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110-465: 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 the value of their quotient ⁠ a / b ⁠ . Equal quotients correspond to equal ratios. A statement expressing

220-675: A and b with b ≠ 0 there are natural numbers q and r such that The number q is called the quotient and r is called the remainder of the division of a by  b . The numbers q and r are uniquely determined by a and  b . This Euclidean division is key to the several other properties ( divisibility ), algorithms (such as the Euclidean algorithm ), and ideas in number theory. The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: Two important generalizations of natural numbers arise from

330-425: A + c = b . This order is compatible with the arithmetical operations in the following sense: if a , b and c are natural numbers and a ≤ b , then a + c ≤ b + c and ac ≤ bc . An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number ; for

440-466: A + 1 = S ( a ) and a × 1 = a . Furthermore, ( N ∗ , + ) {\displaystyle (\mathbb {N^{*}} ,+)} has no identity element. In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where

550-399: 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, the limit of the sequence of these rational ratios is the irrational golden ratio. Similarly, the silver ratio of

660-461: 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 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

770-488: 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 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

880-588: A tally mark for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set. The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all powers of 10 up to over 1 million. A stone carving from Karnak , dating back from around 1500 BCE and now at

990-401: A × ( b + c ) = ( a × b ) + ( a × c ) . These properties of addition and multiplication make the natural numbers an instance of a commutative semiring . Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that N {\displaystyle \mathbb {N} }

1100-404: A × 0 = 0 and a × S( b ) = ( a × b ) + a . This turns ( N ∗ , × ) {\displaystyle (\mathbb {N} ^{*},\times )} into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers . Addition and multiplication are compatible, which is expressed in the distribution law :

1210-480: 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 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

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1320-421: A bold N or blackboard bold ⁠ N {\displaystyle \mathbb {N} } ⁠ . Many other number sets are built from the natural numbers. For example, the integers are made by adding 0 and negative numbers. The rational numbers add fractions, and the real numbers add infinite decimals. Complex numbers add the square root of −1 . This chain of extensions canonically embeds

1430-766: A complete English phrase is in 1763. The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article. Starting at 0 or 1 has long been a matter of definition. In 1727, Bernard Le Bovier de Fontenelle wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889, Giuseppe Peano used N for the positive integers and started at 1, but he later changed to using N 0 and N 1 . Historically, most definitions have excluded 0, but many mathematicians such as George A. Wentworth , Bertrand Russell , Nicolas Bourbaki , Paul Halmos , Stephen Cole Kleene , and John Horton Conway have preferred to include 0. Mathematicians have noted tendencies in which definition

1540-465: 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,

1650-400: 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 , a point with coordinates α, β, γ is the point upon which a weightless sheet of metal in the shape and size of

1760-637: A geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate ( elliptic geometry ). If one takes the fifth postulate as a given, the result is Euclidean geometry . • "To draw a straight line from any point to any point." • "To describe a circle with any center and distance." Euclid, Elements , Book I, Postulates 1 & 3. Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating

1870-460: A glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield , went home to my father's house, and stayed there till I could give any proposition in

1980-399: 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,

2090-534: A line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier. As was common in ancient mathematical texts, when a proposition needed proof in several different cases, Euclid often proved only one of them (often

2200-460: A natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including Russell's paradox . To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements. In 1881, Charles Sanders Peirce provided

2310-526: A need to improve upon the logical rigor in the foundations of mathematics . In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications. Set-theoretical definitions of natural numbers were initiated by Frege . He initially defined

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2420-470: A numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628 CE. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525 CE, without being denoted by a numeral. Standard Roman numerals do not have a symbol for 0; instead, nulla (or the genitive form nullae ) from nullus ,

2530-412: 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

2640-475: 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

2750-424: 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 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

2860-509: A set (because of Russell's paradox ). The standard solution is to define a particular set with n elements that will be called the natural number n . The following definition was first published by John von Neumann , although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to infinite set as a definition of ordinal number , the sets considered below are sometimes called von Neumann ordinals . The definition proceeds as follows: It follows that

2970-401: 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 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

3080-574: A subscript (or superscript) "0" is added in the latter case: This section uses the convention N = N 0 = N ∗ ∪ { 0 } {\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}} . Given the set N {\displaystyle \mathbb {N} } of natural numbers and the successor function S : N → N {\displaystyle S\colon \mathbb {N} \to \mathbb {N} } sending each natural number to

