formulasearchengine - User contributions [en]

Laws of science

2014-01-05T07:09:59Z

67.141.171.76: /* Laws of quantum mechanics */

The ''joule per mole'' (symbol: J·mol−1) is an [[SI]] derived unit of energy per amount of material. Energy is measured in [[joule]]s, and the amount of material is measured in [[mole (unit)|moles]].

Physical quantities measured in J·mol−1 usually describe quantities of energy transferred during [[Phase transition|phase transformations]] or [[chemical reaction]]s. Division by the number of moles facilitates comparison between processes involving different quantities of material and between similar processes involving different types of materials. The meaning of such a quantity is always context-dependent and, particularly for chemical reactions, is dependent on the (possibly arbitrary) definition of a 'mole' for a particular process.

For convenience and due to the range of magnitudes involved, these quantities are almost always reported in kJ·mol−1 rather than in J·mol−1. For example, heats of fusion and vaporization are usually of the order of 10 kJ·mol−1, bond energies are of the order of 100 kJ·mol−1, and ionization energies of the order of 1000 kJ·mol−1.

1 kJ·mol−1 is equal to 0.239 [[Kilocalorie per mole|kcal·mol−1]] or 1.04×10-2 [[Electronvolt|eV]] per particle. At room temperature (25 [[Celsius|°C]], 77 [[Fahrenheit|°F]], or 298.15 [[Kelvin|K]]) 1 kJ·mol−1 is equal to 0.4034 [[kT (energy)|<math>k_B T</math>]].

==References==
{{reflist}}
{{Unreferenced|date=December 2009}}

{{DEFAULTSORT:Joule Per Mole}}
[[Category:SI derived units]]

{{measurement-stub}}

Propeller (aircraft)

2014-01-02T07:18:24Z

67.141.244.25: /* History */ typo (wass -> was)

In the [[mathematics|mathematical]] area of [[graph theory]], a '''triangle-free graph''' is an undirected graph in which no three vertices form a [[triangle graph|triangle]] of edges. Triangle-free graphs may be equivalently defined as graphs with [[clique (graph theory)|clique number]] ≤ 2, graphs with [[girth (graph theory)|girth]] ≥ 4, graphs with no [[induced path|induced 3-cycle]], or [[Neighbourhood (graph theory)|locally independent]] graphs.

By [[Turán's theorem]], the ''n''-vertex triangle-free graph with the maximum number of edges is a [[complete bipartite graph]] in which the numbers of vertices on each side of the bipartition are as equal as possible.

==Triangle finding problem==
The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph.

It is possible to test whether a graph with ''m'' edges is triangle-free in time O(''m''1.41).{{sfnp|Alon|Yuster|Zwick|1994}} Another approach is to find the [[Trace (linear algebra)|trace]] of ''A''3, where ''A'' is the [[adjacency matrix]] of the graph. The trace is zero if and only if the graph is triangle-free. For dense graphs, it is more efficient to use this simple algorithm which relies on [[matrix multiplication]], since it gets the time complexity down to O(''n''2.373), where ''n'' is the number of vertices.

As {{Harvtxt|Imrich|Klavžar|Mulder|1999}} show, triangle-free graph recognition is equivalent in complexity to [[median graph]] recognition; however, the current best algorithms for median graph recognition use triangle detection as a subroutine rather than vice versa.

The [[decision tree complexity]] or [[query complexity]] of the problem, where the queries are to an oracle which stores the adjacency matrix of a graph, is Θ(''n''2). However, for [[quantum algorithm]]s, the best known lower bound is Ω(''n''), but the best known algorithm is O(''n''1.29) due to {{harvtxt|Belovs|2011}}.

