Scientific notation is a way of expressingnumbers that are too large or too small to be conveniently written indecimal form, since to do so would require writing out an inconveniently long string of digits. It may be referred to asscientific form orstandard index form, orstandard form in the United Kingdom. Thisbase ten notation is commonly used by scientists, mathematicians, and engineers, in part because it can simplify certainarithmetic operations. Onscientific calculators, it is usually known as "SCI" display mode.
| Decimal notation | Scientific notation |
|---|---|
| 2 | 2×100 |
| 300 | 3×102 |
| 4321.768 | 4.321768×103 |
| −53000 | −5.3×104 |
| 6720000000 | 6.72×109 |
| 0.2 | 2×10−1 |
| 987 | 9.87×102 |
| 0.00000000751 | 7.51×10−9 |
In scientific notation, nonzero numbers are written in the form
orm times ten raised to the power ofn, wheren is aninteger, and thecoefficientm is a nonzeroreal number (usually between 1 and 10 in absolute value, and nearly always written as aterminating decimal). The integern is called theexponent and the real numberm is called thesignificand ormantissa.[1] The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of thefractional part of thecommon logarithm. If the number is negative then a minus sign precedesm, as in ordinary decimal notation. Innormalized notation, the exponent is chosen so that theabsolute value (modulus) of the significandm is at least 1 but less than 10.
Decimal floating point is a computer arithmetic system closely related to scientific notation.
For performing calculations with aslide rule, standard form expression is required. Thus, the use of scientific notation increased as engineers and educators used that tool. SeeSlide rule#History.
Any real number can be written in the formm×10^n in many ways: for example, 350 can be written as3.5×102 or35×101 or350×100.
Innormalized scientific notation (called "standard form" in the United Kingdom), the exponentn is chosen so that theabsolute value ofm remains at least one and less than ten (1 ≤ |m| < 10). Thus 350 is written as3.5×102. This form allows easy comparison of numbers: numbers with bigger exponents are (due to the normalization) larger than those with smaller exponents, and subtraction of exponents gives an estimate of the number oforders of magnitude separating the numbers. It is also the form that is required when using tables ofcommon logarithms. In normalized notation, the exponentn is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as5×10−1). The 10 and exponent are often omitted when the exponent is 0. For a series of numbers that are to be added or subtracted (or otherwise compared), it can be convenient to use the same value ofn for all elements of the series.
Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized or differently normalized form, such asengineering notation, is desired. Normalized scientific notation is often calledexponential notation – although the latter term is more general and also applies whenm is not restricted to the range 1 to 10 (as in engineering notation for instance) and tobases other than 10 (for example,3.15×2^20).
Engineering notation (often named "ENG" on scientific calculators) differs from normalized scientific notation in that the exponentn is restricted tomultiples of 3. Consequently, the absolute value ofm is in the range 1 ≤ |m| < 1000, rather than 1 ≤ |m| < 10. Though similar in concept, engineering notation is rarely called scientific notation. Engineering notation allows the numbers to explicitly match their correspondingSI prefixes, which facilitates reading and oral communication. For example,12.5×10−9 m can be read as "twelve-point-five nanometres" and written as12.5 nm, while its scientific notation equivalent1.25×10−8 m would likely be read out as "one-point-two-five times ten-to-the-negative-eight metres".
