In mathematics , a ratio ( / ˈ r eɪ ʃ ( i ) oʊ / ) shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).
124-406: In mathematics , a percentage (from Latin per centum 'by a hundred') is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign (%), although the abbreviations pct. , pct , and sometimes pc are also used. A percentage is a dimensionless number (pure number), primarily used for expressing proportions, but percent is nonetheless
248-558: A factor or multiplier . Ratios may also be established between incommensurable quantities (quantities whose ratio, as value of a fraction, amounts to an irrational number ). The earliest discovered example, found by the Pythagoreans , is the ratio of the length of the diagonal d to the length of a side s of a square , which is the square root of 2 , formally a : d = 1 : 2 . {\displaystyle a:d=1:{\sqrt {2}}.} Another example
372-424: A part of a quantity is another quantity that "measures" it and conversely, a multiple of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity (meaning aliquot part ) is a part that, when multiplied by an integer greater than one, gives the quantity. Euclid does not define
496-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
620-404: A unit of measurement in its orthography and usage. For example, 45% (read as "forty-five percent") is equal to the fraction 45 / 100 , the ratio 45:55 (or 45:100 when comparing to the total rather than the other portion), or 0.45. Percentages are often used to express a proportionate part of a total. (Similarly, one can also express a number as a fraction of 1,000, using
744-600: 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
868-412: A comparison of the quantities of a two-entity ratio can be expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is 2 3 {\displaystyle {\tfrac {2}{3}}} that of the second entity. If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to
992-411: A comparison works only when values being compared are consistent, like always expressing width in relation to height. Ratios can be reduced (as fractions are) by dividing each quantity by the common factors of all the quantities. As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers. Thus, the ratio 40:60 is equivalent in meaning to
1116-412: A concentration of 3% w/v usually means 3 g of substance in every 100 mL of solution. This cannot be converted to a dimensionless ratio, as in weight/weight or volume/volume fractions. The locations of points relative to a triangle with vertices A , B , and C and sides AB , BC , and CA are often expressed in extended ratio form as triangular coordinates . In barycentric coordinates ,
1240-402: 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 averages are quoted. As "percent" it is used to describe the grade or slope, the steepness of a road or railway , formula for which is 100 × rise / run which could also be expressed as the tangent of
1364-473: A dedicated ratio character, U+2236 ∶ RATIO . The numbers A and B are sometimes called terms of the ratio , with A being the antecedent and B being the consequent . A statement expressing the equality of two ratios A : B and C : D is called a proportion , written as A : B = C : D or A : B ∷ C : D . This latter form, when spoken or written in the English language,
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#17327803861391488-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
1612-419: A horizontal line, from which the modern "%" symbol is derived. 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 1,250 apples, one first computes the ratio 50 / 1250 = 0.04, and then multiplies by 100 to obtain 4%. The percent value can also be found by multiplying first instead of later, so in this example,
1736-399: A large extent, identified with quotients and their prospective values. However, this is a comparatively recent development, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold: first, there was the previously mentioned reluctance to accept irrational numbers as true numbers, and second,
1860-419: A major misunderstanding when journalists report about election results, for example, expressing both new results and differences with earlier results as percentages. For example, if a party obtains 41% of the vote and this is said to be a 2.5% increase, does that mean the earlier result was 40% (since 41 = 40 × (1 + 2.5 / 100 ) ) or 38.5% (since 41 = 38.5 + 2.5 )? In financial markets, it
1984-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
2108-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of
2232-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it
2356-494: A point with coordinates x : y : z has perpendicular distances to side BC (across from vertex A ) and side CA (across from vertex B ) in the ratio x : y , distances to side CA and side AB (across from C ) in the ratio y : z , and therefore distances to sides BC and AB in the ratio x : z . Since all information is expressed in terms of ratios (the individual numbers denoted by α, β, γ, x, y, and z have no meaning by themselves),
2480-414: A point with coordinates α, β, γ is the point upon which a weightless sheet of metal in the shape and size of the triangle would exactly balance if weights were put on the vertices, with the ratio of the weights at A and B being α : β , the ratio of the weights at B and C being β : γ , and therefore the ratio of weights at A and C being α : γ . In trilinear coordinates ,
2604-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
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#17327803861392728-475: A ratio as between two quantities of the same type , so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities p and q , if there exist integers m and n such that mp > q and nq > p . This condition
2852-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
2976-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
3100-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating
3224-593: A symbol abbreviating the Italian per cento . In some other languages, the form procent or 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
