Scientific notation
Scientific notation is a way of expressing
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
or m times ten raised to the power of n, where n is an
Decimal floating point is a computer arithmetic system closely related to scientific notation.
History
Normalized notation
Any real number can be written in the form m×10 n in many ways: for example, 350 can be written as 3.5×102 or 35×101 or 350×100.
In normalized scientific notation (called "standard form" in the United Kingdom), the exponent n is chosen so that the
Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized or differently normalized form, such as engineering notation, is desired. Normalized scientific notation is often called exponential notation – although the latter term is more general and also applies when m is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (for example, 3.15×2 20).
Engineering notation
Engineering notation (often named "ENG" on scientific calculators) differs from normalized scientific notation in that the exponent n is restricted to
Significant figures
A significant figure is a digit in a number that adds to its precision. This includes all nonzero numbers, zeroes between significant digits, and zeroes indicated 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 number 1230400 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. Thus 1230400 would become 1.2304×106 if it had five significant digits. If the number were known to six or seven significant figures, it would be shown as 1.23040×106 or 1.230400×106. Thus, an additional advantage of scientific notation is that the number of significant figures is unambiguous.
Estimated final digits
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).
Additional information about precision can be conveyed through additional notation. It is often useful to know how exact the final digit or digits are. For instance, the accepted value of the mass of the proton can properly be expressed as 1.67262192369(51)×10−27 kg, which is shorthand for (1.67262192369±0.00000000051)×10−27 kg. However it is still unclear whether the error (5.1×10−37 in this case) is the maximum possible error, standard error, or some other confidence interval.
E notation
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
|
1.6E-35
or 1.6e-35
. (This use of the letter "e" is unrelated to the mathematical constant e or the exponential function ex.) This abbreviated notation – often just called "scientific notation" in computing contexts, or sometimes "E notation" to distinguish it from the standard explicit variant – reduces keystrokes, works with widely available characters, displays the exponent inline at full size, and is more concise.[citation needed] However, it is discouraged in some style guides.[2]Most popular programming languages – including
programming language supports the use of either "E" or "D".The ALGOL 60 (1960) programming language uses a subscript ten "10" character instead of the letter "E", for example: 6.0221023
.[7][8] This presented a challenge for computer systems which did not provide such a character, so ALGOL W (1966) replaced the symbol by a single quote, e.g. 6.022'+23
,[9] and some Soviet Algol variants allowed the use of the Cyrillic letter "ю", e.g. 6.022ю+23
. Subsequently, the ALGOL 68 programming language provided the choice of 4 characters: E
, e
, \
, or 10
.[10] The ALGOL "10" character was included in the Soviet GOST 10859 text encoding (1964), and was added to Unicode 5.2 (2009) as U+23E8 ⏨ DECIMAL EXPONENT SYMBOL.[11]
Some programming languages use other symbols. For instance,
6.022*^23
(reserving the letter E
for the mathematical constant e
The first
6.022 23
, as seen in the HP-25), or a pair of smaller and slightly raised digits were reserved for the exponent (e.g. 6.022 23
, as seen in the Commodore PR100). In 1976, Hewlett-Packard calculator user Jim Davidson coined the term decapower 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.[14] The letters "E" or "D" were used as a scientific-notation separator by Sharp pocket computers released between 1987 and 1995, "E" used for 10-digit numbers and "D" used for 20-digit double-precision numbers.[15] The Texas Instruments TI-83 and TI-84 series of calculators (1996–present) use a small capital E
for the separator.[16]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".[17]
Use of spaces
In normalized scientific notation, in E notation, and in engineering notation, the space (which in typesetting may be represented by a normal width space or a thin space) that is allowed only before and after "×" or in front of "E" is sometimes omitted, though it is less common to do so before the alphabetical character.[18]
Further examples of scientific notation
- An electron's mass is about 0.000000000000000000000000000000910938356 kg.[19] In scientific notation, this is written 9.10938356×10−31 kg.
- The Earth's mass is about 5972400000000000000000000 kg.[20] In scientific notation, this is written 5.9724×1024 kg.
- The Earth's circumference is approximately 40000000 m.[21] In scientific notation, this is 4×107 m. In engineering notation, this is written 40×106 m. In SI writing style, this may be written 40 Mm (40 megametres).
- An nanometer2.5400000×101 mm, or beyond.
- Hyperinflation means that too much money is put into circulation, perhaps by printing banknotes, chasing too few goods. It is sometimes defined as inflation of 50% or more in a single month. In such conditions, money rapidly loses its value. Some countries have had events of inflation of 1 million percent or more in a single month, which usually results in the rapid abandonment of the currency. For exsample, in November 2008 the monthly inflation rate of the Zimbabwean dollar reached 79.6 billion percent; the approximate value with three significant figures would be 7.96×1010 %,[22][23] or more simply a rate of 7.96×108.
Converting numbers
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.
Decimal to scientific
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 number 1,230,400 in normalized scientific notation, the decimal separator would be moved 6 digits to the left and × 106
appended, resulting in 1.2304×106. 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.