| Explicit notation | E notation |
|---|---|
| 2×100 | 2E0 |
| 3×102 | 3E2 |
| 4.321768×103 | 4.321768E3 |
| −5.3×104 | -5.3E4 |
| 6.72×109 | 6.72E9 |
| 2×10−1 | 2E-1 |
| 9.87×102 | 9.87E2 |
| 7.51×10−9 | 7.51E-9 |
Calculators andcomputer programs typically present very large or small numbers using scientific notation, and some can be configured to uniformly present all numbers that way. Becausesuperscript exponents like 107 can be inconvenient to display or type, the letter "E" or "e" (for "exponent") is often used to represent "times ten raised to the power of", so that the notationm E n for a decimal significandm and integer exponentn means the same asm × 10n. For example6.022×1023 is written as6.022E23 or6.022e23, and1.6×10−35 is written as1.6E-35 or1.6e-35. While common in computer output, this abbreviated version of scientific notation is discouraged for published documents by some style guides.[2][3]
Most popular programming languages – includingFortran,C/C++,Python, andJavaScript – use this "E" notation, which comes from Fortran and was present in the first version released for theIBM 704 in 1956.[4] The E notation was already used by the developers ofSHARE Operating System (SOS) for theIBM 709 in 1958.[5] Later versions of Fortran (at least sinceFORTRAN IV as of 1961) also use "D" to signifydouble precision numbers in scientific notation,[6] and newer Fortran compilers use "Q" to signifyquadruple precision.[7] TheMATLAB programming language supports the use of either "E" or "D".
TheALGOL 60 (1960) programming language uses a subscript ten "10" character instead of the letter "E", for example:6.0221023.[8][9] This presented a challenge for computer systems which did not provide such a character, soALGOL W (1966) replaced the symbol by a single quote, e.g.6.022'+23,[10] and some Soviet ALGOL variants allowed the use of the Cyrillic letter "ю", e.g.6.022ю+23[citation needed]. Subsequently, theALGOL 68 programming language provided a choice of characters:E,e,\,⊥, or10.[11] The ALGOL "10" character was included in the SovietGOST 10859 text encoding (1964), and was added toUnicode 5.2 (2009) asU+23E8 ⏨DECIMAL EXPONENT SYMBOL.[12]
Some programming languages use other symbols. For instance,Simula uses& (or&& forlong), as in6.022&23.[13]Mathematica supports the shorthand notation6.022*^23 (reserving the letterE for themathematical constante).
The firstpocket calculators supporting scientific notation appeared in 1972.[14] To enter numbers in scientific notation calculators include a button labeled "EXP" or "×10x", among other variants. The displays of pocket calculators of the 1970s did not display an explicit symbol between significand and exponent; instead, one or more digits were left blank (e.g.6.022 23, as seen in theHP-25), or a pair of smaller and slightly raised digits were reserved for the exponent (e.g.6.02223, as seen in theCommodore PR100). In 1976,Hewlett-Packard calculator user Jim Davidson coined the termdecapower for the scientific-notation exponent to distinguish it from "normal" exponents, and suggested the letter "D" as a separator between significand and exponent in typewritten numbers (for example,6.022D23); these gained some currency in the programmable calculator user community.[15] The letters "E" or "D" were used as a scientific-notation separator bySharppocket computers released between 1987 and 1995, "E" used for 10-digit numbers and "D" used for 20-digit double-precision numbers.[16] TheTexas InstrumentsTI-83 andTI-84 series of calculators (1996–present) use asmall capitalE for the separator.[17]
In 1962, Ronald O. Whitaker of Rowco Engineering Co. proposed a power-of-ten system nomenclature where the exponent would be circled, e.g. 6.022 × 103 would be written as "6.022③".[18]
A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroesindicated to be significant. Leading and trailing zeroes are not significant digits, because they exist only to show the scale of the number. Unfortunately, this leads to ambiguity. The number1230400 is usually read to have five significant figures: 1, 2, 3, 0, and 4, the final two zeroes serving only as placeholders and adding no precision. The same number, however, would be used if the last two digits were also measured precisely and found to equal 0 – seven significant figures.
When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the placeholding zeroes are no longer required. Thus1230400 would become1.2304×106 if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as1.23040×106 or1.230400×106. Thus, an additional advantage of scientific notation is that the number of significant figures is unambiguous.
It is customary in scientific measurement to record all the definitely known digits from the measurement and to estimate at least one additional digit if there is any information at all available on its value. The resulting number contains more information than it would without the extra digit, which may be considered a significant digit because it conveys some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together).
