Engineering notation: Difference between revisions

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{{Short description|A version of scientific notation in which the exponent of ten reflects powers of a thousand}}

{{use dmy dates|date=December 2021|cs1-dates=y}}

'''Engineering notation''' or '''engineering form''' is a version of [[scientific notation]] in which the exponent of ten must be divisible by three (i.e., they are powers of a thousand, but written as, for example, 10⁶ instead of 1000²). 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"/>

{{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 notation that aligns arewith 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 about 1978/1979 (fe.eg. 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 500&nbsp;×&nbsp;10⁻⁶&nbsp;{{val|500E-6|u=m}} cannot express the [[uncertainty]] distinctions between 5&nbsp;×&nbsp;10⁻⁴&nbsp;{{val|5E-4|u=m}}, {{val|5.0&nbsp;×&nbsp;10⁻⁴&nbsp;0E-4|u=m}}, and {{val|5.00&nbsp;×&nbsp;10⁻⁴&nbsp;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;k&Omega; 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 and second) is expressed as {{val|3.00&nbsp;×&nbsp;10⁸&nbsp;00E8|u=m/s}} or {{val|3.00&nbsp;×&nbsp;10⁵&nbsp;00E5|u=km/s}} then it is clear that it is between 299 500&nbsp;{{val|299500|u=km/s}} and 300 500&nbsp;{{val|300500|u=km/s}}, but when using 300&nbsp;×&nbsp;10⁶&nbsp;{{val|300E6|u=m/s}}, or 300&nbsp;×&nbsp;10³&nbsp;{{val|300E3|u=km/s}}, 300 000&nbsp;{{val|300000|u=km/s}}, or the unusual but short {{val|300&nbsp;|u=Mm/s}}, this is not clear. A possibility is using {{val|0.300E9|u=m/s}} or {{val|0.300&nbsp;|u=Gm/s}}.

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|12.5&nbsp;×&nbsp;10⁻⁹&nbsp;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|1.25&nbsp;×&nbsp;10⁻⁸&nbsp;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)|Euler's number e]] or the symbol for the [[Exa-|exa]]-prefix.

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

:3.0 × 10⁻⁹

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 ''e'']] which holds a completely different significance. In the latter case, it would be shown that 3e⁻⁹ ≈ 0.000 370 23.

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

! colspan="5" | [[SI prefix]]es

|-

Line 39 ⟶ 34:

! style="text-align:center" | Base 10

! style="text-align:center" | Value

|-

| [[quetta-|quetta]]

| Q

| style="text-align:left;" | 1000¹⁰

| style="text-align:left;" | &nbsp;[[Orders of magnitude (numbers)#1030|10³⁰]]

| style="text-align:right;" | {{val|1000000000000000000000000000000}}

|-

| [[ronna-|ronna]]

| R

| style="text-align:left;" | 1000⁹

| style="text-align:left;" | &nbsp;[[Orders of magnitude (numbers)#1027|10²⁷]]

| style="text-align:right;" | {{val|1000000000000000000000000000}}

|-

| [[yotta-|yotta]]

Line 140 ⟶ 147:

| style="text-align:left;" | &nbsp;[[Orders of magnitude (numbers)#10.E2.88.9224|10⁻²⁴]]&nbsp;

| style="text-align:left;" | {{val|0.000000000000000000000001}}

|-

| [[ronto-|ronto]]

| r

| style="text-align:left;" | 1000⁻⁹

| style="text-align:left;" | &nbsp;[[Orders of magnitude (numbers)#10.E2.88.9227|10⁻²⁷]]&nbsp;

| style="text-align:left;" | {{val|0.000000000000000000000000001}}

|-

| [[quecto-|quecto]]

| q

| style="text-align:left;" | 1000⁻¹⁰

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

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

|}

=={{anchor|Binary}}Binary engineering notation==

Just like decimal engineering notation can be viewed as a base-1000 scientific notation (10³ = 1000), [[Binary numeral system|binary]] engineering notation relates to a base-1024 scientific notation (2¹⁰ = 1024), where the exponent of two must be divisible by ten. This is closely related to the base-2 [[floating-point]] representation ([[B notation (scientific notation)|B notation]]) commonly used in computer arithmetic, and the usage of IEC [[binary prefix]]es, (e.g. 1B10 for 1&nbsp;×&nbsp;2¹⁰, 1B20 for 1&nbsp;×&nbsp;2²⁰, 1B30 for 1&nbsp;×&nbsp;2³⁰, 1B40 for 1&nbsp;×&nbsp;2⁴⁰ etc.).<ref name="Martin_1968"/>