3190-522: A thousand different editions. Theon's Greek edition was recovered and published in 1533 based on Paris gr. 2343 and Venetus Marcianus 301. In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley . Copies of the Greek text still exist, some of which can be found in the Vatican Library and

3300-530: A unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2). However, in the definition of perfect number which comes shortly afterward, Euclid treats 1 as a number like any other. Independent studies on numbers also occurred at around the same time in India , China, and Mesoamerica . Nicolas Chuquet used the term progression naturelle (natural progression) in 1484. The earliest known use of "natural number" as

3410-509: Is consistent (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong. The five Peano axioms are the following: These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic,

Ratio - Misplaced Pages Continue

3520-505: Is a free monoid on one generator. This commutative monoid satisfies the cancellation property , so it can be embedded in a group . The smallest group containing the natural numbers is the integers . If 1 is defined as S (0) , then b + 1 = b + S (0) = S ( b + 0) = S ( b ) . That is, b + 1 is simply the successor of b . Analogously, given that addition has been defined, a multiplication operator × {\displaystyle \times } can be defined via

3630-491: Is a subset of m . In other words, the set inclusion defines the usual total order on the natural numbers. This order is a well-order . Euclid%27s Elements The Elements ( Ancient Greek : Στοιχεῖα Stoikheîa ) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid c. 300 BC. It is a collection of definitions, postulates , propositions ( theorems and constructions ), and mathematical proofs of

3740-552: Is based on set theory . It defines the natural numbers as specific sets . More precisely, each natural number n is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set S has n elements" means that there exists a one to one correspondence between the two sets n and S . The sets used to define natural numbers satisfy Peano axioms. It follows that every theorem that can be stated and proved in Peano arithmetic can also be proved in set theory. However,

3850-575: Is based on an axiomatization of the properties of ordinal numbers : each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory . One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem . The set of all natural numbers

3960-803: 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

4070-410: Is not closed under subtraction (that is, subtracting one natural from another does not always result in another natural), means that N {\displaystyle \mathbb {N} } is not a ring ; instead it is a semiring (also known as a rig ). If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with

4180-399: 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

4290-403: Is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then

4400-429: Is standardly denoted N or N . {\displaystyle \mathbb {N} .} Older texts have occasionally employed J as the symbol for this set. Since natural numbers may contain 0 or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as: Alternatively, since

4510-504: Is still an active area of research. Campanus of Novara relied heavily on these Arabic translations to create his edition (sometime before 1260) which ultimately came to dominate Latin editions until the availability of Greek manuscripts in the 16th century. There are more than 100 pre-1482 Campanus manuscripts still available today. The first printed edition appeared in 1482 (based on Campanus's translation), and since then it has been translated into many languages and published in about

Ratio - Misplaced Pages Continue

4620-400: 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

4730-505: 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

4840-448: 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

4950-472: 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 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

5060-425: 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 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

5170-488: 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 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

5280-422: Is used, such as algebra texts including 0, number theory and analysis texts excluding 0, logic and set theory texts including 0, dictionaries excluding 0, school books (through high-school level) excluding 0, and upper-division college-level books including 0. There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include division by zero and

5390-701: The Bodleian Library in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available). Ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of

5500-579: The Elements from the Byzantines around 760; this version was translated into Arabic under Harun al-Rashid ( c. 800). The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation. A relatively recent discovery

5610-547: The Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements , encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Furthermore, its logical, axiomatic approach and rigorous proofs remain

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5720-452: The Elements , and applied their knowledge of it to their work. Mathematicians and philosophers, such as Thomas Hobbes , Baruch Spinoza , Alfred North Whitehead , and Bertrand Russell , have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced. The austere beauty of Euclidean geometry has been seen by many in western culture as

5830-705: The Elements : "Euclid, who put together the Elements , collecting many of Eudoxus ' theorems, perfecting many of Theaetetus ', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". Pythagoras ( c. 570–495 BC) was probably the source for most of books I and II, Hippocrates of Chios ( c. 470–410 BC, not the better known Hippocrates of Kos ) for book III, and Eudoxus of Cnidus ( c. 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated

5940-466: The Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although Euclid was known to Cicero , for instance, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received

6050-499: The Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context. A much later advance was the development of

6160-400: The whole numbers are the natural numbers plus zero. In other cases, the whole numbers refer to all of the integers , including negative integers. The counting numbers are another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are six coins on

6270-488: The 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation. No indication is given of the method of reasoning that led to the result, although the Data does provide instruction about how to approach

