|
|
| Line 1: |
Line 1: |
| '''Scientific notation''' (commonly referred to as "standard form") is a way of writing numbers that are too big or too small to be conveniently written in decimal form. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers.
| | I am Hilda from Marseille studying International Relations. I did my schooling, secured 87% and hope to find someone with same interests in Genealogy.<br><br>my blog [http://Srncomm.com/blog/2014/08/09/fifa-15-coin-hack/ FIFA coin Generator] |
| | |
| {| class="wikitable" style="float:right; margin:5px;"
| |
| |-
| |
| !Standard decimal notation
| |
| !Normalized scientific notation
| |
| |-
| |
| | 2
| |
| |{{val|2|e=0}}
| |
| |-
| |
| | 300
| |
| |{{val|3|e=2}}
| |
| |-
| |
| | 4,321.768
| |
| |{{val|4.321768|e=3}}
| |
| |-
| |
| | −53,000
| |
| |{{val|-5.3|e=4}}
| |
| |-
| |
| | 6,720,000,000
| |
| |{{val|6.72|e=9}}
| |
| |-
| |
| | 0.2
| |
| |{{val|2|e=-1}}
| |
| |-
| |
| | 0.000 000 007 51
| |
| |{{val|7.51|e=-9}}
| |
| |}
| |
| In scientific notation all numbers are written in the form of
| |
| :<math>a \times 10^b</math>
| |
| (''a'' times ten raised to the power of ''b''), where the [[exponentiation|exponent]] ''b'' is an [[integer]], and the [[coefficient]] ''a'' is any [[real number]] (however, see [[#Normalized notation|normalized notation]] below), called the ''[[significand]]'' or ''mantissa''. The term "mantissa" may cause confusion, however, because it can also refer to the [[fraction (mathematics)|fractional]] part of the common [[logarithm]].<!-- actually these are floating-point terms, not scientific notation --> If the number is negative then a minus sign precedes ''a'' (as in ordinary decimal notation).
| |
| | |
| [[Decimal floating point]] is a computer arithmetic system closely related to scientific notation.
| |
| | |
| =={{anchor|Normalized notation}}Normalized notation==
| |
| Any given number can be written in the form {{gaps|''a''|e=''b''}} 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 number|''normalized'' scientific notation]], the exponent ''b'' is chosen so that the [[absolute value]] of ''a'' remains at least one but less than ten (1 ≤ |''a''| < 10). Following these rules, 350 would always be written as {{val|3.5|e=2}}. This form allows easy comparison of two numbers of the same sign in ''a'', as the exponent ''b'' gives the number's [[order of magnitude]]. In normalized notation, the exponent ''b'' is negative for a number with absolute value between 0 and 1 (e.g., negative one half is written as {{val|-5|e=-1}}). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 cannot be written in normalized scientific notation since it cannot be expressed as {{gaps|''a''|e=''b''}} for any non-zero ''a''.
| |
| | |
| Normalized scientific form is the typical form of expression of large numbers for many fields, except during intermediate calculations or when an unnormalised form, such as [[engineering notation]], is desired. Normalized scientific notation is often called '''[[exponentiation|exponential]] notation'''—although the latter term is more general and also applies when ''a'' is not restricted to the range 1 to 10 (as in engineering notation for instance) and to [[base (exponentiation)|base]]s other than 10 (as in {{gaps|3.15|base= 2|e=20}}).
| |
| | |
| ==Engineering notation==
| |
| {{Main|Engineering notation}}
| |
| Engineering notation differs from normalized scientific notation in that the exponent ''b'' is restricted to [[multiple (mathematics)|multiples]] of 3. Consequently, the absolute value of ''a'' is in the range 1 ≤ |''a''| < 1000, rather than 1 ≤ |''a''| < 10. Though similar in concept, engineering notation is rarely called scientific notation. This allows the numbers to explicitly match their corresponding [[SI prefixes]], which facilitates reading and oral communication. For example, {{val|12.5|e=-9|u=m}} can be read as "twelve-point-five nanometers" or written as {{val|12.5|u=nm}}, while its scientific notation counterpart {{val|1.25|e=-8|u=m}} would likely be read out as "one-point-two-five times ten-to-the-negative-eight meters".
| |
| | |
| ==Significant figures==
| |
| {{Main|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 [[Significant figures#Identifying significant digits|indicated to be significant.]]
| |
| Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore, 1,230,400 has five significant figures—1, 2, 3, 0, and 4; the two zeroes serve only as placeholders and add no precision to the original number.
| |
| | |
| 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 all of the place holding zeroes are incorporated into the exponent. Following these rules, 1,230,400 becomes 1.2304 x 10<sup>6</sup>.
