '''engineering form''' (also '''technical notation''') is a version of [[scientific notation]] in which the exponent of ten mustis always selected to be divisible by three (to match the common metric prefixes, i.e., theyscientific arenotation that aligns with powers of a thousand, but written as, for example, 10531×10⁶³ instead of 10005.31×10²⁵ (but on calculator displays written in [[E notation]] - with "E" instead of "×10" to save space). As an alternative to writing powers of 10, [[SI prefix]]es can be used,<ref name="Gordon_1969"/> which also usually provide steps of a factor of a thousand.<ref group="nb" name="NB_Cubic"/>

On most calculators, engineering notation is called "ENG" mode as scientific notation is denoted SCI.

=={{anchor|Exponent shift}}History==

An early implementation of engineering notation in the form of range selection and number display with SI prefixes was introduced in the computerized HP&nbsp;5360A [[frequency counter]] by [[Hewlett-Packard]] in 1969.<ref name="Gordon_1969"/>

Based on an idea by Peter D. Dickinson<ref name="Dickinson_1976"/><ref name="Gordon_1969"/> the first [[calculator]] to support engineering notation displaying the power-of-ten exponent values was the [[HP-25]] in 1975.<ref name="Neff_1975"/> It was implemented as a dedicated display mode in addition to scientific notation.

In 1975, [[Commodore Business Machines|Commodore]] introduced a number of scientific calculators (like the [[Commodore SR4148|SR4148]]/SR4148R<ref name="Commodore_SR4148R"/> and [[Commodore SR4190R|SR4190R]]<ref name="Commodore_SR4190R"/>) providing a ''variable scientific notation'', where pressing the {{button|EE↓}} and {{button|EE↑}} keys shifted the exponent and decimal point by ±1<ref group="nb" name="NB_Exp-Shift"/> in ''scientific''<!-- not engineering! --> notation. Between 1976 and 1980 the same ''exponent shift'' facility was also available on some [[Texas Instruments]] calculators of the pre-[[LCD]] era such as early [[TI SR-40|SR-40]],<ref name="SR-40"/><ref name="SR-40_Manual"/> [[TI-30]]<ref name="TI-30"/><ref name="TI-30_Manual"/><ref name="TI-30-BR"/><ref name="TI-30_BR_Manual"/><ref name="TI-30_2"/><ref name="TI-30_RCI"/><ref name="TI-30_1"/><ref name="TI-30_Super"/> and [[TI-45]]<ref name="TI-45"/><ref name="TI-45_Manual"/> model variants utilizing ({{button|INV}}){{button|EE↓}} instead. This can be seen as a precursor to a feature implemented on many [[Casio]] calculators since 1978/1979 (e.g. in the [[Casio FX-501P|FX-501P]]/[[Casio FX-502P|FX-502P]]), where number display in ''engineering'' notation is available on demand by the single press of a ({{button|INV}}){{button|ENG}} button (instead of having to activate a dedicated display mode as on most other calculators), and subsequent button presses would shift the exponent and decimal point of the number displayed by ±3<ref group="nb" name="NB_Exp-Shift"/> in order to easily let results match a desired prefix. Some graphical calculators (for example the [[Casio fx-9860G|fx-9860G]]) in the 2000s also support the display of some SI prefixes (f, p, n, µμ, m, k, M, G, T, P, E) as suffixes in engineering mode.

==Overview==

{{uncited section|date=February 2024}}

Compared to normalized scientific notation, one disadvantage of using SI prefixes and engineering notation is that [[significant figure]]s are not always readily apparent when the smallest significant digit or digits are 0. For example, 500&nbsp;µmμm and {{val|500E-6|u=m}} cannot express the [[uncertainty]] distinctions between {{val|5E-4|u=m}}, {{val|5.0E-4|u=m}}, and {{val|5.00E-4|u=m}}. This can be solved by changing the range of the coefficient in front of the power from the common 1–1000 to 0.001–1.0. In some cases this may be suitable; in others it may be impractical. In the previous example, 0.5&nbsp;mm, 0.50&nbsp;mm, or 0.500&nbsp;mm would have been used to show uncertainty and significant figures. It is also common to state the precision explicitly, such as "{{gaps|47&nbsp;kΩ|±|5%}}"

