
Standard form is a way of expressing numbers that are too large or too small to be conveniently written in decimal form, since to do so would require writing out an inconveniently long string of digits. It may be referred to as scientific form or standard index form, or Scientific notation in the United States. This base ten notation 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.
Decimal notation | Standard Form |
---|---|
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 integer, and the coefficient m is a nonzero real number (usually between 1 and 10 in absolute value, and nearly always written as a terminating decimal). The integer n is called the exponent and the real number m is called the significand or mantissa. The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional 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 (modulus) of the significand m is at least 1 but less than 10.
Decimal floating point is a computer arithmetic system closely related to scientific notation.
History
Styles
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 absolute value of m remains at least one but less than ten (1 ≤ |m| < 10). Thus 350 is written as 3.5×102. 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 of orders of magnitude separating the numbers. It is also the form that is required when using tables of common logarithms. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1 (e.g. 0.5 is written as 5×10−1). 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 of m 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 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 multiples of 3. Consequently, the absolute value of m 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 corresponding SI prefixes, which facilitates reading and oral communication. For example, 12.5×10−9 m can be read as "twelve-point-five nanometres" 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 metres".
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 |
Calculators and computer programs typically present very large or small numbers using scientific notation, and some can be configured to uniformly present all numbers that way. Because superscript 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 notation m E n for a decimal significand m and integer exponent n means the same as m × 10n. For example 6.022×1023 is written as 6.022E23
or 6.022e23
, and 1.6×10−35 is written as 1.6E-35
or 1.6e-35
. While common in computer output, this abbreviated version of scientific notation is discouraged for published documents by some style guides.
Most popular programming languages – including Fortran, C/C++, Python, and JavaScript – use this "E" notation, which comes from Fortran and was present in the first version released for the IBM 704 in 1956. The E notation was already used by the developers of SHARE Operating System (SOS) for the IBM 709 in 1958. Later versions of Fortran (at least since FORTRAN IV as of 1961) also use "D" to signify double precision numbers in scientific notation, and newer Fortran compilers use "Q" to signify quadruple precision. The MATLAB 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
. 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
, and some Soviet ALGOL variants allowed the use of the Cyrillic letter "ю", e.g. 6.022ю+23
[citation needed]. Subsequently, the ALGOL 68 programming language provided a choice of characters: E
, e
, \
, ⊥
, or 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.
Some programming languages use other symbols. For instance, Simula uses &
(or &&
for long), as in 6.022&23
.Mathematica supports the shorthand notation 6.022*^23
(reserving the letter E
for the mathematical constant e).
The first pocket calculators supporting scientific notation appeared in 1972. 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 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. 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. The Texas Instruments TI-83 and TI-84 series of calculators (1996–present) use a small capital E
for the separator.
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".
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.
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.
Further examples of scientific notation
- An electron's mass is about 0.000000000000000000000000000000910938356 kg. In scientific notation, this is written 9.10938356×10−31 kg.
- The Earth's mass is about 5972400000000000000000000 kg. In scientific notation, this is written 5.9724×1024 kg.
- The Earth's circumference is approximately 40000000 m. 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 inch is defined as exactly 25.4 mm. Using scientific notation, this value can be uniformly expressed to any desired precision, from the nearest tenth of a millimeter 2.54×101 mm to the nearest nanometer 2.5400000×101 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 example, in November 2008 the monthly inflation rate of the Zimbabwean dollar reached 79.6 billion percent (470% per day); the approximate value with three significant figures would be 7.96×1010 %, 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, and
Multiplication and division are performed using the rules for operation with exponentiation: and
Some examples are: and
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, 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 × 2d11b or 1.001b × 10b11b using binary numbers (or shorter 1.001 × 1011 if binary context is obvious).[citation needed] 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 of Brookhaven National Laboratory in 1968, as in 1.001bB11b (or shorter: 1.001B11). For comparison, the same number in decimal 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 becomes 1.001b × 10b3d or shorter 1.001B3.
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 "b"), 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 × 10h0h = 1.40H0 = 1.40h0, or 98000 = 2.7732o × 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. 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. 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.Apple's Swift supports it as well. It is also required by the IEEE 754-2008 binary floating-point standard. Example: 1.3DEp42 represents 1.3DEh × 242.
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
- Caliò, Franca; Alessandro, Lazzari (September 2017). Elements of Mathematics with Numerical Applications. Società Editrice Esculapio. pp. 30–32. 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.
- Coghill, Anne M.; Garson, Lorrin R.; American Chemical Society, eds. (2006). The ACS style guide: effective communication of scientific information (3rd ed.). Washington, DC : Oxford; New York: American Chemical Society; Oxford University Press. p. 210. ISBN 978-0-8412-3999-9. OCLC 62872860.