| |
| | |
| ===Ambiguity of the last digit===
| |
| It is customary in scientific measurements to record all the significant digits from the measurements, and to guess one additional digit if there is any information at all available to the observer to make a guess.{{Citation needed|date=March 2012}} The resulting number is considered more valuable than it would be without that extra digit, and it is considered a significant digit because it contains 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 notations. In some cases, it may be useful to know how exact the final significant digit is. For instance, the accepted value of the unit of elementary charge can properly be expressed as {{val|1.602176487|(40)|e=-19|ul=C}},<ref name="cuu-e">{{cite web|url=http://physics.nist.gov/cgi-bin/cuu/Value?e |title=NIST value for the elementary charge |publisher=Physics.nist.gov |date= |accessdate=2012-03-06}}</ref> which is shorthand for {{val|1.602176487|0.000000040|e=-19|u=C}}
| |
| | |
| =={{anchor|E notation}}E notation==
| |
| [[Image:Avogadro's number in e notation.jpg|thumb|upright|A calculator display showing the [[Avogadro constant]] in E notation]]
| |
| | |
| Most [[calculator]]s and many [[computer program]]s present very large and very small results in scientific notation. Because [[subscript and superscript|superscripted]] exponents like 10<sup>7</sup> cannot always be conveniently displayed, the letter ''E'' or ''e'' is often used to represent ''times ten raised to the power of'' (which would be written as "x 10<sup>''b''</sup>") and is followed by the value of the exponent. Note that in this usage the character ''e'' is not related to the [[e (mathematical constant)|mathematical constant ''e'']] or the [[exponential function]] ''e''<sup>''x''</sup> (a confusion that is less likely with capital ''E''); and though it stands for ''exponent'', the notation is usually referred to as ''(scientific) E notation'' or ''(scientific) e notation'', rather than ''(scientific) exponential notation'' (though the latter also occurs). The use of this notation is not encouraged in publications.<ref>{{Citation|last= Edwards |first= John |year= 2009 |title= Submission Guidelines for Authors: HPS 2010 Midyear Proceedings |publisher= Health Physics Society |publication-place= [[McLean, Virginia]] |page= 5 |url= http://hps.org/documents/2010_midyear_author-submission-guidelines.pdf |format= PDF |accessdate= 2013-03-30 }}</ref>
| |
| | |
| ===Examples and other notations ===
| |
| * In the [[Ada (programming language)|Ada]], [[C++]], [[Fortran|FORTRAN]], [[MATLAB]], [[Scilab]], [[Perl]], [[Java (programming language)|Java]],<ref>{{cite web|url=http://download.oracle.com/javase/tutorial/java/nutsandbolts/datatypes.html |title=Primitive Data Types (The Java™ Tutorials > Learning the Java Language > Language Basics) |publisher=Download.oracle.com |date= |accessdate=2012-03-06}}</ref> [[Python (programming language)|Python]] and [[Lua (programming language)|Lua]] programming languages, <code>6.0221418E23</code> or <code>6.0221418e23</code> is equivalent to [[Avogadro constant|{{val|6.0221418|e=23}}]]. FORTRAN also uses "D" to signify double precision numbers.<ref>{{cite web|url=http://www.math.hawaii.edu/lab/197/fortran/fort3.htm#double |title=UH Mānoa Mathematics » Fortran lesson 3: Format, Write, etc |publisher=Math.hawaii.edu |date=2012-02-12 |accessdate=2012-03-06}}</ref>
| |
| * The [[ALGOL|ALGOL 60]] programming language uses a subscript ten "<sub>10</sub>" character instead of the letter E, for example: {{nowrap begin}}<code lang="ALGOL">6.0221415<sub>10</sub>23</code>{{nowrap end}}.<ref>Report on the Algorithmic Language ALGOL 60, Ed. P. Naur, Copenhagen 1960</ref>
| |
| * The [[ALGOL 68]] programming language has the choice of 4 characters: e, E, \, or <sub>10</sub>. By examples: {{nowrap begin}}<code lang="ALGOL">6.0221415e23</code>{{nowrap end}}, {{nowrap begin}}<code lang="ALGOL">6.0221415E23</code>{{nowrap end}}, {{nowrap begin}}<code lang="ALGOL">6.0221415\23</code>{{nowrap end}} or {{nowrap begin}}<code lang="ALGOL">6.0221415<sub>10</sub>23</code>{{nowrap end}}.<ref>{{cite web| title=Revised Report on the Algorithmic Language Algol 68 | url=http://www.springerlink.com/content/k902506t443683p5/ | accessdate=April 30, 2007 |date=September 1973}}</ref>
| |
| {{SpecialChars
| |
| | alt = Decimal Exponent Symbol
| |
| | link = http://mailcom.com/unicode/DecimalExponent.ttf
| |
| | special = Unicode 6.0 "[http://www.unicode.org/charts/PDF/U2300.pdf Miscellaneous Technical]" characters
| |
| | fix = Unicode#External_links
| |
| | characters = something like "₁₀" ([http://mailcom.com/unicode/DecimalExponent.ttf Decimal Exponent Symbol U+23E8 TTF])
| |
| }}
| |
| * ''Decimal Exponent Symbol'' is part of "[http://unicode.org/versions/Unicode6.0.0 The Unicode Standard 6.0]" e.g. {{nowrap begin}}<code lang="ALGOL">6.0221415⏨23</code>{{nowrap end}} - it was included 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 '''<small>E</small>''' character to display ''decimal exponent'' and the <small>10</small> character to denote an equivalent [[Operator (programming)|Operator]][http://education.ti.com/downloads/guidebooks/sdk/83p/sdk83pguide.pdf].
