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Strong Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variable (mathematics)
In mathematics, a variable (from Latin '' variabilis'', "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set. Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula solves any quadratic equation by substituting the numeric values of the coefficients of that equation for the variables that represent them in the quadratic formula. In mathematical logic, a ''variable'' is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation. History In ancient works such as Euclid's ''Elements'', single letters refer to geometric points and shapes. In the 7th century, Brahmagupta used different colours to represent the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francesco Maurolico
Francesco Maurolico (Latin: ''Franciscus Maurolycus''; Italian: ''Francesco Maurolico''; gr, Φραγκίσκος Μαυρόλυκος, 16 September 1494 - 21/22 July 1575) was a mathematician and astronomer from Sicily. He made contributions to the fields of geometry, optics, conics, mechanics, music, and astronomy. He edited the works of classical authors including Archimedes, Apollonius, Autolycus, Theodosius and Serenus. He also composed his own unique treatises on mathematics and mathematical science. Life Francesco was born in Messina with the surname of Marulì, although the surname is sometimes reported as "Mauroli". He was one of seven sons of Antonio Marulì, a government official, and Penuccia. His father was a Greek physician who fled Constantinople when the Ottomans invaded the city. Antonio had studied with the Neoplatonic Hellenist Constantine Lascaris, so Francesco received a "Lascarian" education through his father and from Francesco Faraone and Giacomo Ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chakravala Method
The ''chakravala'' method ( sa, चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE)Hoiberg & Ramchandani – Students' Britannica India: Bhaskaracharya II, page 200Kumar, page 23 although some attribute it to Jayadeva (c. 950 ~ 1000 CE).Plofker, page 474 Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his ''Bijaganita'' treatise. He called it the Chakravala method: ''chakra'' meaning "wheel" in Sanskrit, a reference to the cyclic nature of the algorithm.Goonatilake, page 127 – 128 C.-O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity. This method is also known as the cyclic method and contains traces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bhāskara II
Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiromani (सिद्धांतशिरोमणी), it can be inferred that he was born in 1114 in Vijjadavida (Vijjalavida) and living in the Sahyadri mountain ranges of Western Ghats, believed to be the town of Patan in Chalisgaon, located in present-day Khandesh region of Maharashtra by scholars. He is the only ancient mathematician who has been immortalized on a monument. In a temple in Maharashtra, an inscription supposedly created by his grandson Changadeva, lists Bhaskaracharya's ancestral lineage for several generations before him as well as two generations after him. Colebrooke who was the first European to translate (1817) Bhaskaracharya II's mathematical classics refers to the family as Maharashtrian Brahmins residing on the ban ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squared Triangular Number
In number theory, the sum of the first cubes is the square of the th triangular number. That is, :1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2. The same equation may be written more compactly using the mathematical notation for summation: :\sum_^n k^3 = \bigg(\sum_^n k\bigg)^2. This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). History Nicomachus, at the end of Chapter 20 of his ''Introduction to Arithmetic'', pointed out that if one writes a list of the odd numbers, the first is the cube of 1, the sum of the next two is the cube of 2, the sum of the next three is the cube of 3, and so on. He does not go further than this, but from this it follows that the sum of the first cubes equals the sum of the first n(n+1)/2 odd numbers, that is, the odd numbers from 1 to n(n+1)-1. The average of these numbers is obviously n(n+1)/2, and there are n(n+1)/2 of them, so their sum is \bigl(n(n+1)/2\bigr)^2. Many early mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences. Truth is usually held to be the opposite of falsehood. The concept of truth is discussed and debated in various contexts, including philosophy, art, theology, and science. Most human activities depend upon the concept, where its nature as a concept is assumed rather than being a subject of discussion; these include most of the sciences, law, journalism, and everyday life. Some philosophers view the concept of truth as basic, and unable to be explained in any terms that are more easily understood than the concept of truth itself. Most commonly, truth is viewed as the correspondence of language or thought to a mind-independent world. This is called the correspondence theory of truth. Various theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aryabhata
Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the ''Aryabhatiya'' (which mentions that in 3600 ''Kali Yuga'', 499 CE, he was 23 years old) and the ''Arya-siddhanta.'' Aryabhata created a system of phonemic number notation in which numbers were represented by consonant-vowel monosyllables. Later commentators such as Brahmagupta divide his work into ''Ganita ("Mathematics"), Kalakriya ("Calculations on Time") and Golapada ("Spherical Astronomy")''. His pure mathematics discusses topics such as determination of square and cube roots, geometrical figures with their properties and mensuration, arithmetric progression problems on the shadow of the gnomon, quadratic equations, linear and indeterminate equations. Aryabhata calculated the value of pi (''π)'' to the fourth decimal digit and was likely aware that p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of each term is a specific positive integer depending on and . For example, for , (x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. The coefficient in the term of is known as the binomial coefficient \tbinom or \tbinom (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where \tbinom gives the number of different combinations of elements that can be chosen from an -element set. Therefore \tbinom is often pronounced as " choose ". History Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid ment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a and the common difference of successive members is d, then the n-th term of the sequence (a_n) is given by: :a_n = a + (n - 1)d, If there are ''m'' terms in the AP, then a_m represents the last term which is given by: :a_m = a + (m - 1)d. A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series. Sum Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Al-Karaji
( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian people, Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: ''Al-Badi' fi'l-hisab'' (''Wonderful on calculation''), ''Al-Fakhri fi'l-jabr wa'l-muqabala'' (''Glorious on algebra''), and ''Al-Kafi fi'l-hisab'' (''Sufficient on calculation''). Work Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus) but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book ''al-fakhri fi al-jabr wa al-muqabala'', which survives from the medieval era in at least four copies. In his book "Extraction of hidden waters" he has mentioned that earth is spherical in shape but considers it t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