6380-534: The Latin word for "none", was employed to denote a 0 value. The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes . Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all. Euclid , for example, defined a unit first and then a number as a multitude of units, thus by his definition,

6490-400: The angles sum to less than two right angles. This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician Nikolai Lobachevsky published a description of acute geometry (or hyperbolic geometry ),

6600-439: The chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being 10 3 ( 5 − 5 ) = 5 + 5 6 . {\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.} The spurious Book XV

6710-404: 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 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

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6820-455: 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 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 ,

6930-432: The cornerstone of mathematics. One of the most notable influences of Euclid on modern mathematics is the discussion of the parallel postulate . In Book I, Euclid lists five postulates, the fifth of which stipulates If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles , then the two lines, if extended indefinitely, meet on that side on which

7040-475: 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

7150-602: The errors include Hilbert's geometry axioms and Tarski's . In 2018, Michael Beeson et al. used computer proof assistants to create a new set of axioms similar to Euclid's and generate proofs that were valid with those axioms. Beeson et al. checked only Book I and found these errors: missing axioms, superfluous axioms, gaps in logic (such as failing to prove points were colinear), missing theorems (such as an angle cannot be less than itself), and outright bad proofs. The bad proofs were in Book I, Proof 7 and Book I, Proposition 9. It

7260-422: 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 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

7370-403: The existence of some figure by detailing the steps he used to construct the object using a compass and straightedge . His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct'

7480-457: The figure used as an example to illustrate one given configuration. Euclid's Elements contains errors. Some of the foundational theorems are proved using axioms that Euclid did not state explicitly. A few proofs have errors, by relying on assumptions that are intuitive but not explicitly proven. Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements ]

7590-409: The first axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of Dedekind's axioms in his book The principles of arithmetic presented by a new method ( Latin : Arithmetices principia, nova methodo exposita ). This approach is now called Peano arithmetic . It

7700-511: 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, a concentration of 3% w/v usually means 3 g of substance in every 100 mL of solution. This cannot be converted to

7810-511: The idea that  0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BCE by the Babylonians, who omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BCE , but this usage did not spread beyond Mesoamerica . The use of

7920-504: 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 is extended to four terms p , q , r and s as p : q ∷ q : r ∷ r : s , and so on. Sequences that have

8030-410: 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 a part of

8140-402: The most difficult), leaving the others to the reader. Later editors such as Theon often interpolated their own proofs of these cases. Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles,

8250-536: The most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482, the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements

8360-446: The natural numbers are defined iteratively as follows: It can be checked that the natural numbers satisfy the Peano axioms . With this definition, given a natural number n , the sentence "a set S has n elements" can be formally defined as "there exists a bijection from n to S ." This formalizes the operation of counting the elements of S . Also, n ≤ m if and only if n

8470-458: The natural numbers in the other number systems. Natural numbers are studied in different areas of math. Number theory looks at things like how numbers divide evenly ( divisibility ), or how prime numbers are spread out. Combinatorics studies counting and arranging numbered objects, such as partitions and enumerations . The most primitive method of representing a natural number is to use one's fingers, as in finger counting . Putting down

8580-403: The natural numbers naturally form a subset of the integers (often denoted Z {\displaystyle \mathbb {Z} } ), they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a superscript " ∗ {\displaystyle *} " or "+" is added in the former case, and

8690-435: The natural numbers, this is denoted as ω (omega). In this section, juxtaposed variables such as ab indicate the product a × b , and the standard order of operations is assumed. While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder or Euclidean division is available as a substitute: for any two natural numbers

8800-439: The next one, one can define addition of natural numbers recursively by setting a + 0 = a and a + S ( b ) = S ( a + b ) for all a , b . Thus, a + 1 = a + S(0) = S( a +0) = S( a ) , a + 2 = a + S(1) = S( a +1) = S(S( a )) , and so on. The algebraic structure ( N , + ) {\displaystyle (\mathbb {N} ,+)} is a commutative monoid with identity element  0. It

8910-441: The number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals . The presentation of each result is given in a stylized form, which, although not invented by Euclid,

9020-595: The ordinary natural numbers via the ultrapower construction . Other generalizations are discussed in Number § Extensions of the concept . Georges Reeb used to claim provocatively that "The naïve integers don't fill up N {\displaystyle \mathbb {N} } ". There are two standard methods for formally defining natural numbers. The first one, named for Giuseppe Peano , consists of an autonomous axiomatic theory called Peano arithmetic , based on few axioms called Peano axioms . The second definition

9130-559: 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 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,