- However, E notation was not included in the preliminary specification of Fortran, as of 1954. Backus, John Warner, ed. (1954-11-10). Specifications for: The IBM Mathematical FORmula TRANSlating System, FORTRAN (PDF) (Preliminary report). New York: Programming Research Group, Applied Science Division, International Business Machines Corporation. Retrieved 2022-07-04. (29 pages)
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. Retrieved 2022-07-04. (2+51+1 pages)
- DiGri, Vincent J.; King, Jane E. (April 1959) [1958-06-11]. "The SHARE 709 System: Input-Output Translation". Journal of the ACM. 6 (2): 141–144. doi:10.1145/320964.320969. 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". Communications of the ACM. 3 (5). Copenhagen: 299–311. doi:10.1145/367236.367262.
- 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.
- "Revised Report on the Algorithmic Language Algol 68". Acta Informatica. 5 (1–3): 1–236. September 1973. CiteSeerX 10.1.1.219.3999. doi:10.1007/BF00265077. 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.
- Such as the TI SR-10. Texas Instruments electronic slide rule calculator SR-10 (PDF). Dallas: 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 65 Notes newsletter for Hewlett-Packard HP-65 users, and Richard C. Vanderburgh promoted these in the newsletter for Texas Instruments SR-52 users. Davidson, Jim (January 1976). Nelson, Richard J. (ed.). "[title unknown]". 65 Notes. 3 (1). Santa Ana, CA: 4. V3N1P4.
Vanderburgh, 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 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" 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. Retrieved 2018-05-07. (NB. The term decapower 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). - Specifically, models (1987), (1987), (1987), PC-1480U (1988), PC-1490U (1990), PC-1490UII (1991), PC-E500 (1988), PC-E500S (1995), PC-E550 (1990), PC-E650 (1993), and 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.
- Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (July–September 2016). "CODATA recommended values of the fundamental physical constants: 2014". Reviews of Modern Physics. 88 (3): 035009. arXiv:1507.07956. Bibcode:2016RvMP...88c5009M. CiteSeerX 10.1.1.150.1225. doi:10.1103/RevModPhys.88.035009. S2CID 1115862.
- Luzum, Brian; Capitaine, Nicole; Fienga, Agnès; Folkner, William; Fukushima, Toshio; Hilton, James; Hohenkerk, Catherine; Krasinsky, George; Petit, Gérard; Pitjeva, Elena; Soffel, Michael; Wallace, Patrick (August 2011). "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): 293–304. Bibcode:2011CeMDA.110..293L. doi:10.1007/s10569-011-9352-4.
- Various (2000). Lide, David R. (ed.). Handbook of Chemistry and Physics (81st ed.). CRC. 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.
- electronic hexadecimal calculator/converter SR-22 (PDF) (Revision A ed.). 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.)
- Schwartz, Jake; Grevelle, Rick (2003-10-20) [April 1993]. HP16C Emulator Library for the HP48S/SX. 1.20 (1 ed.). Retrieved 2015-08-15. (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.)
- Martin, Bruce Alan (October 1968). "Letters to the editor: On binary notation". Communications of the ACM. 11 (10): 658. doi:10.1145/364096.364107. S2CID 28248410.
- 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.
- "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.
- Beebe, Nelson H. F. (2017-08-22). The Mathematical-Function Computation Handbook – Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City: Springer. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. 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.
Standard form is a way of expressing numbers that are too large or too small to be conveniently written in decimal form since to do so would require writing out an inconveniently long string of digits It may be referred to as scientific form or standard index form or Scientific notation in the United States This base ten notation 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 Decimal notation Standard Form2 2 100300 3 1024321 768 4 321768 103 53000 5 3 1046720 000 000 6 72 1090 2 2 10 1987 9 87 1020 000000 007 51 7 51 10 9 In scientific notation nonzero numbers are written in the form m 10n or m times ten raised to the power of n where n is an integer and the coefficient m is a nonzero real number usually between 1 and 10 in absolute value and nearly always written as a terminating decimal The integer n is called the exponent and the real number m is called the significand or mantissa The term mantissa can be ambiguous where logarithms are involved because it is also the traditional 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 modulus of the significand m is at least 1 but less than 10 Decimal floating point is a computer arithmetic system closely related to scientific notation HistoryStylesNormalized 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 absolute value of m remains at least one but less than ten 1 m lt 10 Thus 350 is written as 3 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 of orders of magnitude separating the numbers It is also the form that is required when using tables of common logarithms In normalized notation the exponent n is negative for a number with absolute value between 0 and 1 e g 0 5 is written as 5 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 of m 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 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 multiples of 3 Consequently the absolute value