==Independence number and Ramsey theory==
An [[Independent set (graph theory)|independent set]] of √''n'' vertices in an ''n''-vertex triangle-free graph is easy to find: either there is a vertex with greater than √''n'' neighbors (in which case those neighbors are an independent set) or all vertices have fewer than √''n'' neighbors (in which case any [[maximal independent set]] must have at least √''n'' vertices).<ref>{{harvtxt|Boppana|Halldórsson|1992}} p. 184, based on an idea from an earlier coloring approximation algorithm of [[Avi Wigderson]].</ref> This bound can be tightened slightly: in every triangle-free graph there exists an independent set of <math>\Omega(\sqrt{n\log n})</math> vertices, and in some triangle-free graphs every independent set has <math>O(\sqrt{n\log n})</math> vertices.{{sfnp|Kim|1995}} One way to generate triangle-free graphs in which all independent sets are small is the ''triangle-free process''<ref>{{harvtxt|Erdős|Suen|Winkler|1995}}; {{harvtxt|Bohman|2008}}</ref> in which one generates a maximal triangle-free graph by repeatedly adding randomly chosen edges that do not complete a triangle. With high probability, this process produces a graph with independence number <math>O(\sqrt{n\log n})</math>. It is also possible to find [[regular graph]]s with the same properties.{{sfnp|Alon|Ben-Shimon|Krivelevich|2008}}

These results may also be interpreted as giving asymptotic bounds on the [[Ramsey number]]s R(3,''t'') of the form <math>\Theta(\tfrac{t^2}{\log t})</math>: if the edges of a complete graph on <math>\Omega(\tfrac{t^2}{\log t})</math> vertices are colored red and blue, then either the red graph contains a triangle or, if it is triangle-free, then it must have an independent set of size ''t'' corresponding to a clique of the same size in the blue graph.

==Coloring triangle-free graphs==
Much research about triangle-free graphs has focused on [[graph coloring]]. Every [[bipartite graph]] (that is, every 2-colorable graph) is triangle-free, and [[Grötzsch's theorem]] states that every triangle-free [[planar graph]] may be 3-colored.<ref>{{Harvtxt|Grötzsch|1959}}; {{Harvtxt|Thomassen|1994}}).</ref> However, nonplanar triangle-free graphs may require many more than three colors.

{{harvtxt|Mycielski|1955}} defined a construction, now called the [[Mycielskian]], for forming a new triangle-free graph from another triangle-free graph. If a graph has [[chromatic number]] ''k'', its Mycielskian has chromatic number ''k'' + 1, so this construction may be used to show that arbitrarily large numbers of colors may be needed to color nonplanar triangle-free graphs. In particular the [[Grötzsch graph]], an 11-vertex graph formed by repeated application of Mycielski's construction, is a triangle-free graph that cannot be colored with fewer than four colors, and is the smallest graph with this property.{{sfnp|Chvátal|1974}} {{harvtxt|Gimbel &Thomassen||2000}} and {{harvtxt|Nilli|2000}} showed that the number of colors needed to color any ''m''-edge triangle-free graph is
:<math>O \left(\frac{m^{1/3}}{(\log m)^{2/3}} \right)</math>
and that there exist triangle-free graphs that have chromatic numbers proportional to this bound.

There have also been several results relating coloring to minimum degree in triangle-free graphs. {{harvtxt|Andrásfai|Erdős|Sós|1974}} proved that any ''n''-vertex triangle-free graph in which each vertex has more than 2''n''/5 neighbors must be bipartite. This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2''n''/5 neighbors per vertex. Motivated by this result, {{Harvtxt|Erdős|Simonovits|1973}} conjectured that any ''n''-vertex triangle-free graph in which each vertex has at least ''n''/3 neighbors can be colored with only three colors; however, {{Harvtxt|Häggkvist|1981}} disproved this conjecture by finding a counterexample in which each vertex of the Grötzsch graph is replaced by an independent set of a carefully chosen size. {{Harvtxt|Jin|1995}} showed that any ''n''-vertex triangle-free graph in which each vertex has more than 10''n''/29 neighbors must be 3-colorable; this is the best possible result of this type, because Häggkvist's graph requires four colors and has exactly 10''n''/29 neighbors per vertex. Finally, {{Harvtxt|Brandt|Thomassé|2006}} proved that any ''n''-vertex triangle-free graph in which each vertex has more than ''n''/3 neighbors must be 4-colorable. Additional results of this type are not possible, as Hajnal<ref>see {{Harvtxt|Erdős|Simonovits|1973}}.</ref> found examples of triangle-free graphs with arbitrarily large chromatic number and minimum degree (1/3 − ε)''n'' for any ε > 0.