3348-450: A very common but ambiguous way to say that an interest rate rose from 10% per annum to 15% per annum, for example, is to say that the interest rate increased by 5%, which could theoretically mean that it increased from 10% per annum to 10.5% per annum. It is clearer to say that the interest rate increased by 5 percentage points (pp). The same confusion between the different concepts of percent(age) and percentage points can potentially cause
3472-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to
3596-405: Is 25 ⁄ 100 / 100 = 0.0025 . A term such as 100 / 100 % would also be incorrect, since it would be read as 1 percent, even if the intent was to say 100%. Whenever communicating 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
3720-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
3844-399: Is flat " and "a field is always a ring ". Ratio The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive . A ratio may be specified either by giving both constituting numbers, written as " a to b " or " a:b ", or by giving just
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3968-402: Is 10, and 20% of 50 is 10. The calculation of percentages is carried out and taught in different ways depending on the prerequisites and requirements. In this way, the usual formulas can be obtained with proportions, which saves them from having to remember them. In so-called mental arithmetic, the intermediary question is usually asked what 100% or 1% is (corresponds to). Example: 42 kg
4092-418: Is 7%. How much is (corresponds to) 100%? Given are W (percentage) and p % (percentage). We are looking for G (basic value). 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
4216-528: 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 Chicago Manual of Style —generally state that the number and percent sign are written without any space in between. However, the International System of Units and the ISO 31-0 standard require a space. The word "percentage" is often a misnomer in
4340-409: Is based on the ideas present in definition 5. In modern notation it says that given quantities p , q , r and s , p : q > r : s if there are positive integers m and n so that np > mq and nr ≤ ms . As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms p , q and r to be in proportion when p : q ∷ q : r . This
4464-428: Is common to refer to an increase of one percentage point (e.g. from 3% per annum to 4% per annum) as an increase of "100 basis points". In most forms of English , percent is usually written as two words ( per cent ), although percentage and percentile are written as one word. In American English , percent is the most common variant (but per mille is written as two words). In the early 20th century, there
4588-525: 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
4712-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
4836-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
4960-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
5084-447: Is extended to four terms p , q , r and s as p : q ∷ q : r ∷ r : s , and so on. Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions . Definitions 9 and 10 apply this, saying that if p , q and r are in proportion then p : r is the duplicate ratio of p : q and if p , q , r and s are in proportion then p : s is the triplicate ratio of p : q . In general,
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5208-404: Is followed by a decrease of x percent, and the initial amount was p , the final amount is p (1 + 0.01 x )(1 − 0.01 x ) = p (1 − (0.01 x )) ; hence the net change is an overall decrease by x percent of x 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 x = 10 percent ,
5332-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
5456-410: 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 changes : In general, a change of x percent in a quantity results in a final amount that is 100 + x percent of the original amount (equivalently, (1 + 0.01 x ) times
5580-810: Is known as the Archimedes property . Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modern notation, Euclid's definition of equality is that given quantities p , q , r and s , p : q ∷ r : s if and only if, for any positive integers m and n , np < mq , np = mq , or np > mq according as nr < ms , nr = ms , or nr > ms , respectively. This definition has affinities with Dedekind cuts as, with n and q both positive, np stands to mq as p / q stands to
5704-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
5828-506: Is no mathematical restriction and percentages may take on other values. For example, it is common to refer to 111% or −35%, especially for percent changes and comparisons. In Ancient Rome , long before the existence of the decimal system, computations were often made in fractions in the 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
5952-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
6076-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
6200-400: Is often expressed as A , B , C and D are called the terms of the proportion. A and D are called its extremes , and B and C are called its means . The equality of three or more ratios, like A : B = C : D = E : F , is called a continued proportion . Ratios are sometimes used with three or even more terms, e.g., the proportion for the edge lengths of a " two by four " that
6324-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
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#17327803861396448-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
6572-512: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
6696-400: Is ten inches long is therefore a good concrete mix (in volume units) is sometimes quoted as For a (rather dry) mixture of 4/1 parts in volume of cement to water, it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement. The meaning of such a proportion of ratios with more than two terms
6820-513: Is that the ratio of any two terms on the left-hand side is equal to the ratio of the corresponding two terms on the right-hand side. It is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος ( logos ). Early translators rendered this into Latin as ratio ("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning. Medieval writers used
6944-452: Is the dimensionless quotient between two physical quantities measured with the same unit . A quotient of two quantities that are measured with different units may be called a rate . The ratio of numbers A and B can be expressed as: When a ratio is written in the form A : B , the two-dot character is sometimes the colon punctuation mark. In Unicode , this is U+003A : COLON , although Unicode also provides