of m is in the range 1 m lt 1000 rather than 1 m lt 10 Though similar in concept engineering notation is rarely called scientific notation 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 nanometres 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 metres E notation Explicit notation E notation2 100 2E03 102 3E24 321768 103 4 321768E3 5 3 104 5 3E46 72 109 6 72E92 10 1 2E 19 87 102 9 87E27 51 10 9 7 51E 9 Calculators and computer programs typically present very large or small numbers using scientific notation and some can be configured to uniformly present all numbers that way Because superscript 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 notation m E n for a decimal significand m and integer exponent n means the same as m 10n For example 6 022 1023 is written as 6 022E23 or 6 022e23 and 1 6 10 35 is written as 1 6E 35 or 1 6e 35 While common in computer output this abbreviated version of scientific notation is discouraged for published documents by some style guides Most popular programming languages including Fortran C C Python and JavaScript use this E notation which comes from Fortran and was present in the first version released for the IBM 704 in 1956 The E notation was already used by the developers of SHARE Operating System SOS for the IBM 709 in 1958 Later versions of Fortran at least since FORTRAN IV as of 1961 also use D to signify double precision numbers in scientific notation and newer Fortran compilers use Q to signify quadruple precision The MATLAB 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 6 022 sub 10 sub 23 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 and some Soviet ALGOL variants allowed the use of the Cyrillic letter yu e g 6 022yu 23 citation needed Subsequently the ALGOL 68 programming language provided a choice of characters E e or sub 10 sub 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 Some programming languages use other symbols For instance Simula uses amp or amp amp for long as in 6 022 amp 23 Mathematica supports the shorthand notation 6 022 23 reserving the letter E for the mathematical constant e A Texas Instruments TI 84 Plus calculator display showing the Avogadro constant to three significant figures in E notation The first pocket calculators supporting scientific notation appeared in 1972 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 the HP 25 or a pair of smaller and slightly raised digits were reserved for the exponent e g 6 022 sup 23 sup 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 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 The Texas Instruments TI 83 and TI 84 series of calculators 1996 present use a small capital small E small for the separator 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 6 022 Significant figuresA 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 1230 400 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 1230 400 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 672621 923 69 51 10 27 kg which is shorthand for 1 672621 923 69 0 000000 000 51 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 Use of spacesIn 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 Further examples of scientific notationAn electron s mass is about 0 000000 000 000 000 000 000 000 000 000 910 938 356 kg In scientific notation this is written 9 109383 56 10 31 kg The Earth s mass is about 5972 400 000 000 000 000 000 000 kg In scientific notation this is written 5 9724 1024 kg The Earth s circumference is approximately 40000 000 m 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 inch is defined as exactly 25 4 mm Using scientific notation this value can be uniformly expressed to any desired precision from the nearest tenth of a millimeter 2 54 101 mm to the nearest nanometer 2 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 example in November 2008 the monthly inflation rate of the Zimbabwean dollar reached 79 6 billion percent 470 per day the approximate value with three significant figures would be 7 96 1010 or more simply a rate of 7 96 108 Converting numbersConverting 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 10 i sup n sup i to the right 10 i sup n sup i To represent the number 1 230 400 in normalized scientific notation the decimal separator would be moved 6 digits to the left and 10 sup 6 sup 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 10 i sup n sup i 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 1 234 103 12 34 102 123 4 101 1234Basic operationsGiven two numbers in scientific notation x0 m0 10n0 displaystyle x 0 m 0 times 10 n 0 and x1 m1 10n1 displaystyle x 1 m 1 times 10 n 1 Multiplication and division are performed using the rules for operation with exponentiation x0x1 m0m1 10n0 n1 displaystyle x 0 x 1 m 0 m 1 times 10 n 0 n 1 and x0x1 m0m1 10n0 n1 displaystyle frac x 0 x 1 frac m 0 m 1 times 10 n 0 n 1 Some examples are 5 67 10 5 2 34 102 13 3 10 5 2 13 3 10 3 1 33 10 2 displaystyle 5 67 times 10 5 times 2 34 times 10 2 approx 13 3 times 10 5 2 13 3 times 10 3 1 33 times 10 2 and 2 34 1025 67 10 5 0 413 102 5 0 413 107 4 13 106 displaystyle frac 2 34 times 10 2 5 67 times 10 5 approx 0 413 times 10 2 5 0 413 times 10 7 4 13 times 10 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 x0 m0 10n0 displaystyle x 0 m 0 times 10 n 0 and x1 m1 10n1 displaystyle x 1 m 1 times 10 n 1 with n0 n1 displaystyle n 0 n 1 Next add or subtract the significands x0 x1 m0 m1 10n0 displaystyle x 0 pm x 1 m 0 pm m 1 times 10 n 0 An example 2 34 10 5 5 67 10 6 2 34 10 5 0 567 10 5 2 