==See also==
*[[Monochromatic triangle]] problem, the problem of partitioning the edges of a given graph into two triangle-free graphs

== References ==
;Notes
{{reflist|30em}}

;Sources
{{refbegin|30em}}
*{{cite arxiv
| last1 = Alon | first1 = N. | author1-link = Noga Alon
| last2 = Ben-Shimon | first2 = S.
| last3 = Krivelevich | first3 = M.
| eprint = 0812.2386
| title = A note on regular Ramsey graphs
| year = 2008}}.
*{{citation
| last1 = Alon | first1 = N. | author1-link = Noga Alon
| last2 = Yuster | first2 = R.
| last3 = Zwick | first3 = U.
| contribution = Finding and counting given length cycles
| pages = 354–364
| title = Proceedings of the 2nd [[European Symposium on Algorithms]], Utrecht, The Netherlands
| year = 1994}}.
*{{citation
| last1 = Andrásfai | first1 = B.
| last2 = Erdős | first2 = P. | author2-link = Paul Erdős
| last3 = Sós | first3 = V. T. | author3-link = Vera Sós
| doi = 10.1016/0012-365X(74)90133-2
| issue = 3
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| pages = 205–218
| title = On the connection between chromatic number, maximal clique and minimal degree of a graph
| volume = 8
| year = 1974}}.
*{{cite arxiv
| last = Bohman | first = T.
| eprint = 0806.4375
| title = The triangle-free process
| year = 2008}}.
*{{citation
| last1 = Boppana | first1 = Ravi
| last2 = Halldórsson | first2 = Magnús M.
| doi = 10.1007/BF01994876
| issue = 2
| journal = BIT
| mr = 1172185
| pages = 180–196
| title = Approximating maximum independent sets by excluding subgraphs
| volume = 32
| year = 1992}}.
*{{citation
| last1 = Brandt | first1 = S.
| last2 = Thomassé | first2 = S.
| year = 2006
| title = Dense triangle-free graphs are four-colorable: a solution to the Erdős-Simonovits problem
| url = http://www.lirmm.fr/~thomasse/liste/vega11.pdf}}.
*{{citation
| last1 = Chiba | first1 = N.
| last2 = Nishizeki | first2 = T.
| doi = 10.1137/0214017
| journal = [[SIAM Journal on Computing]]
| pages = 210–223
| title = Arboricity and subgraph listing algorithms
| issue = 1
| volume = 14
| year = 1985}}.
*{{citation
| last = Chvátal | first = Vašek | author-link = Vasek Chvatal
| contribution = The minimality of the Mycielski graph
| pages = 243–246
| publisher = Springer-Verlag
| series = Lecture Notes in Mathematics
| title = Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973)
| volume = 406
| year = 1974}}.
*{{citation
| last1 = Erdős | first1 = P. | author1-link = Paul Erdős
| last2 = Simonovits | first2 = M.
| doi = 10.1016/0012-365X(73)90126-X
| journal = Discrete Mathematics
| pages = 323–334
| issue = 4
| title = On a valence problem in extremal graph theory
| volume = 5
| year = 1973}}.
*{{citation
| doi = 10.1002/rsa.3240060217
| last1 = Erdős | first1 = P. | author1-link = Paul Erdős
| last2 = Suen | first2 = S.
| last3 = Winkler | first3 = P. | author3-link = Peter Winkler
| issue = 2–3
| journal = Random Structures and Algorithms
| pages = 309–318
| title = On the size of a random maximal graph
| volume = 6
| year = 1995}}.
*{{citation
| last1 = Gimbel | first1 = John
