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Number Theory
Number
Number
theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline.[1] Number
Number
theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory)
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Numerology
Numerology
Numerology
is any belief in the divine or mystical relationship between a number and one or more coinciding events.[2] It is also the study of the numerical value of the letters in words, names and ideas. It is often associated with the paranormal, alongside astrology and similar divinatory arts.[3] Despite the long history of numerological ideas, the word "numerology" is not recorded in English before c.1907.[4] The term numerologist can be used for those who place faith in numerical patterns and draw pseudo-scientific inferences from them, even if those people do not practice traditional numerology
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Figurate Numbers
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers)
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Pythagoras
Pythagoras
Pythagoras
of Samos
Samos
(US: /pɪˈθæɡərəs/,[2] UK: /paɪˈθæɡərəs/;[3] Ancient Greek: Πυθαγόρας ὁ Σάμιος, translit. Pythagóras ho Sámios, lit. ' Pythagoras
Pythagoras
the Samian', or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 – c. 495 BC)[Notes 1][4] was an Ionian Greek
Ionian Greek
philosopher and the eponymous founder of the Pythagoreanism
Pythagoreanism
movement. His political and religious teachings were well-known in Magna Graecia
Magna Graecia
and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Knowledge of Pythagoras's life is largely clouded by legend and obfuscation, but he appears to have been the son of Mnesarchus, a seal engraver on the island of Samos
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Thales
Thales
Thales
of Miletus
Miletus
(/ˈθeɪliːz/; Greek: Θαλῆς (ὁ Μῑλήσιος), Thalēs; c. 624 – c. 546 BC) was a pre-Socratic Greek philosopher, mathematician, and astronomer from Miletus
Miletus
in Asia Minor
Asia Minor
(present-day Milet
Milet
in Turkey). He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him as the first philosopher in the Greek tradition,[1][2] and he is otherwise historically recognized as the first individual in Western civilization known to have entertained and engaged in scientific philosophy.[3][4] Thales
Thales
is recognized for breaking from the use of mythology to explain the world and the universe, and instead explaining natural objects and phenomena by theories and hypotheses, i.e. science
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Egypt
Coordinates: 26°N 30°E / 26°N 30°E / 26; 30Arab Republic
Republic
of Egyptجمهورية مصر العربيةArabic: Jumhūrīyat Miṣr al-ʿArabīyahEgyptian: Gomhoreyet Maṣr El ʿArabeyahFlagCoat of armsAnthem: "Bilady, Bilady, Bilady" "بلادي، بلادي، بلادي" "My country, my country, my country"Capital and largest city Cairo 30°2′N 31°13′E / 30.033°N 31.217°E / 30.033; 31.217Official languages Arabic[a]National language Egyptian ArabicReligion90% Islam 9% Orthodox Christian 1% Other Christian[1]Demonym EgyptianGovernment Unitary semi-presidential republic• PresidentAbdel Fattah el-Sisi• Prime MinisterSherif IsmailLegislature House of RepresentativesEstablishment• Unification of Upper and Lower Egypt[2][3][b]c
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Square Root Of 2
The square root of 2, or the (1/2)th power of 2, written in mathematics as √2 or 2​1⁄2, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The rational approximation of the square root of two, 665,857/470,832, derived from the fourth step in the Babylonian algorithm starting with a0 = 1, is too large by approx. 6988160000000000000♠1.6×10−12: its square is 7000200000000000450♠2.0000000000045… The rational approximation 99/70 (≈ 1.4142857) is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 6995720000000000000♠+0.72×10−4)
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Irrational Number
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of 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 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;[1][2][3] in fact all square roots of natural numbers, other than of perfect squares, are irrational. It can be shown that irrational numbers, when expressed in a positional numeral system (e.g
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Theodorus Of Cyrene
Theodorus of Cyrene (Greek: Θεόδωρος ὁ Κυρηναῖος) was an ancient Libyan Greek and lived during the 5th century BC. The only first-hand accounts of him that survive are in three of Plato's dialogues: the Theaetetus, the Sophist, and the Statesman. In the former dialogue, he posits a mathematical theorem now known as the Spiral of Theodorus.Contents1 Life 2 Work in mathematics 3 See also 4 ReferencesLife[edit] Little is known of Theodorus' biography beyond what can be inferred from Plato's dialogues. He was born in the northern African colony of Cyrene, and apparently taught both there and in Athens.[1] He complains of old age in the Theaetetus, whose dramatic date of 399 BC suggests his period of flourishing to have occurred in the mid-5th century
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Hippasus Of Metapontum
Metapontum
Metapontum
or Metapontium (Ancient Greek: Μεταπόντιον, translit. Metapontion) was an important city of Magna Graecia, situated on the gulf of Tarentum, between the river Bradanus
Bradanus
and the Casuentus (modern Basento). It was distant about 20 km from Heraclea and 40 from Tarentum. The ruins of Metapontum
Metapontum
are located in the frazione of Metaponto, in the comune of Bernalda, in the Province of Matera, Basilicata
Basilicata
region, Italy.Contents1 Foundation 2 Early history 3 Peloponnesian War 4 Pyrrhic War and Roman domination 5 Decline 6 Coinage 7 Notes 8 References 9 Further reading 10 External linksFoundation[edit] Though Metapontum
Metapontum
was an ancient Greek Achaean colony,[1] various traditions assigned to it a much earlier origin
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Polygonal Number
In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.Contents1 Definition and examples1.1 Triangular numbers 1.2 Square numbers 1.3 Pentagonal numbers 1.4 Hexagonal numbers2 Formula2.1 Every hexagonal number is also a triangular number3 Table of values 4 Combinations 5 See also 6 Notes 7 References 8 External linksDefinition and examples[edit] The number 10 for example, can be arranged as a triangle (see triangular number):But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number):By convention, 1 is the first polygonal number for any number of sides
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Egyptian Mathematics
Ancient Egyptian
Ancient Egyptian
mathematics is the mathematics that was developed and used in Ancient Egypt
Ancient Egypt
c. 3000 to c. 300 BC, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical problems, often involving multiplication and fractions. Evidence for Egyptian mathematics is limited to a scarce amount of surviving sources written on papyri
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Brute Force Method
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds.[1] This is a method of direct proof. A proof by exhaustion contains two stages:A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. A proof of each of the cases.The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion. Computer
Computer
expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite
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Vedic
DivisionsSamhita Brahmana Aranyaka UpanishadsUpanishads Rig vedicAitareya KaushitakiSama vedicChandogya KenaYajur vedicBrihadaranyaka Isha Taittiriya Katha Shvetashvatara MaitriAtharva vedicMundaka Mandukya PrashnaOther scripturesBhagavad Gita AgamasRelated Hindu
Hindu
textsVedangasShiksha Chandas Vyakarana Nirukta Kalpa JyotishaPuranas Brahma
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Sunzi Suanjing
Sunzi Suanjing
Sunzi Suanjing
(Chinese: 孙子算经; pinyin: Sūnzĭ Suànjīng; Wade–Giles: Sun Tzu
Sun Tzu
Suan Ching; literally: "The Mathematical Classic of Master Sun") was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still unknown but he lived much later than eponymous Sun Tzu, author of The Art of War
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Āryabhaṭa
Aryabhata
Aryabhata
(IAST: Āryabhaṭa) or Aryabhata
Aryabhata
I[2][3] (476–550 CE)[4][5] was the first of the major mathematician-astronomers from the classical age of Indian mathematics
Indian mathematics
and Indian astronomy
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