Another example: when the [[speed of light]] (exactly {{val|299792458|u=m/s}}<ref name="CUU_2014_c"/> by the definition of the meter) is expressed as {{val|3.00E8|u=m/s}} or {{val|3.00E5|u=km/s}} then it is clear that it is between {{val|299500|u=km/s}} and {{val|300500|u=km/s}}, but when using {{val|300E6|u=m/s}}, or {{val|300E3|u=km/s}}, {{val|300000|u=km/s}}, or the unusual but short {{val|300|u=Mm/s}}, this is not clear. A possibility is using {{val|0.300E9|u=m/s}} or {{val|0.300|u=Gm/s}}.

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On the other hand, 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 nanometers" (10⁻⁹ being ''nano'') and written as 12.5&nbsp;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 meters".

Engineering notation, like scientific notation generally, can use the [[E notation]], such that {{val|3.0E-9}} can be written as 3.0E−9 or 3.0e−9. The E (or e) should not be confused with the [[E (mathematical constant)|exponential 'Euler's number e'']] whichor holdsthe asymbol completelyfor differentthe significance[[Exa-|exa]]-prefix.

:{| class="wikitable" style="padding:0; text-align:center; width:0"

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| q

| style="text-align:left;" | 1000⁻¹⁰

| style="text-align:left;" | &nbsp;[[Orders of magnitude (numbers)#10.E2.88.923010−30|10⁻³⁰]]&nbsp;

| style="text-align:left;" | {{val|0.000000000000000000000000000001}}

|}

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<ref name="TI-30-BR">{{Cite web|url=http://www.datamath.org/Sci/MAJESTIC/TI-30_BR.htm|title = Datamath}}</ref>

<ref name="TI-30_BR_Manual">http://www.datamath.net/Manuals/TI-30_BR.pdf {{Bare URL PDF|date=March 2022}}</ref>

<ref name="TI-30_2">{{Cite web|url=http://www.datamath.org/Sci/MAJESTIC/TI-30_2.htm|title=DatamathDATAMATH|website=www.datamath.org}}</ref>

<ref name="TI-30_RCI">{{Cite web|url=http://www.datamath.org/Sci/MAJESTIC/TI-30_RCI1380.htm|title=DatamathDATAMATH|website=www.datamath.org}}</ref>

<ref name="TI-30_1">{{Cite web|url=http://www.datamath.org/SCISci/MAJESTIC/TI-30_1.htm|title=DatamathDATAMATH|website=www.datamath.org}}</ref>

<ref name="TI-30_Super">{{Cite web|url=http://www.datamath.org/Others/KohINoor/TI-30.htm|title = Datamath}}</ref>

<ref name="TI-45">{{Cite web|url=http://www.datamath.org/Sci/MAJESTIC/TI-45.htm|title = Datamath}}</ref>

<ref name="TI-45_Manual">http://www.datamath.net/Manuals/TI-45_EU.pdf {{Bare URL PDF|date=March 2022}}</ref>

<ref name="CUU_2014_c">{{cite web |title=CODATA Value: Speed of light in vacuum ''c'', ''c''₀ |work=[[CODATA 2014]]: The NIST Reference on Constants, Units, and Uncertainty: Fundamental Physical Constants |publisher=[[NIST]] |date=2017-05-24 |url=http://physics.nist.gov/cgi-bin/cuu/Value?c |access-date=2017-05-25 |url-status=live |archive-url=https://web.archive.org/web/20170625090639/http://physics.nist.gov/cgi-bin/cuu/Value?c |archive-date=2017-06-25}}</ref>

<ref name="Martin_1968">{{cite journal |title=Letters to the editor: On binary notation |author-first=Bruce Alan |author-last=Martin |publisher=[[Associated Universities Inc.]] |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>

}}