907 10 5 displaystyle 2 34 times 10 5 5 67 times 10 6 2 34 times 10 5 0 567 times 10 5 2 907 times 10 5 Other basesWhile base ten is normally used for scientific notation powers of other bases can be used too 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 2d11b or 1 001b 10b11b using binary numbers or shorter 1 001 1011 if binary context is obvious citation needed 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 of Brookhaven National Laboratory in 1968 as in 1 001bB11b or shorter 1 001B11 For comparison the same number in decimal 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 becomes 1 001b 10b3d or shorter 1 001B3 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 b 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 10h0h 1 40H0 1 40h0 or 98000 2 7732o 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 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 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 Apple s Swift supports it as well It is also required by the IEEE 754 2008 binary floating point standard Example 1 3DEp42 represents 1 3DEh 242 Engineering notation can be viewed as a base 1000 scientific notation See alsoPositional 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 1000ReferencesCalio Franca Alessandro Lazzari September 2017 Elements of Mathematics with Numerical Applications Societa Editrice Esculapio pp 30 32 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 Coghill Anne M Garson Lorrin R American Chemical Society eds 2006 The ACS style guide effective communication of scientific information 3rd ed Washington DC Oxford New York American Chemical Society Oxford University Press p 210 ISBN 978 0 8412 3999 9 OCLC 62872860 However E notation was not included in the preliminary specification of Fortran as of 1954 Backus John Warner ed 1954 11 10 Specifications for The IBM Mathematical FORmula TRANSlating System FORTRAN PDF Preliminary report New York Programming Research Group Applied Science Division International Business Machines Corporation Retrieved 2022 07 04 29 pages 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 Retrieved 2022 07 04 2 51 1 pages DiGri Vincent J King Jane E April 1959 1958 06 11 The SHARE 709 System Input Output Translation Journal of the ACM 6 2 141 144 doi 10 1145 320964 320969 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 Manoa 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 Communications of the ACM 3 5 Copenhagen 299 311 doi 10 1145 367236 367262 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 Revised Report on the Algorithmic Language Algol 68 Acta Informatica 5 1 3 1 236 September 1973 CiteSeerX 10 1 1 219 3999 doi 10 1007 BF00265077 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 Such as the TI SR 10 Texas Instruments electronic slide rule calculator SR 10 PDF Dallas 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 65 Notes newsletter for Hewlett Packard HP 65 users and Richard C Vanderburgh promoted these in the newsletter for Texas Instruments SR 52 users Davidson Jim January 1976 Nelson Richard J ed title unknown 65 Notes 3 1 Santa Ana CA 4 V3N1P4 Vanderburgh 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 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 from the decapower for typewritten numbers as Jim also suggests For example 123 sup 45 sup sic 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 EEX and TI s EE could be changed to ED for enter decapower 1 Decapower 52 Notes Newsletter of the SR 52 Users Club Vol 1 no 6 Dayton OH November 1976 p 1 Retrieved 2018 05 07 NB The term decapower was frequently used in subsequent issues of this newsletter up to at least 1978 Specifically models 1987 1987 1987 PC 1480U 1988 PC 1490U 1990 PC 1490UII 1991 PC E500 1988 PC E500S 1995 PC E550 1990 PC E650 1993 and 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 Mohr Peter J Newell David B Taylor Barry N July September 2016 CODATA recommended values of the fundamental physical constants 2014 Reviews of Modern Physics 88 3 035009 arXiv 1507 07956 Bibcode 2016RvMP 88c5009M CiteSeerX 10 1 1 150 1225 doi 10 1103 RevModPhys 88 035009 S2CID 1115862 Luzum Brian Capitaine Nicole Fienga Agnes Folkner William Fukushima Toshio Hilton James Hohenkerk Catherine Krasinsky George Petit Gerard Pitjeva Elena Soffel Michael Wallace Patrick August 2011 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 4 293 304 Bibcode 2011CeMDA 110 293L doi 10 1007 s10569 011 9352 4 Various 2000 Lide David R ed Handbook of Chemistry and Physics 81st ed CRC 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 electronic hexadecimal calculator converter SR 22 PDF Revision A ed 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 Schwartz Jake Grevelle Rick 2003 10 20 April 1993 HP16C Emulator Library for the HP48S SX 1 20 1 ed Retrieved 2015 08 15 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 Martin Bruce Alan October 1968 Letters to the editor On binary notation Communications of the ACM 11 10 658 doi 10 1145 364096 364107 S2CID 28248410 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 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 Beebe Nelson H F 2017 08 22 The Mathematical Function Computation Handbook Programming Using the MathCW Portable Software Library 1 ed Salt Lake City Springer doi 10 1007 978 3 319 64110 2 ISBN 978 3 319 64109 6 LCCN 2017947446 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 linksLook up scientific notation in Wiktionary the free dictionary 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