Engineering notation: Difference between revisions

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, which also usually provide steps of a factor of a thousand. 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.

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 EE↓ and EE↑ keys shifted the exponent and decimal point by ±1 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 (INV)EE↓ 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 (INV)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 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 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 Euler's number e or the symbol for the exa-prefix.

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.

IEC prefixes
Prefix		Representations
Name	Symbol	Base 1024	Base 2	Value
quebi	Qi	102410	2100	1267650600228229401496703205376
robi	Ri	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

See also

• Significant figures
• Scientific notation
• Binary prefix
• International System of Units (SI)
• RKM code

External links

• Engineering Prefix User Defined Function for Excel
• Perl CPAN module for converting number to engineering notation
• Java functions for converting between a string and a double type