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scientific notation
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{{About|a numeric notation|the musical notation|Scientific pitch notation}}Scientific notation (also referred to as scientific form or standard index form, or standard form in the UK) is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as "SCI" display mode.{| class="wikitable" style="float:right; margin:5px;"!Decimal notation!Scientific notation
| 2
2|e=0}}
| 300
3|e=2}}
| 4,321.768
4.321768|e=3}}
| −53,000
-5.3|e=4}}
| 6,720,000,000
6.72|e=9}}
| 0.2
2|e=-1}}
| 0.000 000 007 51
7.51|e=-9}}
| -0.001 14
-1.14|e=-3}}
In scientific notation all numbers are written in the form
m Ã— 10{{sup|n}}
(m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any real number, called the significand or mantissa.{{Citation |title=Computer Systems Architecture |series=Chapman & Hall/CRC Textbooks in Computing |author-first=Aharon |author-last=Yadin |page=68−69 |publisher=CRC Press |date=2016 |isbn=978-1482231069 |url=https://books.google.com/books?id=yMDBDAAAQBAJ&pg=PA68 |dead-url=no |archive-url=https://web.archive.org/web/20161122012329weblink |archive-date=2016-11-22}} However, the term "mantissa" may cause confusion because it is the name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m (as in ordinary decimal notation). In normalized notation, the exponent is chosen so that the absolute value of the coefficient is at least one but less than ten.Decimal floating point is a computer arithmetic system closely related to scientific notation.