More detailed information about the precision of a value written in scientific notation can be conveyed through additional notation. For instance, the accepted value of the mass of theproton can be expressed as1.67262192595(52)×10−27 kg, which is shorthand for(1.67262192595±0.00000000052)×10−27 kg. However, it is unclear whether an error expressed in this way (5.2×10−37 in this case) is the maximum possible error,standard error, or some otherconfidence interval.
In normalized scientific notation, in E notation, and in engineering notation, thespace (which intypesetting may be represented by a normal width space or athin space) that is allowedonly before and after "×" or in front of "E" is sometimes omitted, though it is less common to do so before the alphabetical character.[19]
Converting a number in these cases means to either convert the number into scientific notation form, convert it back into decimal form or to change the exponent part of the equation. None of these alter the actual number, only how it's expressed.
First, move the decimal separator point sufficient places,n, to put the number's value within a desired range, between 1 and 10 for normalized notation. If the decimal was moved to the left, append× 10n; to the right,× 10−n. To represent the number1,230,400 in normalized scientific notation, the decimal separator would be moved 6 digits to the left and× 106 appended, resulting in1.2304×106. The number−0.0040321 would have its decimal separator shifted 3 digits to the right instead of the left and yield−4.0321×10−3 as a result.
Converting a number from scientific notation to decimal notation, first remove the× 10n on the end, then shift the decimal separatorn digits to the right (positiven) or left (negativen). The number1.2304×106 would have its decimal separator shifted 6 digits to the right and become1,230,400, while−4.0321×10−3 would have its decimal separator moved 3 digits to the left and be−0.0040321.
Conversion between different scientific notation representations of the same number with different exponential values is achieved by performing opposite operations of multiplication or division by a power of ten on the significand and an subtraction or addition of one on the exponent part. The decimal separator in the significand is shiftedx places to the left (or right) andx is added to (or subtracted from) the exponent, as shown below.
Given two numbers in scientific notation,and
Multiplication anddivision are performed using the rules for operation withexponentiation:and
Some examples are:and
Addition andsubtraction require the numbers to be represented using the same exponential part, so that the significand can be simply added or subtracted:
Next, add or subtract the significands:
An example:
While base ten is normally used for scientific notation, powers of other bases can be used too,[25] base 2 being the next most commonly used one.
For example, in base-2 scientific notation, the number 1001b inbinary (=9d) is written as1.001b × 2d11b or1.001b × 10b11b using binary numbers (or shorter1.001 × 1011 if binary context is obvious).[citation needed] In E notation, this is written as1.001bE11b (or shorter: 1.001E11) with the letter "E" now standing for "times two (10b) to the power" here. In order to better distinguish this base-2 exponent from a base-10 exponent, a base-2 exponent is sometimes also indicated by using the letter "B" instead of "E",[26] a shorthand notation originally proposed byBruce Alan Martin ofBrookhaven National Laboratory in 1968,[27] as in1.001bB11b (or shorter: 1.001B11). For comparison, the same number indecimal representation:1.125 × 23 (using decimal representation), or 1.125B3 (still using decimal representation). Some calculators use a mixed representation for binary floating point numbers, where the exponent is displayed as decimal number even in binary mode, so the above becomes1.001b × 10b3d or shorter 1.001B3.[26]
This is closely related to the base-2floating-point representation commonly used in computer arithmetic, and the usage of IECbinary prefixes (e.g. 1B10 for 1×210 (kibi), 1B20 for 1×220 (mebi), 1B30 for 1×230 (gibi), 1B40 for 1×240 (tebi)).
Similar to "B" (or "b"[28]), the letters "H"[26] (or "h"[28]) and "O"[26] (or "o",[28] or "C"[26]) are sometimes also used to indicatetimes 16 or 8 to the power as in 1.25 =1.40h × 10h0h = 1.40H0 = 1.40h0, or 98000 =2.7732o × 10o5o = 2.7732o5 = 2.7732C5.[26]
Another similar convention to denote base-2 exponents is using a letter "P" (or "p", for "power"). In this notation the significand is always meant to be hexadecimal, whereas the exponent is always meant to be decimal.[29] This notation can be produced by implementations of theprintf family of functions following theC99 specification and (Single Unix Specification)IEEE Std 1003.1POSIX standard, when using the%a or%A conversion specifiers.[29][30][31] Starting withC++11,C++ I/O functions could parse and print the P notation as well. Meanwhile, the notation has been fully adopted by the language standard sinceC++17.[32]Apple'sSwift supports it as well.[33] It is also required by theIEEE 754-2008 binary floating-point standard. Example: 1.3DEp42 represents1.3DEh × 242.
