# Engineering notation

Engineering notation or engineering form (also technical notation) is a version of scientific notation in which the exponent of ten is always selected to be divisible by three to match the common metric prefixes, i.e. scientific notation that aligns with powers of a thousand, for example, 531×103 instead of 5.31×105 (but on calculator displays written without the ×10 to save space). As an alternative to writing powers of 10, SI prefixes can be used,[1] which also usually provide steps of a factor of a thousand.[nb 1] On most calculators, engineering notation is called "ENG" mode as scientific notation is denoted SCI.

## 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 5360A frequency counter by Hewlett-Packard in 1969.[1]

Based on an idea by Peter D. Dickinson[2][1] the first calculator to support engineering notation displaying the power-of-ten exponent values was the HP-25 in 1975.[3] It was implemented as a dedicated display mode in addition to scientific notation.

In 1975, Commodore introduced a number of scientific calculators (like the SR4148/SR4148R[4] and SR4190R[5]) providing a variable scientific notation, where pressing the and keys shifted the exponent and decimal point by ±1[nb 2] in scientific 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 SR-40,[6][7] TI-30[8][9][10][11][12][13][14][15] and TI-45[16][17] model variants utilizing () instead. This can be seen as a precursor to a feature implemented on many Casio calculators since 1978/1979 (e.g. in the FX-501P/FX-502P), where number display in engineering notation is available on demand by the single press of a () 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[nb 2] in order to easily let results match a desired prefix. Some graphical calculators (for example the fx-9860G) in the 2000s also support the display of some SI prefixes (f, p, n, µ, m, k, M, G, T, P, E) as suffixes in engineering mode.

## Overview

Compared to normalized scientific notation, one disadvantage of using SI prefixes and engineering notation is that significant figures are not always readily apparent when the smallest significant digit or digits are 0. For example, 500 µm and 500×10−6 m cannot express the uncertainty distinctions between 5×10−4 m, 5.0×10−4 m, and 5.00×10−4 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 mm, 0.50 mm, or 0.500 mm would have been used to show uncertainty and significant figures. It is also common to state the precision explicitly, such as "47 kΩ±5%"

Another example: when the speed of light (exactly 299792458 m/s[18] by the definition of the meter) is expressed as 3.00×108 m/s or 3.00×105 km/s then it is clear that it is between 299500 km/s and 300500 km/s, but when using 300×106 m/s, or 300×103 km/s, 300000 km/s, or the unusual but short 300 Mm/s, this is not clear. A possibility is using 0.300×109 m/s or 0.300 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, 12.5×10−9 m can be read as "twelve-point-five nanometers" (10−9 being nano) and written as 12.5 nm, while its scientific notation equivalent 1.25×10−8 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 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 exponential e which holds a completely different significance.

SI prefixes
Prefix Representations
Name Symbol Base 1000 Base 10 Value
quetta Q 100010  1030 1000000000000000000000000000000
ronna R 10009  1027 1000000000000000000000000000
yotta Y 10008  1024 1000000000000000000000000
zetta Z 10007  1021 1000000000000000000000
exa E 10006  1018 1000000000000000000
peta P 10005  1015 1000000000000000
tera T 10004  1012 1000000000000
giga G 10003  109 1000000000
mega M 10002  106 1000000
kilo k 10001  103 1000
10000  100 1
milli m 1000−1  10−3 0.001
micro μ 1000−2  10−6 0.000001
nano n 1000−3  10−9 0.000000001
pico p 1000−4  10−12 0.000000000001
femto f 1000−5  10−15 0.000000000000001
atto a 1000−6  10−18 0.000000000000000001
zepto z 1000−7  10−21 0.000000000000000000001
yocto y 1000−8  10−24  0.000000000000000000000001
ronto r 1000−9  10−27  0.000000000000000000000000001
quecto q 1000−10  10−30  0.000000000000000000000000000001

## Binary engineering notation

Just like decimal engineering notation can be viewed as a base-1000 scientific notation (103 = 1000), 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) commonly used in computer arithmetic, and the usage of IEC binary prefixes, e.g. 1B10 for 1 × 210, 1B20 for 1 × 220, 1B30 for 1 × 230, 1B40 for 1 × 240 etc.[19]

IEC prefixes
Prefix Representations
Name Symbol Base 1024 Base 2 Value
quebi[nb 3] Qi[nb 3] 102410  2100 1267650600228229401496703205376
robi[nb 3] Ri[nb 3] 10249  290 1237940039285380274899124224
yobi Yi 10248  280 1208925819614629174706176
zebi Zi 10247  270 1180591620717411303424
exbi Ei 10246  260 1152921504606846976
pebi Pi 10245  250 1125899906842624
tebi Ti 10244  240 1099511627776
gibi Gi 10243  230 1073741824
mebi Mi 10242  220 1048576
kibi Ki 10241  210 1024
10240  20 1

## Notes

1. ^ 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.
2. ^ a b 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 would inverse the action in the other direction.
3. ^ a b c d 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). As of 2022, these binary prefixes have not been adopted by the IEC and ISO.

## References

1. ^ a b c Gordon, Gary B.; Reeser, Gilbert A. (May 1969). "Introducing the Computing Counter - Here is the most significant advance in electronic counters in recent years" (PDF). Hewlett-Packard Journal. Hewlett-Packard Company. 20 (9): 2–16. Archived (PDF) from the original on 2017-06-04. Retrieved 2017-06-04. […] 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 (hertz, second, 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.)
2. ^ US 3987290, Dickinson, Peter D., "Calculator Apparatus for Displaying Data in Engineering Notation", published 1976-10-19, assigned to Hewlett-Packard Company . "[…] 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)
3. ^ Neff, Randall B.; Tillman, Lynn (November 1975). "Three New Pocket Calculators: Smaller, less Costly, More Powerful" (PDF). Hewlett-Packard Journal. Hewlett-Packard Company. 27 (3): 1–7. Archived (PDF) from the original on 2017-06-10. Retrieved 2017-06-10. [1]
4. ^
5. ^ commodore - Multi-Function Preprogrammed Rechargeable Scientific Notation Calculator - Model SR4190R - Owner's Manual (PDF). Commodore. 1975. pp. 10–11. Archived (PDF) from the original on 2017-06-24. Retrieved 2017-06-24. Variable scientific notation: 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 and will algebraically increment or decrement the value of the exponent by one for each depression, moving accordingly the decimal point of the mantissa.
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18. ^ "CODATA Value: Speed of light in vacuum c, c0". CODATA 2014: The NIST Reference on Constants, Units, and Uncertainty: Fundamental Physical Constants. NIST. 2017-05-24. Archived from the original on 2017-06-25. Retrieved 2017-05-25.
19. ^ Martin, Bruce Alan (October 1968). "Letters to the editor: On binary notation". Communications of the ACM. Associated Universities Inc. 11 (10): 658. doi:10.1145/364096.364107. S2CID 28248410.