Normalized notation

Any given real number can be written in the form {{gaps|m|e=n}} in many ways: for example, 350 can be written as {{val|3.5|e=2}} or {{val|35|e=1}} or {{val|350|e=0}}.In normalized scientific notation (called "standard form" in the UK), the exponent n is chosen so that the absolute value of m remains at least one but less than ten (1 â‰¤ |m|  Learning the Java Language > Language Basics) |publisher=Oracle Corporation |access-date=2012-03-06 |dead-url=no |archiveurl=https://web.archive.org/web/20111117041930weblink |archive-date=2011-11-17}} Python, Lua, JavaScript, and others).
  • After the introduction of the first pocket calculators supporting scientific notation in 1972 (HP-35, SR-10) the term decapower was sometimes used in the emerging user communities for the power-of-ten multiplier in order to better distinguish it from "normal" exponents. Likewise, the letter "D" was used in typewritten numbers. This notation was proposed by Jim Davidson and published in the January 1976 issue of Richard J. Nelson's Hewlett-Packard newsletter 65 NotesJOURNAL, unknown, Jim, Davidson, Richard J., Nelson, 65 Notes, January 1976, 3, 1, 4, V3N1P4, for HP-65 users, and it was adopted and carried over into the Texas Instruments community by Richard C. Vanderburgh, the editor of the 52-Notes newsletter for SR-52 users in November 1976.JOURNAL, Richard C., Vanderburgh, Decapower, 52-Notes - Newsletter of the SR-52 Users Club, 1, 6, 1, November 1976, V1N6P1,weblink 2017-05-28, no,weblink" title="web.archive.org/web/20170528031508weblink">weblink 2017-05-28, Decapower - In the January 1976 issue of 65 Notes, 65-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 (floating point number), mantissa" from the decapower for typewritten numbers, as Jim also suggests. For example, {{sic, 123, −45, 123×10−45, which is displayed in scientific notation as 1.23 -43 will now be written 1.23D-43. Perhaps, as this notation gets more and more usage, the calculator manufacturers will change their keyboard abbreviations. HP's {{button|EEX}} and TI's {{button|EE}} could be changed to {{button|ED}} (for enter decapower).}} weblink {{webarchive|url=https://web.archive.org/web/20140803040138weblink |date=2014-08-03}} (NB. The term decapower was frequently used in subsequent issues of this newsletter up to at least 1978.)
  • FORTRAN (at least since FORTRAN IV as of 1961) also uses "D" to signify double precision numbers in scientific notation.WEB,weblink UH Mānoa Mathematics » Fortran lesson 3: Format, Write, etc., Math.hawaii.edu, 2012-02-12, 2012-03-06, no,weblink" title="web.archive.org/web/20111208180836weblink">weblink 2011-12-08,
  • Similar, a "D" was used by Sharp pocket computers PC-1280, PC-1470U, PC-1475, PC-1480U, PC-1490U, PC-1490UII, PC-E500, PC-E500S, PC-E550, PC-E650 and PC-U6000 to indicate 20-digit double-precision numbers in scientific notation in BASIC between 1987 and 1995.BOOK, SHARP Taschencomputer Modell PC-1280 Bedienungsanleitung, German, SHARP Pocket Computer Model PC-1280 Operation Manual, 1987, Sharp Corporation, 7M 0.8-I(TINSG1123ECZZ)(3), 56–60,weblink 2017-03-06, no,weblink" title="web.archive.org/web/20170306145930weblink">weblink 2017-03-06, BOOK, SHARP Taschencomputer Modell PC-1475 Bedienungsanleitung, German, SHARP Pocket Computer Model PC-1475 Operation Manual, 105–108, 131–134, 370, 375, 1987, Sharp Corporation,weblink 2017-02-25,weblink" title="web.archive.org/web/20170225090805weblink">weblink 2017-02-25, BOOK, SHARP Pocket Computer Model PC-E500 Operation Manual, Sharp Corporation, 1989, 9G1KS(TINSE1189ECZZ), BOOK, SHARP Taschencomputer Modell PC-E500S Bedienungsanleitung, German, SHARP Pocket Computer Model PC-E500S Operation Manual, Sharp Corporation, 1995, 6J3KS(TINSG1223ECZZ),weblink 2017-02-24, no,weblink" title="web.archive.org/web/20170224235944weblink">weblink 2017-02-24, BOOK, 電言板5 PC-1490UII PROGRAM LIBRARY, University Co-op, Japanese, 1991, 5, (NB. University Co-operative {{webarchive|url=https://web.archive.org/web/20170727143822weblink |date=2017-07-27}})BOOK, 電言板6 PC-U6000 PROGRAM LIBRARY, University Co-op, Japanese, 1993, 6, (NB. University Co-operative {{webarchive|url=https://web.archive.org/web/20170727143822weblink |date=2017-07-27}})
  • The ALGOL 60 (1960) programming language uses a subscript ten "10" character instead of the letter E, for example: {{nowrap begin}}6.0221023{{nowrap end}}.BOOK, Report on the Algorithmic Language ALGOL 60, Peter, Naur, Peter Naur, Copenhagen, 1960, WEB, Computer Arithmetic, The Early Days of Hexadecimal, John J. G., Savard, 2018, 2005, quadibloc,weblink 2018-07-16, no,weblink" title="web.archive.org/web/20180716102439weblink">weblink 2018-07-16,
  • The use of the "10" in the various Algol standards provided a challenge on some computer systems that did not provide such a "10" character. As a consequence Stanford University Algol-W required the use of a single quote, e.g. 6.02486'+23,WEB,weblink Archived copy, 2017-04-08, no,weblink" title="web.archive.org/web/20150909195915weblink">weblink 2015-09-09, and some Soviet Algol variants allowed the use of the Cyrillic character "ÑŽ" character, e.g. 6.022ÑŽ+23.
  • Subsequently, the ALGOL 68 programming language provided the choice of 4 characters: E, e, , or 10. By examples: {{nowrap begin}}6.022E23{{nowrap end}}, {{nowrap begin}}6.022e23{{nowrap end}}, {{nowrap begin}}6.02223{{nowrap end}} or {{nowrap begin}}6.0221023{{nowrap end}}.JOURNAL, Revised Report on the Algorithmic Language Algol 68, Acta Informatica, 5, 1–236,weblink 2007-04-30, September 1973, 10.1007/BF00265077, 10.1.1.219.3999,
{{SpecialChars |alt=Decimal Exponent Symbol |link=http://mailcom.com/unicode/DecimalExponent.ttf |special=Unicode "Miscellaneous Technical" characters |fix=Unicode#External_links |characters="₁₀" (Decimal Exponent Symbol U+23E8 TTF)}}
  • Decimal Exponent Symbol is part of the Unicode Standard,HTTPS://WWW.UNICODE.ORG/VERSIONS/UNICODE7.0.0/, The Unicode Standard, e.g. {{nowrap begin}}6.022⏨23{{nowrap end}}. It is included as {{unichar|23E8|DECIMAL EXPONENT SYMBOL}} to accommodate usage in the programming languages Algol 60 and Algol 68.
  • The TI-83 series and TI-84 Plus series of calculators use a stylized E character to display decimal exponent and the 10 character to denote an equivalent ×10^ operator.WEB,weblink Archived copy, 2010-03-09, no,weblink" title="web.archive.org/web/20100214195930weblink">weblink 2010-02-14,
  • The Simula programming language requires the use of & (or && for long), for example: {{nowrap begin}}6.022&23{{nowrap end}} {{nowrap begin}}(or 6.022&&23){{nowrap end}}.WEB, SIMULA standard as defined by the SIMULA Standards Group - 3.1 Numbers,weblink 2009-10-06, August 1986, no,weblink" title="web.archive.org/web/20110724175858weblink">weblink 2011-07-24,
  • The Wolfram Language (utilized in Mathematica) allows a shorthand notation of 6.022^23. (Instead, E denotes the mathematical constant e).