Scientific to decimal
Converting a number from scientific notation to decimal notation, first remove the × 10n
on the end, then shift the decimal separator n digits to the right (positive n) or left (negative n). The number 1.2304×106 would have its decimal separator shifted 6 digits to the right and become 1,230,400, while −4.0321×10−3 would have its decimal separator moved 3 digits to the left and be −0.0040321.
Exponential
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 shifted x places to the left (or right) and x is added to (or subtracted from) the exponent, as shown below.
Basic operations
Given two numbers in scientific notation,
Multiplication and division are performed using the rules for operation with exponentiation:
Some examples are:
Addition and subtraction 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:
Other bases
While base ten is normally used for scientific notation, powers of other bases can be used too,[24] base 2 being the next most commonly used one.
For example, in base-2 scientific notation, the number 1001b in
This is closely related to the base-2
Similar to "B" (or "b"[27]), the letters "H"[25] (or "h"[27]) and "O"[25] (or "o",[27] or "C"[25]) are sometimes also used to indicate times 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.[25]
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.
Engineering notation can be viewed as a base-1000 scientific notation.
See also
- Positional notation
- ISO/IEC 80000 – an international standard which guides the use of physical quantities and units of measurement in science
- Suzhou numerals – a Chinese numeral system formerly used in commerce, with order of magnitude written below the significand
- RKM code – a notation to specify resistor and capacitor values, with symbols for powers of 1000
References
- ISBN 978-8-89385052-0.
- ^ Edwards, John (2009). Submission Guidelines for Authors: HPS 2010 Midyear Proceedings (PDF). McLean, VA: Health Physics Society. p. 5. Retrieved 2013-03-30.
- ^ However, E notation was not included in the preliminary specification of Fortran, as of 1954.
International Business Machines Corporation. Retrieved 2022-07-04. (29 pages)International Business Machines Corporation. pp. 9, 27. Retrieved 2022-07-04. (2+51+1 pages)
- S2CID 19660148.
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.) - ^ "UH Mānoa Mathematics » Fortran lesson 3: Format, Write, etc". Math.hawaii.edu. 2012-02-12. Retrieved 2012-03-06.
- ^
For instance, DEC FORTRAN 77 (f77), Intel Fortran, Compaq/Digital Visual Fortran, and GNU Fortran(gfortran) "Double Precision, REAL**16". DEC Fortran 77 Manual. Digital Equipment Corporation. Retrieved 2022-12-21.
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 on Alpha systems.
Intel Fortran: Language Reference (PDF).Intel Corporation. 2005 [2003]. pp. 3-7–3-8, 3–10. 253261-003. Retrieved 2022-12-22. (858 pages) Compaq Visual Fortran – Language Reference (PDF). Houston:Compaq Computer Corporation. August 2001. Retrieved 2022-12-22. (1441 pages)"6. Extensions: 6.1 Extensions implemented in GNU Fortran: 6.1.8 Q exponent-letter". The GNU Fortran Compiler. 2014-06-12. Retrieved 2022-12-21.
- ^ Naur, Peter, ed. (1960). Report on the Algorithmic Language ALGOL 60. Copenhagen.
{{cite book}}
: CS1 maint: location missing publisher (link) - ^ Savard, John J. G. (2018) [2005]. "Computer Arithmetic". quadibloc. The Early Days of Hexadecimal. Retrieved 2018-07-16.
- ^ Bauer, Henry R.; Becker, Sheldon; Graham, Susan L. (January 1968). "ALGOL W – Notes For Introductory Computer Science Courses" (PDF). Stanford University, Computer Science Department. Retrieved 2017-04-08.
- S2CID 2490556.
- ^ Broukhis, Leonid (2008-01-22), "Revised proposal to encode the decimal exponent symbol" (PDF), unicode.org (Working Group Document), L2/08-030R
"The Unicode Standard" (v. 7.0.0 ed.). Retrieved 2018-03-23.
- ^ "SIMULA standard as defined by the SIMULA Standards Group – 3.1 Numbers". August 1986. Retrieved 2009-10-06.
- Texas Instruments Incorporated. 1973. 1304-739-266. Retrieved 2023-01-01. (1+1+45+1 pages) (NB. Although this manual is dated 1973, presumably version 1 of this calculator was introduced in November 1972 according to other sources.)
- ^
Jim Davidson coined decapower and recommended the "D" separator in the SR-52users. Davidson, Jim (January 1976). Nelson, Richard J. (ed.). "unknown".65 Notes. 3 (1). Santa Ana, CA: 4. V3N1P4.)
{{cite journal}}
: Cite uses generic title (helpVanderburgh, Richard C., ed. (November 1976). "Decapower" (PDF). 52-Notes – Newsletter of the SR-52 Users Club. 1 (6). Dayton, OH: 1. V1N6P1. Retrieved 2017-05-28.