| last2 = Thomassen | first2 = Carsten
| doi = 10.1016/S0012-365X(00)00087-X
| issue = 1-3
| journal =Discrete Mathematics
| pages = 275–277
| title = Coloring triangle-free graphs with fixed size
| volume = 219
| year = 2000}}.
*{{citation
| last = Grötzsch | first = H. | authorlink = Herbert Grötzsch
| journal = Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe
| pages = 109–120
| title = Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel
| volume = 8
| year = 1959}}.
*{{citation
| last = Häggkvist | first = R.
| contribution = Odd cycles of specified length in nonbipartite graphs
| pages = 89–99
| title = Graph Theory (Cambridge, 1981)
| year = 1981}}.
*{{citation
| last1 = Imrich | first1 = Wilfried
| last2 = Klavžar | first2 = Sandi
| last3 = Mulder | first3 = Henry Martyn
| doi = 10.1137/S0895480197323494
| mr = 1666073
| issue = 1
| journal = [[SIAM Journal on Discrete Mathematics]]
| pages = 111–118
| title = Median graphs and triangle-free graphs
| volume = 12
| year = 1999}}.
*{{citation
| last1 = Itai | first1 = A.
| last2 = Rodeh | first2 = M.
| doi = 10.1137/0207033
| journal = [[SIAM Journal on Computing]]
| pages = 413–423
| title = Finding a minimum circuit in a graph
| issue = 4
| volume = 7
| year = 1978}}.
*{{citation
| last = Jin | first = G.
| doi = 10.1016/0012-365X(94)00063-O
| journal = Discrete Mathematics
| pages = 151–170
| title = Triangle-free four-chromatic graphs
| issue = 1-3
| volume = 145
| year = 1995}}.
*{{citation
| last = Kim | first = J. H.
| edition = 3
| journal = Random Structures and Algorithms
| pages = 173–207
| title = The Ramsey number <math>R(3,t)</math> has order of magnitude <math>\tfrac{t^2}{\log t}</math>
| volume = 7
| year = 1995}}.
*{{cite arxiv
| last1 = Magniez | first1 = Frederic
| last2 = Santha | first2 = Miklos
| last3 = Szegedy | first3 = Mario | authorlink3 = Mario Szegedy
| eprint = quant-ph/0310134
| title = Quantum Algorithms for the Triangle Problem
| year = 2003}}.
*{{citation
| last = Mycielski | first = J. | authorlink = Jan Mycielski
| journal = Colloq. Math.
| pages = 161–162
| title = Sur le coloriage des graphes
| volume = 3
| year = 1955}}.
*{{citation
| last = Nilli | first = A.
| doi = 10.1016/S0012-365X(99)00109-0
| issue = 1–3
| journal = Discrete Mathematics
| pages = 261–262
| title = Triangle-free graphs with large chromatic numbers
| volume = 211
| year = 2000}}.
*{{citation
| last = Shearer | first = J. B.
| doi = 10.1016/0012-365X(83)90273-X
| issue = 1
| journal = Discrete Mathematics
| pages = 83–87
| title = Note on the independence number of triangle-free graphs
| volume = 46
| year = 1983}}.
*{{citation
| last = Thomassen | first = C.
| doi = 10.1006/jctb.1994.1069
| journal = [[Journal of Combinatorial Theory, Series B]]
| pages = 268–279
| title = Grötzsch's 3-color theorem
| issue = 2
| volume = 62
| year = 1994}}.
*{{cite arxiv
| last1 = Belovs | first1 = Aleksandrs
| eprint = 1105.4024
| title = Span Programs for Functions with Constant-Sized 1-certificates
| year = 2011}}.
{{refend}}