Engineering notation can be viewed as a base-1000 scientific notation.
Sayre, David, ed. (1956-10-15).The FORTRAN Automatic Coding System for the IBM 704 EDPM: Programmer's Reference Manual(PDF). New York: Applied Science Division and Programming Research Department,International Business Machines Corporation. pp. 9, 27. Retrieved2022-07-04. (2+51+1 pages)
It tells the input translator that the field to be converted is a decimal number of the form ~X.XXXXE ± YY where E implies that the value of ~x.xxxx is to be scaled by ten to the ±YY power.(4 pages) (NB. This was presented at the ACM meeting 11–13 June 1958.)
Digital Fortran 77 also allows the syntax Qsnnn, if the exponent field is within the T_floating double precision range. […] A REAL*16 constant is a basic real constant or an integer constant followed by a decimal exponent. A decimal exponent has the form: Qsnn […] s is an optional sign […] nn is a string of decimal digits […] This type of constant is only available onAlpha systems.Intel Fortran: Language Reference(PDF).Intel Corporation. 2005 [2003]. pp. 3-7 –3-8,3–10. 253261-003. Retrieved2022-12-22. (858 pages)Compaq Visual Fortran – Language Reference(PDF). Houston:Compaq Computer Corporation. August 2001. Retrieved2022-12-22. (1441 pages)
"6. Extensions: 6.1 Extensions implemented in GNU Fortran: 6.1.8 Q exponent-letter".The GNU Fortran Compiler. 2014-06-12. Retrieved2022-12-21.
"The Unicode Standard" (v. 7.0.0 ed.). Retrieved2018-03-23.
Vanderburgh, Richard C., ed. (November 1976)."Decapower"(PDF).52-Notes – Newsletter of the SR-52 Users Club.1 (6). Dayton, OH: 1. V1N6P1. Archived fromthe original(PDF) on 2017-05-28. Retrieved2017-05-28.Decapower – In the January 1976 issue of65-Notes (V3N1p4) Jim Davidson (HP-65 Users Club member #547) suggested the term "decapower" as a descriptor for the power-of-ten multiplier used in scientific notation displays. I'm going to begin using it in place of "exponent" which is technically incorrect, and the letter D to separate the "mantissa" from the decapower for typewritten numbers, as Jim also suggests. For example,
[1]"Decapower".52-Notes – Newsletter of the SR-52 Users Club. Vol. 1, no. 6. Dayton, OH. November 1976. p. 1. Retrieved2018-05-07. (NB. The termdecapower was frequently used in subsequent issues of this newsletter up to at least 1978.)123−45 [sic] which is displayed in scientific notation as1.23 -43 will now be written1.23D-43. Perhaps, as this notation gets more and more usage, the calculator manufacturers will change their keyboard abbreviations. HP's EEX and TI's EE could be changed to ED (for enter decapower).
電言板6 PC-U6000 PROGRAM LIBRARY [Telephone board 6 PC-U6000 program library] (in Japanese). Vol. 6. University Co-op. 1993.
"TI-83 Programmer's Guide"(PDF). Retrieved2010-03-09.
"INTOUCH 4GL a Guide to the INTOUCH Language". Archived fromthe original on 2015-05-03.
The hexadecimal floating-point literals were not part of C++ until C++17, although they can be parsed and printed by the I/O functions since C++11: both C++ I/O streams when std::hexfloat is enabled and the C I/O streams: std::printf, std::scanf, etc. See std::strtof for the format description.