Decapower – In the January 1976 issue of
(NB. The term decapower was frequently used in subsequent issues of this newsletter up to at least 1978.)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,. 52-Notes – Newsletter of the SR-52 Users Club. Vol. 1, no. 6. Dayton, OH. November 1976. p. 1. Retrieved 2018-05-07.123−45
[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). [1] "Decapower" - ^
Specifically, models PC-U6000(1993). SHARP Taschencomputer Modell PC-1280 Bedienungsanleitung [SHARP Pocket Computer Model PC-1280 Operation Manual] (PDF) (in German). Sharp Corporation. 1987. pp. 56–60. 7M 0.8-I(TINSG1123ECZZ)(3). Retrieved 2017-03-06. SHARP Taschencomputer Modell PC-1475 Bedienungsanleitung [SHARP Pocket Computer Model PC-1475 Operation Manual] (PDF) (in German). Sharp Corporation. 1987. pp. 105–108, 131–134, 370, 375. Archived from the original (PDF) on 2017-02-25. Retrieved 2017-02-25. SHARP Pocket Computer Model PC-E500 Operation Manual. Sharp Corporation. 1989. 9G1KS(TINSE1189ECZZ). SHARP Taschencomputer Modell PC-E500S Bedienungsanleitung [SHARP Pocket Computer Model PC-E500S Operation Manual] (PDF) (in German). Sharp Corporation. 1995. 6J3KS(TINSG1223ECZZ). Archived from the original (PDF) on 2017-02-24. Retrieved 2017-02-24. 電言板5 PC-1490UII PROGRAM LIBRARY [Telephone board 5 PC-1490UII program library] (in Japanese). Vol. 5. University Co-op. 1991.
電言板6 PC-U6000 PROGRAM LIBRARY [Telephone board 6 PC-U6000 program library] (in Japanese). Vol. 6. University Co-op. 1993.
- ^ Also see TI calculator character sets.
"TI-83 Programmer's Guide" (PDF). Retrieved 2010-03-09.
- ^ Whitaker, Ronald O. (1962-06-15). "Numerical Prefixes" (PDF). Crosstalk. Electronics. p. 4. Retrieved 2022-12-24. (1 page)
- ^ Samples of usage of terminology and variants:
Moller, Donald A. (June 1976). "A Computer Program For The Design And Static Analysis Of Single-Point Sub-Surface Mooring Systems: NOYFB" (PDF) (Technica Report). WHOI Document Collection. Woods Hole, MA: Woods Hole Oceanographic Institution. WHOI-76-59. Retrieved 2015-08-19.
"Cengage – the Leading Provider of Higher Education Course Materials". Archived from the original on 2007-10-19.
"Bryn Mawr College: Survival Skills for Problem Solving – Scientific Notation". Retrieved 2007-04-07.
"Scientific Notation". Retrieved 2007-04-07.
[2]
"INTOUCH 4GL a Guide to the INTOUCH Language". Archived from the original on 2015-05-03.
- S2CID 1115862.
- .
- ISBN 978-0-8493-0481-1.
- ^ Kadzere, Martin (2008-10-09). "Zimbabwe: Inflation Soars to 231 Million Percent". Harare, Zimbabwe: The Herald. Retrieved 2008-10-10.
- ^ "Zimbabwe inflation hits new high". BBC News. 2008-10-09. Archived from the original on 2009-05-14. Retrieved 2009-10-09.
- Texas Instruments Incorporated. 1974. p. 7. 1304-389 Rev A. Retrieved 2017-03-20. (NB. This calculator supports floating point numbers in scientific notation in bases 8, 10 and 16.)
- ^ floating-point numbersin scientific notation in addition to the usual decimal floating-point numbers.)
- S2CID 28248410.
- ^ a b c Schwartz, Jake; Grevelle, Rick (2003-10-21). HP16C Emulator Library for the HP48 – Addendum to the Operator's Manual. 1.20 (1 ed.). Retrieved 2015-08-15.
- ^ a b "Rationale for International Standard – Programming Languages – C" (PDF). 5.10. April 2003. pp. 52, 153–154, 159. Retrieved 2010-10-17.
- ^ The IEEE and The Open Group (2013) [2001]. "dprintf, fprintf, printf, snprintf, sprintf – print formatted output". The Open Group Base Specifications (Issue 7, IEEE Std 1003.1, 2013 ed.). Retrieved 2016-06-21.
- S2CID 30244721.
- ^ "floating point literal". cppreference.com. Retrieved 2017-03-11.
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.
- ^ "The Swift Programming Language (Swift 3.0.1)". Guides and Sample Code: Developer: Language Reference. Apple Corporation. Lexical Structure. Retrieved 2017-03-11.
External links
- Decimal to Scientific Notation Converter
- Scientific Notation to Decimal Converter
- Scientific Notation in Everyday Life
- An exercise in converting to and from scientific notation
- Scientific Notation Converter
- Scientific Notation chapter from Lessons In Electric Circuits Vol 1 DC free ebook and Lessons In Electric Circuits series.