== External links ==

* {{Citation | url=http://www.graphclasses.org/classes/gc_371.html | title=triangle-free
| work = [http://www.graphclasses.org/index.html Information System on Graph Classes and their Inclusions]}}

[[Category:Graph families]]

Biorhythm

2013-11-29T16:47:16Z

67.141.213.133:

[[Image:Percent 18e.svg|right|146x146px]]
In [[mathematics]], a '''percentage''' is a number or [[ratio]] expressed as a [[fraction (mathematics)|fraction]] of 100. It is often denoted using the [[percent sign]], "%", or the abbreviation "pct."

For example, 45% (read as "forty-five percent") is equal to [[fraction (mathematics)|45/100]], or [[decimals|0.45]]. A related system which expresses a number as a fraction of 1,000 uses the terms "[[per mil]]" and "millage". Percentages are used to express how large or small one quantity is relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity. For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15/2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase. The word 'percent' means 'out of 100' or 'per 100'.

Percentages are usually used to express values between zero and one. However, it is possible to express any ratio as a percentage; for example, 111% is 1.11 and −35% is −0.35.

== History ==
In [[Ancient Rome]], long before the existence of the decimal system, computations were often made in fractions which were multiples of 1/100. For example [[Augustus]] levied a tax of 1/100 on goods sold at auction known as ''centesima rerum venalium''. Computation with these fractions was similar to computing percentages. As denominations of money grew in the [[Middle Ages]], computations with a denominator of 100 become more standard and from the late 15th century to the early 16th century it became common for arithmetic texts to include such computations. Many of these texts applied these methods to profit and loss, interest rates, and the [[Rule of three (mathematics)|Rule of Three]]. By the 17th century it was standard to quote interest rates in hundredths.<ref>{{cite book|last=Smith|first=D.E.|title=History of Mathematics|isbn=0-486-20430-8
|publisher=Courier Dover Publications|year=1951∨1958|volume=2|pages=247–249}}</ref>

===Percent sign===
{{main|percent sign}}
The word "percent" is derived from the Latin ''per centum'' meaning "by the hundred".<ref>American Heritage Dictionary of the English Language, 3rd ed. (1992) Houghton Mifflin</ref>
The [[percentage sign|percent sign]] evolved by gradual contraction of the [[Italian language|Italian]] term ''per cento'', meaning "for a hundred". The "per" was often abbreviated as "p." and eventually disappeared entirely. The "cento" was contracted to two circles separated by a horizontal line from which the modern "%" is derived.<ref>Smith p. 250</ref>

== Calculations ==
The percent value is computed by multiplying the numeric value of the ratio by 100. For example, to find 50 apples as a percentage of 1250 apples, first compute the ratio 50/1250 = .04, and then multiply by 100 to obtain 4%. The percent value can also be found by multiplying first, so in this example the 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1250 to give 4%.

To calculate a percentage of a percentage, convert both percentages to fractions of 100, or to decimals, and multiply them. For example, 50% of 40% is:
:{{nowrap|1= (50/100) × (40/100) = 0.50 × 0.40 = 0.20 = 20/100 = 20%.}}
It is not correct to divide by 100 and use the percent sign at the same time. (E.g. {{nowrap|1= 25% = 25/100 = 0.25}}, not {{nowrap|25% / 100}}, which actually is {{nowrap|1= (25/100) / 100 = 0.0025}}. A term such as {{nowrap|(100/100)%}} would also be incorrect, this would be read as (1) percent even if the intent was to say 100%.)

Whenever we talk about a percentage, it is important to specify what it is relative to, i.e. what is the total that corresponds to 100%. The following problem illustrates this point.

:In a certain college 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female?

We are asked to compute the [[ratio]] of female computer science majors to all computer science majors. We know that 60% of all students are female, and among these 5% are computer science majors, so we conclude that (60/100) × (5/100) = 3/100 or 3% of all students are female computer science majors. Dividing this by the 10% of all students that are computer science majors, we arrive at the answer: 3%/10% = 30/100 or 30% of all computer science majors are female.

This example is closely related to the concept of [[conditional probability]].

== Percentage increase and decrease ==

Sometimes due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the ''initial value'' of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%).

Some other examples of [[Percent difference#Percentage change|percent changes]]:
* An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial); in other words, the quantity has doubled.
* An increase of 800% means the final amount is 9 times the original (100% + 800% = 900% = 9 times as large).
* A decrease of 60% means the final amount is 40% of the original (100% − 60% = 40%).
* A decrease of 100% means the final amount is ''zero'' (100% − 100% = 0%).