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

Line 156 ⟶ 175:

! style="text-align:center" | Base 2

! style="text-align:center" | Value

|-

| [[quebi-|quebi]]<ref group="nb" name="NB_NewBinPrefix"/>

| Qi<ref group="nb" name="NB_NewBinPrefix"/>

:3.0| × 101024⁻⁹¹⁰

| align="left" | &nbsp;[[1267650600228229401496703205376 (number)|2¹⁰⁰]]

| align="right" | {{val|1267650600228229401496703205376}}

|-

| [[robi-|robi]]<ref group="nb" name="NB_NewBinPrefix"/>

| Ri<ref group="nb" name="NB_NewBinPrefix"/>

| 1024⁹

| align="left" | &nbsp;[[1237940039285380274899124224 (number)|2⁹⁰]]

| align="right" | {{val|1237940039285380274899124224}}

|-

| [[yobi-|yobi]]

Line 223 ⟶ 254:

<ref group="nb" name="NB_Cubic">Except in the case of square and cubic units: in this case the SI prefixes provide only steps of a factor of one million or one billion respectively.</ref>

<ref group="nb" name="NB_Exp-Shift">One ''exponent shift'' action would decrease the exponent by the same amount as the decimal point would be moved to the right, so that the value of the displayed number does not change. Preceding the keypress with {{button|INV}} would inverse the action in the other direction.</ref>

<ref group="nb" name="NB_NewBinPrefix">Natural binary counterparts to the ''[[ronna-]]'' and ''[[quetta-]]'' decimal prefixes introduced in 2022 were suggested in a consultation paper of the [[International Committee for Weights and Measures]]' Consultative Committee for Units (CCU) as ''[[robi-]]'' (Ri, 1024⁹) and ''[[quebi-]]'' (Qi, 1024¹⁰). {{Asof|2022}}, these binary prefixes have not been adopted by the IEC and ISO.<!-- {{cite journal |title=A further short history of the SI prefixes |journal=[[Metrologia]] |department=Letter to the editor |author-first=Richard J. C. |author-last=Brown |date=2023 |orig-date=2022-02-08, 2022-04-01, 2022-11-24 |volume=60 |publisher=[[BIPM]] & [[IOP Publishing Ltd]] |id=013001 |doi=10.1088/1681-7575/ac6afd |url=https://iopscience.iop.org/article/10.1088/1681-7575/ac6afd/pdf |access-date=2022-12-20 |url-status= |archive-url= |archive-date=}} (1+4 pages){{cite journal |title=Reply to 'Facing a shortage of the Latin letters for the prospective new SI symbols: alternative proposal for the new SI prefixes' |author-last=Brown |author-first=Richard J. C. |date=2022-04-27 |journal={{ill|Accreditation and Quality Assurance|de}} |volume=27 |issue= |pages=143–144 |doi=10.1007/s00769-022-01499-7}} --></ref>

}}

== References ==

{{reflist|refs=

<ref name="Gordon_1969">{{cite journal |title=Introducing the Computing Counter - Here is the most significant advance in electronic counters in recent years |author-first1=Gary B. |author-last1=Gordon |author-first2=Gilbert A. |author-last2=Reeser |journal=[[Hewlett-Packard Journal]] |publisher=[[Hewlett-Packard Company]] |volume=20 |number=9 |date=May 1969 |pages=2–16 |url=http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1969-05.pdf |access-date=2017-06-04 |url-status=live |archive-url=https://web.archive.org/web/20170604195957/http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1969-05.pdf |archive-date=2017-06-04 |quote=[…] Measurements are displayed around a stationary decimal point and the display tubes are grouped in threes to make the display more readable. The numerical display is accompanied by appropriate measurement units (e.g.hertz, Hz, Secsecond, etc.) and a prefix multiplier which is computed by the counter (e.g., k for kilo, M for mega, etc.). There are 12 digital display tubes, to permit shifting the displayed value (11 digits maximum) around the fixed decimal point. Insignificant digits and leading zeros are automatically blanked so only significant digits are displayed, or any number of digits from 3 to 11 can be selected manually. Internally, however, the computer always carries 11 digits. […]}} (NB. Introduces the HP 5360A Computing Counter.)</ref>

