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66-401: OEIS records information on integer sequences of interest to both professional and amateur mathematicians , and is widely cited. As of February 2024, it contains over 370,000 sequences, and is growing by approximately 30 entries per day. Each entry contains the leading terms of the sequence, keywords , mathematical motivations, literature links, and more, including the option to generate

132-435: A graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence , or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input. Neil Sloane started collecting integer sequences as a graduate student in 1964 to support his work in combinatorics . The database

198-477: A is not equal to 0 or 1, and b is not a rational number, then any value of a is a transcendental number (there can be more than one value if complex number exponentiation is used). An example that provides a simple constructive proof is The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, log 2 ⁡ 3 {\displaystyle \log _{\sqrt {2}}3} ,

264-487: A ( n ) = n -th term of sequence A n or –1 if A n has fewer than n terms". This sequence spurred progress on finding more terms of A000022 . A100544 lists the first term given in sequence A n , but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a (1) of sequence A n might seem a good alternative if it were not for the fact that some sequences have offsets of 2 and greater. This line of thought leads to

330-566: A Moroccan mathematician from Fez specializing in Islamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, 3 1 5 3 {\displaystyle {\frac {3\quad 1}{5\quad 3}}} ." This same fractional notation appears soon after in

396-535: A consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum ), who probably discovered them while identifying sides of the pentagram . The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as

462-642: A decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence . For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic ), and in many other ways. As

528-406: A kind of reductio ad absurdum that "...established the deductive organization on the basis of explicit axioms..." as well as "...reinforced the earlier decision to rely on deductive reasoning for proof". This method of exhaustion is the first step in the creation of calculus. Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because

594-409: A proof may be found in quadratic irrationals . The proof for the irrationality of the square root of two can be generalized using the fundamental theorem of arithmetic . This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in

660-484: A proof to show that π is irrational, whence it follows immediately that π is irrational also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different method , which showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed

726-403: A remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats. Conversely, suppose we are faced with a repeating decimal , we can prove that it is a fraction of two integers. For example, consider: Here the repetend is 162 and

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792-459: A website (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year. Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the omnibus database. In 2004, Sloane celebrated the addition of

858-692: A week of discussion on the SeqFan mailing list, following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a special sequence for A200000. A300000 was defined in February 2018, and by end of January 2023 the database contained more than 360,000 sequences. Besides integer sequences, the OEIS also catalogs sequences of fractions , the digits of transcendental numbers , complex numbers and so on by transforming them into integer sequences. Sequences of fractions are represented by two sequences (named with

924-459: Is x 0  = (2  + 1) . It is clearly algebraic since it is the root of an integer polynomial, ( x 3 − 1 ) 2 = 2 {\displaystyle (x^{3}-1)^{2}=2} , which is equivalent to ( x 6 − 2 x 3 − 1 ) = 0 {\displaystyle (x^{6}-2x^{3}-1)=0} . This polynomial has no rational roots, since

990-400: Is 0. This special usage has a solid mathematical basis in certain counting functions; for example, the totient valence function N φ ( m ) ( A014197 ) counts the solutions of φ( x ) = m . There are 4 solutions for 4, but no solutions for 14, hence a (14) of A014197 is 0—there are no solutions. Other values are also used, most commonly −1 (see A000230 or A094076 ). The OEIS maintains

1056-488: Is a ratio of integers and therefore a rational number. Dov Jarden gave a simple non- constructive proof that there exist two irrational numbers a and b , such that a is rational: Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that √ 2 is transcendental , hence irrational. This theorem states that if a and b are both algebraic numbers , and

1122-428: Is a real root of a polynomial with integer coefficients. Those that are not algebraic are transcendental . The real algebraic numbers are the real solutions of polynomial equations where the coefficients a i {\displaystyle a_{i}} are integers and a n ≠ 0 {\displaystyle a_{n}\neq 0} . An example of an irrational algebraic number

1188-496: Is as follows: Greek mathematicians termed this ratio of incommensurable magnitudes alogos , or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.' Another legend states that Hippasus

1254-688: Is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 ( A006842 ) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 ( A006843 ). Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions (here 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, ... ( A000796 )), binary expansions (here 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, ... ( A004601 )), or continued fraction expansions (here 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, ... ( A001203 )). The OEIS

1320-524: Is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e , the golden ratio φ , and the square root of two . In fact, all square roots of natural numbers , other than of perfect squares , are irrational. Like all real numbers, irrational numbers can be expressed in positional notation , notably as

1386-443: Is not a finite number of nonzero digits), unlike any rational number. The same is true for binary , octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases. To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m , there can never be a remainder greater than or equal to m . If 0 appears as