{{Anchor|Dex}}Order of magnitude

Scientific notation also enables simpler order-of-magnitude comparisons. A proton's mass is {{val|0.0000000000000000000000000016726|u=kg}}. If written as {{val|1.6726|e=-27|u=kg}}, it is easier to compare this mass with that of an electron, given below. The order of magnitude of the ratio of the masses can be obtained by comparing the exponents instead of the more error-prone task of counting the leading zeros. In this case, −27 is larger than −31 and therefore the proton is roughly four orders of magnitude ({{val|10000|fmt=commas}} times) more massive than the electron.Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as billion, which might indicate either 109 or 1012.In physics and astrophysics, the number of orders of magnitude between two numbers is sometimes referred to as "dex", a contraction of "decimal exponent". For instance, if two numbers are within 1 dex of each other, then the ratio of the larger to the smaller number is less than 10. Fractional values can be used, so if within 0.5 dex, the ratio is less than 100.5, and so on.

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.Samples of usage of terminology and variants: WEB,weblink Archived copy, 2015-08-19, no,weblink 2008-12-17, , weblink, WEB,weblink Archived copy, 2007-04-07, no,weblink" title="web.archive.org/web/20070404125922weblink">weblink 2007-04-04, , WEB, Archived copy,weblink 2007-04-07, no,weblink" title="web.archive.org/web/20070225085247weblink">weblink 2007-02-25, , weblink" title="https:/-/web.archive.org/web/20061029195358weblink">weblink, weblink {{webarchive|url=https://web.archive.org/web/20150503005623weblink |date=2015-05-03}}