7068-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
7192-417: Is the ratio of a circle 's circumference to its diameter, which is called π , and is not just an irrational number , but a transcendental number . Also well known is the golden ratio of two (mostly) lengths a and b , which is defined by the proportion Taking the ratios as fractions and a : b {\displaystyle a:b} as having the value x , yields the equation which has
7316-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
7440-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
7564-411: Is usual either to reduce terms to the lowest common denominator , or to express them in parts per hundred ( percent ). If a mixture contains substances A, B, C and D in the ratio 5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by
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#17327803861397688-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
7812-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
7936-451: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
8060-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
8184-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
8308-409: 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 . Because of the commutative property of multiplication, reversing expressions does not change the result; for example, 50% of 20
8432-471: The 17th century, it was standard to quote interest rates in hundredths. The term "percent" is derived from the Latin per centum , meaning "hundred" or "by the hundred". The sign for "percent" evolved by gradual contraction of the Italian term per cento , meaning "for a hundred". The "per" was often abbreviated as "p."—eventually disappeared entirely. The "cento" was contracted to two circles separated by
8556-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
8680-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
8804-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
8928-495: The 50 would be multiplied by 100 to give 5,000, and this result would be divided by 1,250 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: It is not correct to divide by 100 and use the percent sign at the same time; it would literally imply division by 10,000. For example, 25% = 25 / 100 = 0.25 , not 25% / 100 , which actually
9052-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
9176-476: The amount of orange juice concentrate is 1/5 of the total liquid. In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason. Fractions can also be inferred from ratios with more than two entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction can only compare two quantities. A separate fraction can be used to compare
9300-441: The angle of inclination times 100. This 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 . Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for
9424-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
9548-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
9672-456: The context of sports statistics, when the referenced number is expressed as a decimal proportion, not a percentage: "The 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
9796-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is
9920-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
10044-556: The description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
10168-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
10292-417: The final amount, $ 198, was 10% of 10%, or 1%, less than the initial amount of $ 200. The net change is the same for a decrease of x percent, followed by an increase of x percent; the final amount is p (1 - 0.01 x )(1 + 0.01 x ) = p (1 − (0.01 x )) . This can be expanded for a case where one does not have the same percent change. If the initial amount p leads to a percent change x , and
10416-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",
10540-422: The form 1: x or x :1, where x is not necessarily an integer, to enable comparisons of different ratios. For example, the ratio 4:5 can be written as 1:1.25 (dividing both sides by 4) Alternatively, it can be written as 0.8:1 (dividing both sides by 5). Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the ratio symbol (:), though, mathematically, this makes it
10664-409: The gain" and not " 3 + 1 ⁄ 2 percent of the gain". However the titles of bonds issued by governments and other issuers use the fractional form, e.g. " 3 + 1 ⁄ 2 % Unsecured Loan Stock 2032 Series 2". (When interest rates are very low, the number 0 is included if the interest rate is less than 1%, e.g. " 0 + 3 ⁄ 4 % Treasury Stock", not " 3 ⁄ 4 % Treasury Stock".) It
10788-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
10912-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
11036-400: The lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century. Book V of Euclid's Elements has 18 definitions, all of which relate to ratios. In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that
11160-408: The limit of the sequence of these rational ratios is the irrational golden ratio. Similarly, the silver ratio of a and b is defined by the proportion This equation has the positive, irrational solution x = a b = 1 + 2 , {\displaystyle x={\tfrac {a}{b}}=1+{\sqrt {2}},} so again at least one of the two quantities a and b in
11284-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
11408-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
11532-409: The needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves
11656-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains
11780-579: The original amount). Percent changes applied sequentially do not add up in the usual way. 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), then the final price will be $ 198— not the original price of $ 200. The reason for this apparent discrepancy is that the two percent changes (+10% and −10%) are measured relative to different initial values ($ 200 and $ 220, respectively), and thus do not "cancel out". In general, if an increase of x percent
11904-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