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1452-511: Is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc. " In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to

1518-422: Is rational. For some positive integers m and n , we have It follows that The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made

1584-418: Is why ratios of incommensurable [quantities] appear... .[Q]uantities are, in other words, continuous". What this means is that contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. In fact, these divisions of quantity must necessarily be infinite . For example, consider a line segment: this segment can be split in half, that half split in half,

1650-592: The Yuktibhāṣā . In the Middle Ages , the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as algebraic objects . Middle Eastern mathematicians also merged the concepts of " number " and " magnitude " into a more general idea of real numbers , criticized Euclid's idea of ratios , developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude. In his commentary on Book 10 of

1716-532: The Elements , the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows: "It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value

1782-488: The Fibonacci sequence , the lazy caterer's sequence , and the coefficients in the series expansion of ζ ( n + 2 ) ζ ( n ) {\displaystyle \textstyle {{\zeta (n+2)} \over {\zeta (n)}}} . In OEIS lexicographic order, they are: whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2. Very early in

1848-411: The lexicographical order of the sequences, so each sequence has a predecessor and a successor (its "context"). OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also the sign of each element. Sequences of weight distribution codes often omit periodically recurring zeros. For example, consider: the prime numbers , the palindromic primes ,

1914-404: The n th term of the sequence. Zero is often used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n consecutive primes to form an n × n magic square of least magic constant , or 0 if no such magic square exists." The value of a (1) (a 1 × 1 magic square) is 2; a (3) is 1480028129. But there is no such 2 × 2 magic square, so a (2)

1980-401: The rational root theorem shows that the only possibilities are ±1, but x 0 is greater than 1. So x 0 is an irrational algebraic number. There are countably many algebraic numbers, since there are countably many integer polynomials. Almost all irrational numbers are transcendental . Examples are e and π , which are transcendental for all nonzero rational  r. Because

2046-559: The 100,000th sequence to the database, A100000 , which counts the marks on the Ishango bone . In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000 , was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after

On-Line Encyclopedia of Integer Sequences - Misplaced Pages Continue

2112-479: The 12th century Bhāskara II evaluated some of these formulas and critiqued them, identifying their limitations. During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain irrational values of trigonometric functions . Jyeṣṭhadeva provided proofs for these infinite series in

2178-684: The 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. Historian Carl Benjamin Boyer , however, writes that "such claims are not well substantiated and unlikely to be true". Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots. Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In

2244-672: The algebra he used could not be applied to the square root of 17. Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India. There are references to such calculations in the Samhitas , Brahmanas , and the Shulba Sutras (800 BC or earlier). It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since

2310-464: The algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3 π  + 2, π  +  √ 2 and e √ 3 are irrational (and even transcendental). The decimal expansion of an irrational number never repeats (meaning the decimal expansion does not repeat the same number or sequence of numbers) or terminates (this means there

2376-482: The applications of the subject. Johann Heinrich Lambert proved (1761) that π cannot be rational, and that e is irrational if n is rational (unless n  = 0). While Lambert's proof is often called incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous. Adrien-Marie Legendre (1794), after introducing the Bessel–;Clifford function , provided

2442-484: The concept of irrationality, as he attributes the following to irrational magnitudes: "their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it." The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation in

2508-437: The denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact k th power of another integer, then that first integer's k th root is irrational. Perhaps the numbers most easy to prove irrational are certain logarithms . Here is a proof by contradiction that log 2  3 is irrational (log 2  3 ≈ 1.58 > 0). Assume log 2  3

2574-464: The field of mathematics. Irrational number In mathematics , the irrational numbers ( in- + rational ) are all the real numbers that are not rational numbers . That is, irrational numbers cannot be expressed as the ratio of two integers . When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable , meaning that they share no "measure" in common, that is, there

2640-521: The form of square roots and fourth roots . In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions. Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century . Al-Hassār ,

2706-418: The half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxes , which demonstrated the contradictions inherent in the mathematical thought of

On-Line Encyclopedia of Integer Sequences - Misplaced Pages Continue

2772-408: The hands of Weierstrass, Leopold Kronecker (Crelle, 101), and Charles Méray . Continued fractions , closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence through the writings of Joseph-Louis Lagrange . Dirichlet also added to the general theory, as have numerous contributors to

2838-414: The history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. "I resisted adding these sequences for a long time, partly out of a desire to maintain the dignity of the database, and partly because A22 was only known to 11 terms!", Sloane reminisced. One of the earliest self-referential sequences Sloane accepted into the OEIS was A031135 (later A091967 ) "

2904-493: The keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence , 1 5 , 1 4 , 1 3 , 2 5 , 1 2 , 3 5 , 2 3 , 3 4 , 4 5 {\displaystyle \textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}} ,