Further examples of scientific notation

  • An electron's mass is about {{gaps|0.000|000|000|000|000|000|000|000|000|000|910|938|356}} kg.JOURNAL, Mohr, Peter J., Newell, David B., Taylor, Barry N., July–September 2016, CODATA recommended values of the fundamental physical constants: 2014,weblink Reviews of Modern Physics, 88, 3, 035009, 10.1103/RevModPhys.88.035009, 1507.07956, no,weblink 2017-01-23, 2016RvMP...88c5009M, 10.1.1.150.1225, In scientific notation, this is written {{val|9.10938356|e=-31|u=kg}} (in SI units).
  • The Earth's mass is about {{gaps|5|972|400|000|000|000|000|000|000}} kg.JOURNAL, Luzum, Brian, Capitaine, Nicole, Fienga, Agnès, Folkner, William, Fukushima, Toshio, Hilton, James, Hohenkerk, Catherine, Krasinsky, George, Petit, Gérard, Pitjeva, Elena, Soffel, Michael, Wallace, Patrick, The IAU 2009 system of astronomical constants: The report of the IAU working group on numerical standards for Fundamental Astronomy, Celestial Mechanics and Dynamical Astronomy, 110, 4, August 2011, 293–304, 2011CeMDA.110..293L, 10.1007/s10569-011-9352-4, In scientific notation, this is written {{val|5.9724|e=24|u=kg}}.
  • The Earth's circumference is approximately {{gaps|40|000|000}} m.BOOK, Various, David R., Lide, 2000, Handbook of Chemistry and Physics, 81st, CRC, 978-0-8493-0481-1, In scientific notation, this is {{val|4|e=7|u=m}}. In engineering notation, this is written {{val|40|e=6|u=m}}. In SI writing style, this may be written {{val|40|u=Mm}}"(40 megameters).
  • An inch is defined as exactly 25.4 mm. Quoting a value of 25.400 mm shows that the value is correct to the nearest micrometer. An approximated value with only two significant digits would be {{val|2.5|e=1|u=mm}} instead. As there is no limit to the number of significant digits, the length of an inch could, if required, be written as (say) {{val|2.54000000000|e=1|u=mm}} instead.

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 {{val|1230400|fmt=commas}} in normalized scientific notation, the decimal separator would be moved 6 digits to the left and "× 106" appended, resulting in {{val|1.2304|e=6}}. The number {{val|-0.0040321}} would have its decimal separator shifted 3 digits to the right instead of the left and yield {{val|-4.0321|e=-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 {{val|1.2304|e=6}} would have its decimal separator shifted 6 digits to the right and become {{val|1230400|fmt=commas}}, while {{val|-4.0321|e=-3}} would have its decimal separator moved 3 digits to the left and be {{val|-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.
{{val|1.234|e=3}} = {{val|12.34|e=2}} = {{val|123.4|e=1}} = 1234

Basic operations

Given two numbers in scientific notation,
x_0=m_0times10^{n_0}
and
x_1=m_1times10^{n_1}
Multiplication and division are performed using the rules for operation with exponentiation:
x_0 x_1=m_0 m_1times10^{n_0+n_1}
and
frac{x_0}{x_1}=frac{m_0}{m_1}times10^{n_0-n_1}
Some examples are:
5.67times10^{-5} times 2.34times10^2 approx 13.3times10^{-5+2} = 13.3times10^{-3} = 1.33times10^{-2}
and
frac{2.34times10^2}{5.67times10^{-5}} approx 0.413times10^{2-(-5)} = 0.413times10^{7} = 4.13times10^6
Addition and subtraction require the numbers to be represented using the same exponential part, so that the significand can be simply added or subtracted:
x_0 = m_0 times10^{n_0} and x_1 = m_1 times10^{n_1} with n_0 = n_1
Next, add or subtract the significands:
x_0 pm x_1=(m_0pm m_1)times10^{n_0}
An example:
2.34times10^{-5} + 5.67times10^{-6} = 2.34times10^{-5} + 0.567times10^{-5} = 2.907times10^{-5}

