# Engineering notation

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 prefixes can be used, which also usually provide steps of a factor of a thousand.[nb 1]

On most calculators, engineering notation is called "ENG" mode.

## History

An early implementation of engineering notation in form of range selection and number display with SI prefixes was introduced in the computerized HP 5360A frequency counter by Hewlett-Packard in 1969.

Based on an idea by Peter D. Dickinson the first calculator to support engineering notation displaying the power-of-ten exponent values was the HP-25 in 1975. 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 and SR4190R) 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, TI-30 and TI-45 model variants utilizing () instead. This can be seen as a precursor to a feature implemented on many Casio calculators since about 1978/1979 (f.e. 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. 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 by the definition of the meter and second) is expressed as 3.00 × 108 m/s or 3.00 × 105 km/s then it is clear that it is between 299 500 km/s and 300 500 km/s, but when using 300 × 106 m/s, or 300 × 103 km/s, 300 000 km/s, or the unusual but short 300 Mm/s, this is not clear. A possibility is using 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" 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. In the latter case, it would be shown that 3e−9 ≈ 0.000 370 23.

SI prefixes
Prefix Representations
Name Symbol Base 1000 Base 10 Value
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

## 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 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.).

IEC prefixes
Prefix Representations
Name Symbol Base 1024 Base 2 Value
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