| |
| * The [[Simula]] programming language requires the use of & (or && for [[Double precision|long]]), for example: {{nowrap begin}}<code lang="Simula">6.0221415&23</code>{{nowrap end}} {{nowrap begin}}(or <code lang="Simula">6.0221415&&23</code>){{nowrap end}}.<ref>{{cite web| title=SIMULA Standard As defined by the SIMULA Standards Group - 3.1 Numbers | url=http://prosjekt.ring.hibu.no/simula/Standard/chap_1.htm | accessdate=October 6, 2009 |date=August 1986}}</ref>
| |
| | |
| ==Order of magnitude==
| |
| {{Main|Order of magnitude}}
| |
| Scientific notation also enables simpler order-of-magnitude comparisons. A [[proton]]'s mass is {{gaps|0.000|000|000|000|000|000|000|000|001|6726}} 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 (about {{gaps|10|000}} times) more massive than the electron.
| |
| | |
| Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as ''[[Long and short scales|billion]]'', which might indicate either 10<sup>9</sup> or 10<sup>12</sup>.
| |
| | |
| In physics and astrophysics, the number of orders of magnitude between two numbers is referred to as "dex", a contraction of "decimal exponent". So. 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 <math>\scriptstyle \sqrt{10}</math>, and so on.<ref>[http://en.wiktionary.org/wiki/dex definition of dex in wiktionary.org]</ref>
| |
| | |
| ==Use of spaces==
| |
| In normalized scientific notation, in E notation, and in engineering notation, the [[Space (punctuation)|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" or "e" is sometimes omitted, though it is less common to do so before the alphabetical character.<ref>Samples of usage of terminology and variants: [https://darchive.mblwhoilibrary.org/bitstream/1912/665/1/WHOI-76-59.pdf], [http://www.brookscole.com/physics_d/templates/student_resources/003026961X_serway/review/expnot.html], [http://www.brynmawr.edu/nsf/tutorial/ss/ssnot.html], [http://www.lasalle.edu/~smithsc/Astronomy/Units/sci_notation.html], [http://www.gnsphysics.com/mathreview.pdf], [http://www.ttinet.com/doc/language_v44_003.html#heading_3.2.4.2]</ref>
| |
| | |
| ===Examples===
| |
| *An [[electron]]'s mass is about {{gaps|0.000|000|000|000|000|000|000|000|000|000|910|938|22}} kg. In scientific notation, this is written {{val|9.1093822|e=-31|u=kg}} (in SI units).
| |
| *The [[Earth]]'s [[mass]] is about {{gaps|5|973|600|000|000|000|000|000|000}} kg. In scientific notation, this is written {{val|5.9736|e=24|u=kg}}.
| |
| * The [[Earth#Shape|Earth's circumference]] is approximately {{gaps|40|000|000}} m. In scientific notation, this is {{val|4|e=7|u=m}}. In engineering notation, this is written {{val|40|e=6|u=m}}. In [[International System of Units|SI writing style]], this may be written "{{val|40|u=Mm}}" (''40 megameters'').
| |
| * An [[inch]] is {{gaps|25|400}} [[micrometre|micrometers]]. Describing an inch as {{val|2.5400|e=4|u=µm}} unambiguously states that this conversion is correct to the nearest micrometer. An approximated value with only three significant digits would be {{val|2.54|e=4|u=µm}} instead. In this example, the number of significant zeros is actually infinite (which is not the case with most scientific measurements, which have a limited degree of precision). It can be properly written with the minimum number of significant zeros used with other numbers in the application (no need to have more significant digits that other factors or addends).{{Clarify| reason = This whole paragraph is a total muddle. It looks like one person wrote one thing, and then someone else came along and contradicted it.|date=February 2012}} Or a bar can be written over a single zero, indicating that it repeats forever. The bar symbol is just as valid in scientific notation as it is in decimal notation.