12028-509: The positive, irrational solution x = a b = 1 + 5 2 . {\displaystyle x={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{2}}.} Thus at least one of a and b has to be irrational for them to be in the golden ratio. An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive Fibonacci numbers : even though all these ratios are ratios of two integers and hence are rational,
12152-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
12276-405: The quantities of any two of the entities covered by the ratio: for example, from a ratio of 2:3:7 we can infer that the quantity of the second entity is 3 7 {\displaystyle {\tfrac {3}{7}}} that of the third entity. If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3:2 is the same as 12:8. It
12400-408: The ratio 2:3, the latter being obtained from the former by dividing both quantities by 20. Mathematically, we write 40:60 = 2:3, or equivalently 40:60∷2:3. The verbal equivalent is "40 is to 60 as 2 is to 3." A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in simplest form or lowest terms. Sometimes it is useful to write a ratio in
12524-399: The rational number m / n (dividing both terms by nq ). Definition 6 says that quantities that have the same ratio are proportional or in proportion . Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog". Definition 7 defines what it means for one ratio to be less than or greater than another and
12648-554: The same dimension , even if their units of measurement are initially different. For example, the ratio one minute : 40 seconds can be reduced by changing the first value to 60 seconds, so the ratio becomes 60 seconds : 40 seconds . Once the units are the same, they can be omitted, and the ratio can be reduced to 3:2. On the other hand, there are non-dimensionless quotients, also known as rates (sometimes also as ratios). In chemistry, mass concentration ratios are usually expressed as weight/volume fractions. For example,
12772-399: The second percent change is y , then the final amount is p (1 + 0.01 x )(1 + 0.01 y ) . To change the above example, after an increase of x = 10 percent and decrease of y = −5 percent , the final amount, $ 209, is 4.5% more than the initial amount of $ 200. As shown above, percent changes can be applied in any order and have the same effect. In the case of interest rates ,
12896-429: The silver ratio must be irrational. Odds (as in gambling) are expressed as a ratio. For example, odds of "7 to 3 against" (7:3) mean that there are seven chances that the event will not happen to every three chances that it will happen. The probability of success is 30%. In every ten trials, there are expected to be three wins and seven losses. Ratios may be unitless , as in the case they relate quantities in units of
13020-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
13144-506: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
13268-445: The term " per mille " or the symbol " ‰ ".) If 50% of the total number of students in the class are male, that means that 50 out of every 100 students are male. If there are 500 students, then 250 of them are male. 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 a 6% increase. While many percentage values are between 0 and 100, there
13392-507: The term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the second. These definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII. Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself. Euclid defines
13516-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
13640-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
13764-548: The total and multiply by 100, we have converted to percentages : 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25:45:20:10). If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, 2 5 {\displaystyle {\tfrac {2}{5}}} , or 40% of
13888-417: The total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of the pieces of fruit are oranges. If orange juice concentrate is to be diluted with water in the ratio 1:4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1/4 the amount of water, while
14012-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
14136-612: The validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers ) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus . The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables. The existence of multiple theories seems unnecessarily complex since ratios are, to
14260-608: The value of their quotient a / b . Equal quotients correspond to equal ratios. A statement expressing the equality of two ratios is called a proportion . Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers , are rational numbers , and may sometimes be natural numbers. A more specific definition adopted in physical sciences (especially in metrology ) for ratio
14384-408: The whole is apples and 3 5 {\displaystyle {\tfrac {3}{5}}} , or 60% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion. If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction. For example, older televisions have a 4:3 aspect ratio , which means that
14508-433: The width is 4/3 of the height (this can also be expressed as 1.33:1 or just 1.33 rounded to two decimal places). More recent widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places. One of the popular widescreen movie formats is 2.35:1 or simply 2.35. Representing ratios as decimal fractions simplifies their comparison. When comparing 1.33, 1.78 and 2.35, it is obvious which format offers wider image. Such
14632-471: The word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios. Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers. The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on
14756-491: Was a dotted abbreviation form " per cent. ", as opposed to " per cent ". The form " per cent. " is still in use in 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. The term has been attributed to Latin per centum . The symbol for percent (%) evolved from
14880-511: Was equivalent to computing percentages. As denominations of money grew in the Middle Ages , computations with a denominator of 100 became increasingly standard, such that 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 . By
15004-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
15128-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
15252-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
15376-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
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