9240-413: 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, a comparison of the quantities of a two-entity ratio can be expressed as

9350-399: The propositions. The books cover plane and solid Euclidean geometry , elementary number theory , and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science , and its logical rigor was not surpassed until the 19th century. Euclid's Elements has been referred to as

9460-418: 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), a triangle analysis using barycentric or trilinear coordinates applies regardless of

9570-410: 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 is based on

9680-471: The same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence . A countable non-standard model of arithmetic satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from

9790-494: The six books of Euclid at sight". Edna St. Vincent Millay wrote in her sonnet " Euclid alone has looked on Beauty bare ", "O blinding hour, O holy, terrible day, / When first the shaft into his vision shone / Of light anatomized!". Albert Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book". The success of

9900-399: The size of the empty set . Computer languages often start from zero when enumerating items like loop counters and string- or array-elements . Including 0 began to rise in popularity in the 1960s. The ISO 31-11 standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as ISO 80000-2 . In 19th century Europe, there

10010-431: The size of the triangle. Positive integer In mathematics , the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers 0, 1, 2, 3, ... , while others start with 1, defining them as the positive integers 1, 2, 3, ... . Some authors acknowledge both definitions whenever convenient. Sometimes,

10120-433: The successor of x {\displaystyle x} is x + 1 {\displaystyle x+1} . Intuitively, the natural number n is the common property of all sets that have n elements. So, it seems natural to define n as an equivalence class under the relation "can be made in one to one correspondence ". This does not work in all set theories , as such an equivalence class would not be

10230-419: 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 the whole is apples and 3 5 {\displaystyle {\tfrac {3}{5}}} , or 60% of the whole is oranges. This comparison of

10340-422: The table", in which case they are called cardinal numbers . They are also used to put things in order, like "this is the third largest city in the country", which are called ordinal numbers . Natural numbers are also used as labels, like jersey numbers on a sports team, where they serve as nominal numbers and do not have mathematical properties. The natural numbers form a set , commonly symbolized as

10450-506: 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

10560-609: The text. Also of importance are the scholia , or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study. The Elements is still considered a masterpiece in the application of logic to mathematics . In historical context, it has proven enormously influential in many areas of science . Scientists Nicolaus Copernicus , Johannes Kepler , Galileo Galilei , Albert Einstein and Sir Isaac Newton were all influenced by

10670-455: 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 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,

10780-404: 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 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

10890-463: 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 , 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

11000-402: The two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not provable inside Peano arithmetic. A probable example is Fermat's Last Theorem . The definition of the integers as sets satisfying Peano axioms provide a model of Peano arithmetic inside set theory. An important consequence is that, if set theory

11110-423: The two uses of counting and ordering: cardinal numbers and ordinal numbers . The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0 ) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω . For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by

11220-419: The types of problems encountered in the first four books of the Elements . Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition, is general, valid, and does not depend on

11330-474: The use of letters to refer to figures. Other similar works are also reported to have been written by Theudius of Magnesia , Leon , and Hermotimus of Colophon. In the 4th century AD, Theon of Alexandria produced an edition of Euclid which was so widely used that it became the only surviving source until François Peyrard 's 1808 discovery at the Vatican of a manuscript not derived from Theon's. This manuscript,

11440-600: 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

11550-464: 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

11660-934: Was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the Almagest to Latin. The Euclid manuscript is extant and quite complete. After the translation by Adelard of Bath (known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period include Herman of Carinthia who wrote an edition around 1140, Robert of Chester (his manuscripts are referred to collectively as Adelard II, written on or before 1251), Johannes de Tinemue, possibly also known as John of Tynemouth (his manuscripts are referred to collectively as Adelard III), late 12th century, and Gerard of Cremona (sometime after 1120 but before 1187). The exact details concerning these translations

11770-430: Was mathematical and philosophical discussion about the exact nature of the natural numbers. Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act. Leopold Kronecker summarized his belief as "God made the integers, all else is the work of man". The constructivists saw

11880-413: Was not uncommon in ancient times to attribute to celebrated authors works that were not written by them. It is by these means that the apocryphal books XIV and XV of the Elements were sometimes included in the collection. The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by Apollonius . The book continues Euclid's comparison of regular solids inscribed in spheres, with

11990-460: Was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read. Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians. Proclus (412–485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on

12100-538: Was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose." Later editors have added Euclid's implicit axiomatic assumptions in their list of formal axioms. For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Known errors in Euclid date to at least 1882, when Pasch published his missing axiom . Early attempts to find all

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