| |
| | |
| ==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 the required amount, ''n'', to make the number's value within a desired range, between 1 and 10 for normalized notation. If the decimal was moved to the left, append ''x 10<sup>n</sup>''; to the right, ''x 10<sup>-n</sup>''. To represent the number 1,230,400 in normalized scientific notation, the decimal separator would be moved 6 digits to the left and ''x 10<sup>6</sup>'' appended, resulting in {{val|1.2304|e=6}}. The number -0.004 0321 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 ''x 10<sup>n</sup>'' 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 1 230 400, while {{val|-4.0321|e=-3}} would have its decimal separator moved 3 digits to the left and be {{gaps|-0.004|0321}}.
| |
| | |
| === 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 1''x'' is added to (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==
| |
| <!-- This section is linked from [[Addition]] -->
| |
| Given two numbers in scientific notation,
| |
| | |
| :<math>x_0=a_0\times10^{b_0}</math>
| |
| and
| |
| :<math>x_1=a_1\times10^{b_1}</math>
| |
| | |
| [[Multiplication]] and [[division (mathematics)|division]] are performed using the rules for operation with [[exponential functions]]:
| |
| | |
| :<math>x_0 x_1=a_0 a_1\times10^{b_0+b_1}</math>
| |
| and
| |
| :<math>\frac{x_0}{x_1}=\frac{a_0}{a_1}\times10^{b_0-b_1}</math>
| |
| | |
| Some examples are:
| |
| :<math>5.67\times10^{-5} \times 2.34\times10^2 \approx 13.3\times10^{-3} = 1.33\times10^{-2} </math>
| |
| and
| |
| :<math>\frac{2.34\times10^2}{5.67\times10^{-5}} \approx 0.413\times10^{7} = 4.13\times10^6 </math>
| |
| | |
| [[Addition]] and [[subtraction]] require the numbers to be represented using the same exponential part, so that the significand can be simply added or subtracted. :<math>x_1 = c \times10^{b_0}</math>
| |
| | |
| Next, add or subtract the significands:
| |
| | |
| :<math>x_0 \pm x_1=(a_0\pm c)\times10^{b_0}</math>
| |
| | |
| An example:
| |
| | |
| :<math>2.34\times10^{-5} + 5.67\times10^{-6} = 2.34\times10^{-5} + 0.567\times10^{-5} \approx 2.907\times10^{-5}</math>
| |
| | |
| ==Other bases==
| |
| While base 10 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 1001 in [[binary numeral system|binary]] (=9) is written as:
| |
| :{{val|1.001|e=11}} using binary numbers, or, in E notation,
| |
| :1.001 E11 (with the letter ''E'' now standing for ''times two to the power''), or
| |
| :1.125 × 2<sup>3</sup> (using [[decimal representation]]).
| |
| This is closely related to the base-2 [[floating-point]] representation commonly used in computer arithmetic.
| |
| | |
| [[Engineering notation]] can be viewed as base-1000 scientific notation.
| |
| | |
| ==See also==
| |
| * [[Binary prefix]]
| |
| * [[Engineering notation]]
| |
| * [[Floating point]]
| |
| * [[ISO 31-0]]
| |
| * [[ISO 31-11]]
| |
| * [[Scientific pitch notation]]
| |
| * [[Significant figure]]
| |
| | |
| ==Notes and references==
| |
| {{reflist}}
| |
| | |
| ==External links==
| |
| {{Wiktionary|scientific notation}}
| |
| * [http://www.miniwebtool.com/decimal-to-scientific-notation-converter/ Decimal to Scientific Notation Converter]
| |
| * [http://www.miniwebtool.com/scientific-notation-to-decimal-converter/ Scientific Notation to Decimal Converter]
| |
| * [http://www.math.toronto.edu/mathnet/plain/questionCorner/scinot.html Scientific Notation in Everyday Life]
| |
| * [http://science.widener.edu/svb/tutorial/scinot.html An exercise in converting to and from scientific notation]
| |
| | |
| {{DEFAULTSORT:Scientific Notation}}
| |
| [[Category:Numeral systems]]
| |
| [[Category:Measurement]]
| |
| [[Category:Notation]]
| |
| | |
| {{Link GA|pt}}
| |