<ref name="Dickinson_1976">{{cite patent |inventor-last=Dickinson |inventor-first=Peter D. |pubdate=1976-10-19 |fdate=1975-05-19 |pridate=1975-05-19 |title=Calculator Apparatus for Displaying Data in Engineering Notation |assign1=[[Hewlett-Packard Company]] |country-code=US |patent-number=3987290 |url=https://www.google.ch/patents/US3987290}}. "[…] A computing counter […] has been developed that displays data in engineering notation with the exponent expressed in alphabetic form rather than in numeric form, such as f in place of −15, p in place of −12, n in place of −9, μ in place of −6, m in place of −3, k in place of +3, M in place of +6, G in place of +9, and T in place of +12. This device, however, is limited to displaying only those numeric quantities for which there exists a commonly accepted alphabetic exponent notation. This device is also limited in the range of data that it can display because the size of the exponent display area is limited, and would be unduly large if required to contain all of the alphabetic characters necessary to represent every exponent that is a multiple of three, for example, in the range −99 to +99. […]" (US 05/578,775)</ref>

<ref name="Neff_1975">{{cite journal |title=Three New Pocket Calculators: Smaller, less Costly, More Powerful |author-first1=Randall B. |author-last1=Neff |author-first2=Lynn |author-last2=Tillman |journal=[[Hewlett-Packard Journal]] |publisher=[[Hewlett-Packard Company]] |volume=27 |number=3 |date=November 1975 |pages=1–7 |url=http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1975-11.pdf |access-date=2017-06-10 |url-status=live |archive-url=https://web.archive.org/web/20170610140406/http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1975-11.pdf |archive-date=2017-06-10}} [http://www.hpmuseum.org/journals/wooda.htm<!-- https://web.archive.org/web/20170610135819/http://www.hpmuseum.org/journals/wooda.htm -->]</ref>

<ref name="Commodore_SR4148R">http://www.wass.net/manuals/Commodore%20SR4148R.pdf {{Bare URL PDF|date=March 2022}}</ref>

<ref name="Commodore_SR4190R">{{cite book |title=commodore - Multi-Function Preprogrammed Rechargeable Scientific Notation Calculator - Model SR4190R - Owner's Manual |publisher=[[Commodore International|Commodore]] |date=1975 |url=http://www.wass.net/manuals/Commodore%20SR4190R.pdf |access-date=2017-06-24 |url-status=live |archive-url=https://web.archive.org/web/20170624161616/http://www.wass.net/manuals/Commodore%20SR4190R.pdf |archive-date=2017-06-24 |pages=10–11 |quote=Variable scientific notation: [[Commodore Business Machines|Commodore]] scientific calculators offer the possibility of changing the exponent at will, therefore allowing the full choice of the unit in which the display may be read. The {{button|EE↑}} and {{button|EE↓}} will algebraically increment or decrement the value of the exponent by one for each depression, moving accordingly the decimal point of the mantissa.}}</ref>

<ref name="SR-40">{{Cite web|url=http://www.datamath.org/SCI/MAJESTIC/sr-40.htm|title = Datamath}}</ref>

<ref name="SR-40_Manual">http://www.datamath.net/Manuals/SR-40_US.pdf {{Bare URL PDF|date=March 2022}}</ref>

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

<ref name="TI-30_Manual">http://www.datamath.net/Manuals/TI-30_1976_US.pdf {{Bare URL PDF|date=March 2022}}</ref>

<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=DATAMATH|website=www.datamath.org}}</ref>

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

<ref name="TI-30_1">{{Cite web|url=http://www.datamath.org/SCISci/MAJESTIC/TI-30_1.htm|title=DATAMATH|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'', ''c₀c''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.]] |workjournal=[[Communications of the ACM]] |volume=11 |issue=10 |date=October 1968 |page=658 |doi=10.1145/364096.364107|s2cid=28248410 |doi-access=free }}</ref>

}}

==External links==

* [https://www.youtube.com/watch?v=guZ7crT68h0 Engineering Prefix User Defined Function for Excel]

* [http://search.cpan.org/search?query=Number%3A%3AFormatEng Perl CPAN module for converting number to engineering notation]

* [http://www.labbookpages.co.uk/software/java/engNotation.html Java functions for converting between a string and a double type]