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Arithmetical Operations
Arithmetic
Arithmetic
(from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis
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Arithmetic (song)
"Arithmetic" is a single by Brooke Fraser
Brooke Fraser
released in 2004. The song is the first track Fraser's debut album What To Do With Daylight, which takes its name from this song in the line "Wondering what to do with daylight until I can make you mine." The song was later included on the Sony BMG
Sony BMG
compilation More Nature, a collection of songs from the New Zealand Sony BMG
Sony BMG
catalogue (in particular, those who promote nature and conservation). The song debuted on the New Zealand Singles Chart at number thirty eight on 26 July 2004 and peaked at number eight
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Archimedes
Archimedes
Archimedes
of Syracuse (/ˌɑːrkɪˈmiːdiːz/;[2] Greek: Ἀρχιμήδης; c. 287 – c. 212 BC) was a Greek mathematician, physicist, engineer, inventor, and astronomer.[3] Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Generally considered the greatest mathematician of antiquity and one of the greatest of all time,[4][5] Archimedes
Archimedes
anticipated modern calculus and analysis by applying concepts of infinitesimals and the method of exhaustion to derive and rigorously prove a range of geometrical theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola.[6] Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the spiral bearing his name, and creating a system using exponentiation for expressing very large numbers
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Babylonian Numerals
Babylonian numerals
Babylonian numerals
were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their astronomical observations and calculations (aided by their invention of the abacus), used a sexagesimal (base-60) positional numeral system inherited from either the Sumerian or the Eblaite civilizations.[1] Neither of the predecessors was a positional system (having a convention for which ‘end’ of the numeral represented the units).Contents1 Origin 2
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Vigesimal
The vigesimal or base 20 numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten).Contents1 Places1.1 Converting table2 Fractions 3 Cyclic numbers 4 Real numbers 5 Use5.1 Africa 5.2 Americas 5.3 Asia 5.4 In Europe5.4.1 Etymology 5.4.2 Examples5.5 Related observations6 Examples in Mesoamerican languages6.1 Powers of twenty in Yucatec
Yucatec
Maya and Nahuatl 6.2 Counting in units of twenty7 Further reading 8 NotesPlaces[edit] In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the usual decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A20 (the 20 means base 20), to write nineteen as J20, and the numbers between with the corresponding letters of the alphabet. This is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters "A–F"
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Maya Numerals
The Mayan numeral system was the system to represent numbers and calendar dates in the Maya civilization. It was a vigesimal (base-20) positional numeral system. The numerals are made up of three symbols; zero (shell shape, with the plastron uppermost), one (a dot) and five (a bar). For example, thirteen is written as three dots in a horizontal row above two horizontal bars; sometimes it is also written as three vertical dots to the left of two vertical bars. With these three symbols each of the twenty vigesimal digits could be written.400s20s1s33 429 5125Numbers after 19 were written vertically in powers of twenty. The Mayan used powers of twenty, just as our Hindu-Arabic numeral system uses powers of tens.[1] For example, thirty-three would be written as one dot, above three dots atop two bars. The first dot represents "one twenty" or "1×20", which is added to three dots and two bars, or thirteen. Therefore, (1×20) + 13 = 33
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Hellenistic Civilization
The Hellenistic
Hellenistic
period covers the period of Mediterranean
Mediterranean
history between the death of Alexander the Great
Alexander the Great
in 323 BC and the emergence of the Roman Empire
Roman Empire
as signified by the Battle of Actium
Battle of Actium
in 31 BC[1] and the subsequent conquest of Ptolemaic Egypt
Egypt
the following year.[2] The Ancient Greek
Ancient Greek
word Hellas (Ἑλλάς, Ellás) is the original word for Greece, from which the word "Hellenistic" was derived.[3] At this time, Greek cultural influence and power was at its peak in Europe, North Africa
North Africa
and Western Asia, experiencing prosperity and progress in the arts, exploration, literature, theatre, architecture, music, mathematics, philosophy, and science
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Euclid
Euclid
Euclid
(/ˈjuːklɪd/; Greek: Εὐκλείδης Eukleidēs [eu̯.klěː.dɛːs]; fl. 300 BC), sometimes given the name Euclid
Euclid
of Alexandria[1] to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry"[1] or the "father of geometry". He was active in Alexandria
Alexandria
during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.[2][3][4] In the Elements, Euclid
Euclid
deduced the theorems of what is now called Euclidean geometry
Euclidean geometry
from a small set of axioms
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Greek Mathematics
Greek mathematics
Greek mathematics
refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture and language. Greek mathematics of the period following Alexander the Great
Alexander the Great
is sometimes called Hellenistic mathematics
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Nicomachus
Nicomachus of Gerasa[1] (Greek: Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan), and was strongly influenced by Aristotle. He was a Neopythagorean, who wrote about the mystical properties of numbers.[citation needed]Contents1 Life 2 Works2.1 Introduction to Arithmetic 2.2 Manual of Harmonics 2.3 Lost works3 See also 4 Notes 5 References 6 External linksLife[edit] Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa
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Pythagoreanism
Pythagoreanism
Pythagoreanism
originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras
Pythagoras
and his followers, the Pythagoreans, who were considerably influenced by mathematics and mysticism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism
Neopythagoreanism
or Neoplatonism
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Introduction To Arithmetic
The book Introduction to Arithmetic (Greek: Ἀριθμητικὴ εἰσαγωγή, Arithmetike eisagoge) is the only extant work on mathematics by Nicomachus (60–120 AD).Contents1 Summary 2 Editions 3 See also 4 References 5 External linksSummary[edit] The work contains both philosophical prose and basic mathematical ideas. Nicomachus refers to Plato
Plato
quite often, and writes that philosophy can only be possible if one knows enough about mathematics. Nicomachus also describes how natural numbers and basic mathematical ideas are eternal and unchanging, and in an abstract realm
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Greek Numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals
Roman numerals
are still used elsewhere in the West. For ordinary cardinal numbers, however, Greece
Greece
uses Arabic numerals.Contents1 History 2 Description 3 Table 4 Higher numbers 5 Zero 6 See also 7 References 8 External linksHistory[edit] The Minoan and Mycenaean civilizations' Linear A
Linear A
and Linear B alphabets used a different system, called Aegean numerals, which included specialized symbols for numbers: 𐄇 = 1, 𐄐 = 10, 𐄙 = 100, 𐄢 = 1000, and 𐄫 = 10000.[1] Attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set
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Diophantus
Diophantus
Diophantus
of Alexandria
Alexandria
(Ancient Greek: Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 201 and 215; died around 84 years old, probably sometime between AD 285 and 299) was an Alexandrian Hellenistic mathematician, who was the author of a series of books called Arithmetica, many of which are now lost. Sometimes called "the father of algebra", his texts deal with solving algebraic equations. While reading Claude Gaspard Bachet
Bachet
de Méziriac's edition of Diophantus' Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus
Diophantus
had no solutions, and noted in the margin without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem
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Roman Abacus
The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of the previous Babylonian abacus. It was the first portable calculating device for engineers, merchants and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of arithmetic using Roman numerals. As Karl Menninger says on page 315 of his book,[1] "For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. The Etruscan cameo and the Greek predecessors, such as the Salamis Tablet and the Darius Vase, give us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more
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Heron's Method
In numerical analysis, a branch of mathematics, there are several square root algorithms or methods of computing the principal square root of a non-negative real number. For the square roots of a negative or complex number, see below. Finding S displaystyle sqrt S is the same as solving the equation f ( x ) = x 2 − S = 0 displaystyle f(x)=x^ 2 -S=0,! for a positive x displaystyle x
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