2970-430: The length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In this example we would multiply by 10 to obtain: Now we multiply this equation by 10 where r is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 10 : The result of

3036-472: The necessary logical foundation for incommensurable ratios". This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created. As a result of the distinction between number and magnitude, geometry became

3102-476: The only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases, algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive of x and x as x squared and x cubed instead of x to

3168-424: The other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning

3234-432: The plot on the right shows a clear "gap" between two distinct point clouds, the " uninteresting numbers " (blue dots) and the "interesting" numbers that occur comparatively more often in sequences from the OEIS. It contains essentially prime numbers (red), numbers of the form a (green) and highly composite numbers (yellow). This phenomenon was studied by Nicolas Gauvrit , Jean-Paul Delahaye and Hector Zenil who explained

3300-451: The prime numbers. Each n is a member of exactly one of these two sequences, and in principle it can be determined which sequence each n belongs to, with two exceptions (related to the two sequences themselves): This entry, A046970 , was chosen because it comprehensively contains every OEIS field, filled. In 2009, the OEIS database was used by Philippe Guglielmetti to measure the "importance" of each integer number. The result shown in

3366-579: The question "Does sequence A n contain the number n ?" and the sequences A053873 , "Numbers n such that OEIS sequence A n contains n ", and A053169 , " n is in this sequence if and only if n is not in sequence A n ". Thus, the composite number 2808 is in A053873 because A002808 is the sequence of composite numbers, while the non-prime 40 is in A053169 because it is not in A000040 ,

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3432-399: The resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid . The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine ( Crelle's Journal , 74), Georg Cantor (Annalen, 5), and Richard Dedekind . Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to

3498-418: The same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan . The square root of 2 was likely the first number proved irrational. The golden ratio is another famous quadratic irrational number. The square roots of all natural numbers that are not perfect squares are irrational and

3564-502: The second power and x to the third power. Also crucial to Zeno's work with incommensurable magnitudes was the fundamental focus on deductive reasoning that resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory. Out of this necessity, Eudoxus developed his method of exhaustion ,

3630-512: The speed of the two clouds in terms of algorithmic complexity and the gap by social factors based on an artificial preference for sequences of primes, even numbers, geometric and Fibonacci-type sequences and so on. Sloane's gap was featured on a Numberphile video in 2013. List of amateur mathematicians This is a list of amateur mathematicians —people whose primary vocation did not involve mathematics (or any similar discipline) yet made notable, and sometimes important, contributions to

3696-574: The time. While Zeno's paradoxes accurately demonstrated the deficiencies of contemporary mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore, further investigation had to occur. The next step was taken by Eudoxus of Cnidus , who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea

3762-415: The two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000 A matches the tail end of 10 A exactly. Here, both 10,000 A and 10 A have .162 162 162 ... after the decimal point. Therefore, when we subtract the 10 A equation from the 10,000 A equation, the tail end of 10 A cancels out the tail end of 10,000 A leaving us with: Then

3828-410: The work of Leonardo Fibonacci in the 13th century. The 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre , and especially of Leonhard Euler . The completion of the theory of complex numbers in the 19th century entailed the differentiation of irrationals into algebraic and transcendental numbers , the proof of the existence of transcendental numbers, and

3894-528: The year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880, and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers , separating them into two groups having certain characteristic properties. The subject has received later contributions at

3960-441: Was at first stored on punched cards . He published selections from the database in book form twice: These books were well-received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online –first as an email service (August 1994), and soon thereafter as

4026-458: Was brought to light by Zeno of Elea , who questioned the conception that quantities are discrete and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for "whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects", but Zeno found that in fact "[quantities] in general are not discrete collections of units; this

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4092-642: Was limited to plain ASCII text until 2011, and it still uses a linear form of conventional mathematical notation (such as f ( n ) for functions , n for running variables , etc.). Greek letters are usually represented by their full names, e.g. , mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits, almost always referred to with leading zeros, e.g. , A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces. In comments, formulas, etc., a(n) represents

4158-464: Was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that numbers and geometry were inseparable; a foundation of their theory. The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. This

4224-403: Was that log 2  3 is rational (and so expressible as a quotient of integers m / n with n  ≠ 0). The contradiction means that this assumption must be false, i.e. log 2  3 is irrational, and can never be expressed as a quotient of integers m / n with n  ≠ 0. Cases such as log 10  2 can be treated similarly. An irrational number may be algebraic , that

4290-474: Was the distinction between magnitude and number. A magnitude "...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5". Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus

4356-462: Was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. "Eudoxus' theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying

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