In general, a change of <math>x</math> percent in a quantity results in a final amount that is <math>100+x</math> percent of the original amount (equivalently, <math>1+0.01x</math> times the original amount).

==Compounding percentages==

It is important to understand that percent changes, as they have been discussed here, ''do not add'' in the usual way, if applied sequentially. For example, if the 10% increase in price considered earlier (on the $200 item, raising its price to $220) is followed by a 10% decrease in the price (a decrease of $22), the final price will be $198, ''not'' the original price of $200. The reason for the apparent discrepancy is that the two percent changes (+10% and −10%) are measured relative to ''different'' quantities ($200 and $220, respectively), and thus do not "cancel out".

In general, if an increase of <math>x</math> percent is followed by a decrease of <math>x</math> percent, and the initial amount was <math>p</math>, the final amount is <math>p((1+0.01x)(1-0.01x))=p(1-(0.01x)^2)</math>; thus the net change is an overall decrease by <math>x</math> percent ''of'' <math>x</math> percent (the square of the original percent change when expressed as a decimal number). Thus, in the above example, after an increase and decrease of <math>x=10</math> percent, the final amount, $198, was 10% of 10%, or 1%, less than the initial amount of $200.

This can be expanded for a case where you do not have the same percent change. If the initial percent change is <math>x</math> and the second percent change is <math>y</math>, and the initial amount was <math>p</math>, then the final amount is <math>p((1+0.01x)(1+0.01y))</math>. To change the above example, after an increase of <math>x=10</math> and decrease of <math>y=-5</math> percent, the final amount, $209, is 4.5% more than the initial amount of $200.

In the case of [[interest rate]]s, it is a common practice to state the percent change differently. If an interest rate rises from 10% to 15%, for example, it is typical to say, "The interest rate increased by 5%" — rather than by 50%, which would be correct when measured as a percentage of the initial rate (i.e., from 0.10 to 0.15 is an increase of 50%). Such ambiguity can be avoided by using the term "[[percentage point]]s". In the previous example, the interest rate "increased by 5 percentage points" from 10% to 15%. If the rate then drops by 5 percentage points, it will return to the initial rate of 10%, as expected.

== Word and symbol ==
{{main|Percent sign}}

In [[British English]], ''percent'' is sometimes written as two words (''per cent'', although ''percentage'' and ''percentile'' are written as one word).<ref>{{cite web|url=http://www.wsu.edu/~brians/errors/percent1.html|first=Paul|last=Brians|title=Percent/per cent|work=Common Errors in English Usage|publisher=Washington State University|accessdate=22 November 2010}}</ref> In [[American English]], ''percent'' is the most common variant<ref>{{cite web|url=http://oxforddictionaries.com/view/entry/m_en_us1276846?rskey=eX1NKq&result=1#m_en_us1276846|title=Percent (per cent)|publisher=Oxford Dictionarie |accessdate=22 November 2010}}</ref> (but cf. ''per mille'' written as two words).

In the early part of the twentieth century, there was a dotted abbreviation form ''"per cent.",'' as opposed to ''"per cent"''. The form "per cent." is still in use as a part of the highly formal language found in certain documents like commercial loan agreements (particularly those subject to, or inspired by, common law), as well as in the [[Hansard]] transcripts of British Parliamentary proceedings. While the term has been attributed to [[Latin]] ''per centum'', this is a [[Dog Latin|pseudo-Latin]] construction and the term was likely originally adopted from the [[French language|French]] ''pour cent''.{{citation needed|date=December 2010}} The concept of considering values as parts of a hundred is originally [[Ancient Greece|Greek]]. The [[percent sign|symbol for percent]] (%) evolved from a symbol abbreviating the Italian ''per cento''. In some other languages, the form ''prosent'' is used instead. Some languages use both a word derived from ''percent'' and an expression in that language meaning the same thing, e.g. Romanian ''procent'' and ''la sută'' (thus, ''10 %'' can be read or sometimes written ''ten for [each] hundred'', similarly with the English ''one out of ten''). Other abbreviations are rarer, but sometimes seen.