{{anchor|B notation|P notation}}Other bases

While base ten is normally used for scientific notation, powers of other bases can be used too,BOOK, electronic hexadecimal calculator/converter SR-22, Texas Instruments Incorporated, 1974, 7, Revision A, 1304-389 Rev A,weblink 2017-03-20, no,weblink" title="web.archive.org/web/20170320001329weblink">weblink 2017-03-20, (NB. This calculator supports floating point numbers in scientific notation in bases 8, 10 and 16.) base 2 being the next most commonly used one.For example, in base-2 scientific notation, the number 1001b in binary (=9d) is written as 1.001b Ã— 2d11b or 1.001b Ã— 10b11b using binary numbers (or shorter 1.001 Ã— 1011 if binary context is obvious). In E-notation, this is written as 1.001bE11b (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, a shorthand notation originally proposed by Bruce Alan Martin of Brookhaven National Laboratory in 1968,JOURNAL, Letters to the editor: On binary notation, Bruce Alan, Martin, Communications of the ACM, 11, 10, October 1968, 658, 10.1145/364096.364107, as in 1.001bB11b (or shorter: 1.001B11). For comparison, the same number in decimal representation: 1.125 Ã— 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 becomes 1.001b Ã— 10b3d or shorter 1.001B3.BOOK, HP16C Emulator Library for the HP48S/SX, Jake, Schwartz, Rick, Grevelle, 2003-10-20, 1993, 1, 1.20,weblink 2015-08-15, no,weblink" title="web.archive.org/web/20160621211708weblink">weblink 2016-06-21, (NB. This library also works on the HP 48G/GX/G+. Beyond the feature set of the HP-16C, this package also supports calculations for binary, octal, and hexadecimal floating-point numbers in scientific notation in addition to the usual decimal floating-point numbers.)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 (kibi), 1B20 for 1×220 (mebi), 1B30 for 1×230 (gibi), 1B40 for 1×240 (tebi)).Similar to B (or bBOOK, HP16C Emulator Library for the HP48 - Addendum to the Operator's Manual, Jake, Schwartz, Rick, Grevelle, 2003-10-21, 1, 1.20,weblink 2015-08-15, no,weblink" title="web.archive.org/web/20160621211708weblink">weblink 2016-06-21, ), the letters H (or h) and O (or o, or C) are sometimes also used to indicate times 16 or 8 to the power as in 1.25 = 1.40h Ã— 10h0h = 1.40H0 = 1.40h0, or 98000 = 2.7732o Ã— 10o5o = 2.7732o5 = 2.7732C5.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.WEB, Rationale for International Standard - Programming Languages - C, 5.10, April 2003, 52, 153–154, 159,weblink 2010-10-17, no,weblink" title="web.archive.org/web/20160606072228weblink">weblink 2016-06-06, This notation can be produced by implementations of the printf family of functions following the C99 specification and (Single Unix Specification) IEEE Std 1003.1 POSIX standard, when using the %a or %A conversion specifiers.WEB, dprintf, fprintf, printf, snprintf, sprintf - print formatted output, The Open Group Base Specifications, Issue 7, IEEE Std 1003.1, 2013, 2013, 2001, The IEEE and The Open Group,weblink 2016-06-21, no,weblink" title="web.archive.org/web/20160621211105weblink">weblink 2016-06-21, BOOK, Nelson H. F., Beebe, The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library, 2017-08-22, Salt Lake City, UT, USA, Springer International Publishing AG, 1, 2017947446, 978-3-319-64109-6, 10.1007/978-3-319-64110-2, Starting with C++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 since C++17.WEB, cppreference.com, floating point literal,weblink 2017-03-11, no,weblink" title="web.archive.org/web/20170429201447weblink">weblink 2017-04-29, 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., Apple's Swift supports it as well.WEB, The Swift Programming Language (Swift 3.0.1), Lexical Structure, Guides and Sample Code: Developer: Language Reference, Apple Corporation,weblink 2017-03-11, no,weblink 2017-03-11, It is also required by the IEEE 754-2008 binary floating-point standard. Example: 1.3DEp42 represents 1.3DEh Ã— 242.Engineering notation can be viewed as a base-1000 scientific notation.

See also

References

{{reflist}}

External links

{{Wiktionary|scientific notation}}

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