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Arithmetic Sequence
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] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,* * * * * * * * * * Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Heath, Thomas L. 1897. ''Works of Archimedes''. Archimedes' other mathematical achievements include deriving an approximation of pi, defining and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed non-zero number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where ''r'' ≠ 0 is the common ratio and ''a'' ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression terms is called a ''geometric series''. Elementary properties The ''n''-th term of a geometric sequence with initial value ''a'' = ''a''1 and common ratio ''r'' is given by :a_n = a\,r^, and in general :a_n = a_m\,r^. Such a geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tosafot
The Tosafot, Tosafos or Tosfot ( he, תוספות) are medieval commentaries on the Talmud. They take the form of critical and explanatory glosses, printed, in almost all Talmud editions, on the outer margin and opposite Rashi's notes. The authors of the Tosafot are known as Tosafists ( ''Ba'ale haTosafot''); for a listing see ''List of Tosafists''. Meaning of name The word ''tosafot'' literally means "additions". The reason for the title is a matter of dispute among modern scholars. Many of them, including Heinrich Graetz, think the glosses are so-called as additions to Rashi's commentary on the Talmud. In fact, the period of the Tosafot began immediately after Rashi had written his commentary; the first tosafists were Rashi's sons-in-law and grandsons, and the Tosafot consist mainly of strictures on Rashi's commentary. Others, especially Isaac Hirsch Weiss, object that many tosafot — particularly those of Isaiah di Trani — have no reference to Rashi. Weiss, fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Talmud
The Talmud (; he, , Talmūḏ) is the central text of Rabbinic Judaism and the primary source of Jewish religious law (''halakha'') and Jewish theology. Until the advent of modernity, in nearly all Jewish communities, the Talmud was the centerpiece of Jewish cultural life and was foundational to "all Jewish thought and aspirations", serving also as "the guide for the daily life" of Jews. The term ''Talmud'' normally refers to the collection of writings named specifically the Babylonian Talmud (), although there is also an earlier collection known as the Jerusalem Talmud (). It may also traditionally be called (), a Hebrew abbreviation of , or the "six orders" of the Mishnah. The Talmud has two components: the Mishnah (, 200 CE), a written compendium of the Oral Torah; and the Gemara (, 500 CE), an elucidation of the Mishnah and related Tannaitic writings that often ventures onto other subjects and expounds broadly on the Hebrew Bible. The term "Talmud" may refer to eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sacrobosco
Johannes de Sacrobosco, also written Ioannes de Sacro Bosco, later called John of Holywood or John of Holybush ( 1195 – 1256), was a scholar, monk, and astronomer who taught at the University of Paris. He wrote a short introduction to the Hindu-Arabic numeral system. Judging from the number of manuscript copies that survive today, for the next 400 years it became the most widely read book on that subject. He also wrote a short textbook which was widely read and influential in Europe during the later medieval centuries as an introduction to astronomy. In his longest book, on the computation of the date of Easter, Sacrobosco correctly described the defects of the then-used Julian calendar, and recommended a solution similar to the modern Gregorian calendar three centuries before its implementation. Very little is known about the education and biography of Sacrobosco. For one thing, his year of death has been guessed at 1236, 1244, and 1256, each of which is plausible and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci'', was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for ('son of Bonacci'). However, even earlier in 1506 a notary of the Holy Roman Empire, Perizolo mentions Leonardo as "Lionardo Fibonacci". Fibonacci popularized the Indo–Arabic numeral system in the Western world primarily through his composition in 1202 of ''Liber Abaci'' (''Book of Calculation''). He also introduced Europe to the sequence of Fibonacci numbers, which he used as an example in ''Liber Abaci''. Biography Fibonacci was born around 1170 to Guglielmo, an Italian merchant and customs official. Guglielmo directed a trading post in Bugia (Béjaïa) in modern- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dicuil
Dicuilus (or the more vernacular version of the name Dícuil) was an Irish monk and geographer, born during the second half of the 8th century. Background The exact dates of Dicuil's birth and death are unknown. Of his life nothing is known except that he probably belonged to one of the numerous Irish monasteries of the Frankish Kingdom, and became acquainted by personal observation with islands near England and Scotland. From 814 and 816 Dicuil taught in one of the schools of Louis the Pious, where he wrote an astronomical work, and in 825 a geographical work. Dicuil's reading was wide; he quotes from, or refers to, thirty Greek and Latin writers, including the classical Homer, Hecataeus, Herodotus, Thucydides, Virgil, Pliny and King Juba, the late classical Solinus, the patristic St Isidore and Orosius, and his contemporary the Irish poet Sedulius. In particular, he professes to utilize the alleged surveys of the Roman world executed by order of Julius Caesar, Augustus and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alcuin
Alcuin of York (; la, Flaccus Albinus Alcuinus; 735 – 19 May 804) – also called Ealhwine, Alhwin, or Alchoin – was a scholar, clergyman, poet, and teacher from York, Northumbria. He was born around 735 and became the student of Ecgbert of York, Archbishop Ecgbert at York. At the invitation of Charlemagne, he became a leading scholar and teacher at the Carolingian dynasty, Carolingian court, where he remained a figure in the 780s and 790s. Before that, he was also a court chancellor in Aachen. "The most learned man anywhere to be found", according to Einhard's ''Vita Karoli Magni, Life of Charlemagne'' (–833), he is considered among the most important intellectual architects of the Carolingian Renaissance. Among his pupils were many of the dominant intellectuals of the Carolingian era. During this period, he perfected Carolingian minuscule, an easily read manuscript hand using a mixture of upper- and lower-case letters. Latin paleography in the eighth centur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the '' Khaṇḍakhādyaka'' ("edible bite", dated 665), a more practical text. Brahmagupta was the first to give rules for computing with ''zero''. The texts composed by Brahmagupta were in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived. In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it. Life and career Brahmagupta was born in 598 CE according to his own statement. He lived in ''Bhillamāla'' in Gurjaradesa (modern Bhinmal in Rajasthan, India) during the reign of the Chavda dynasty ruler, ... [...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] |
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