Grammar and style guides often differ as to how percentages are to be written. For instance, it is commonly suggested that the word percent (or per cent) be spelled out in all texts, as in "1 percent" and not "1%". Other guides prefer the word to be written out in humanistic texts, but the symbol to be used in scientific texts. Most guides agree that they always be written with a numeral, as in "5 percent" and not "five percent", the only exception being at the beginning of a sentence: "Ten percent of all writers love style guides." Decimals are also to be used instead of fractions, as in "3.5 percent of the gain" and not "3 ½ percent of the gain". It is also widely accepted to use the percent symbol (%) in tabular and graphic material.

In line with common English practice, style guides—such as the [[The Chicago Manual of Style]]—generally state that the number and percent sign are written without any space in between.<ref>
{{cite web
| title = The Chicago Manual of Style
| publisher = [[University of Chicago Press]]
| year = 2003
| url = http://www.chicagomanualofstyle.org/
| accessdate = 2007-01-05}}
</ref>
However, the [[International System of Units]] and the [[ISO 31-0]] standard require a space.<ref>
{{cite web
| title = The International System of Units
| publisher = [[International Bureau of Weights and Measures]]
| year = 2006
| url = http://www.bipm.org/utils/common/pdf/si_brochure_8.pdf
| accessdate = 2007-08-06}}
</ref><ref>
{{cite web
| title = ISO 31-0 — Quantities and units – Part 0: General principles
| publisher = [[International Organization for Standardization]]
| date = 1999-12-22
| url = http://www.iso.org/iso/en/CatalogueDetailPage.CatalogueDetail?CSNUMBER=3621
| accessdate = 2007-01-05}}
</ref>

== Related units ==
* [[Percentage point]]
* [[permille|Per mille]] (‰) 1 part in 1,000
* [[Basis point]] ({{Unicode|‱}}) 1 part in 10,000
* [[Per cent mille]] (pcm) 1 part in 100,000
* [[Parts-per notation]]
* [[Concentration]]
* [[Grade (slope)]]
* [[Per-unit system]]

== Other uses ==

The word "percentage" is often a misnomer in the context of sports statistics, when the referenced number is expressed as a decimal proportion, not a percentage: "The [[Phoenix Suns|Phoenix Suns']] [[Shaquille O'Neal]] led the [[NBA]] with a .609 [[field goal percentage]] (FG%) during the 2008-09 season." (O'Neal made 60.9% of his shots, not 0.609%.) Likewise, the [[winning percentage]] of a team, the fraction of matches that the club has won, is also usually expressed as a decimal proportion; a team that has a .500 winning percentage has won 50% of their matches. The practice is probably related to the similar way that [[batting average]]s are quoted.

As "percent" it is used to describe the steepness of the [[Grade (slope)|slope]] of a [[road]] or [[Rail tracks|railway]], formula for which is <math> 100 \frac{\text{rise}}{\text{run}}</math> which could also be expressed as the tangent of the angle of inclination times 100. The is the ratio of distances a vehicle would advance vertically and horizontally, respectively, when going up- or downhill, expressed in percent.

Percentage is also used to express composition of a mixture by [[mass percent]] and [[mole percent]].

== Practical applications ==
* [[Baker percentage]]
* [[Volume percent]]

==See also==
*[[Annual percentage rate]]
*[[Percent difference]]
*[[Percentage change]]

==References==
{{Reflist}}

== External links ==
{{Wiktionary|percentage}}
*[http://www.khanacademy.org/video/describing-the-meaning-of-percent?playlist=Developmental Math "Describing the Meaning of Percent"] [[Khan Academy]] module (first of seven)
*[http://percentagecalculator.info Percentage Calculator] - an educational tool that helps kids grasp the idea of percentages. It calculates percentages and solves a few related problems on the fly, showing how different formulas relate to each other.

{{Use dmy dates|date=May 2011}}

[[Category:Elementary arithmetic]]
[[Category:Fractions]]