The Maya calendar utilized a base-20 number system which included the 'number' zero (also see Maya numerals). In China, Zu Chongzhi (5th century) of the Southern and Northern Dynasties was the first person to calculate the value of Pi to seven decimal places. ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Early written mathematics 2000 BC - 600 BC, History of mathematics - India 800 BC - 300 BC, History of mathematics - Greece and Hellenistic mathematics 400 BC - 200 BC, History of mathematics - Arab and Persian mathematics 650 - 1200, History of mathematics - Developing the concept of number through equations, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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In Europe at the dawn of the Renaissance, most of what is now called school mathematics -- addition, subtraction, multiplication, division, and geometry -- was known to educated people, though the notation was cumbersome: Roman numerals and words were used, but no symbols: no plus sign, no equal sign, no zero, and no use of x as an unknown. Almost all of the mathematics now taught in college had yet to be d ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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A one-dimensional polynomial spline, S(t), is an example of a piecewise function. In its most general form a polynomial spline, defined on an interval [a,b], consists of polynomial pieces, Pi(t), with each piece defined on one of a number of given subintervals . That is, It is required that the polynomial pieces on the subintervals all have degree n; and it is also required that tw ...

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Spline mathematics, Spline mathematics - Introduction, Spline mathematics - Definition, Spline mathematics - Examples, Spline mathematics - Notes, Spline mathematics - Representations and names, Spline mathematics - History

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Suppose the interval [a,b] is [0,3] and the subintervals are [0,1), [1,2), and [2,3]. Suppose the polynomial pieces are to be of degree 2, and the pieces on [0,1) and [1,2) must join in value and first derivative (at t=1) while the pieces on [1,2) and [2,3] join simply in value (at t=2). This would define a type of spline S(t) for which would be a member of that type, an ...

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Spline mathematics, Spline mathematics - Introduction, Spline mathematics - Definition, Spline mathematics - Examples, Spline mathematics - Notes, Spline mathematics - Representations and names, Spline mathematics - History

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Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attrib ...

Including:

Infinity - History
Infinity - Ancient view of infinity Infinity - Views from the Renaissance to modern times Infinity - Modern philosophical views Infinity - Infinity symbol
Infinity - Mathematical infinity
Infinity - Infinity in real analysis Infinity - Infinity in complex analysis Infinity - Arithmetic properties of infinity Infinity - Infinity in set theory Infinity - Mathematics without infinity
Infinity - Use of infinity in common speech Infinity - Physical infinity
Infinity - Infinity in cosmology
Infinity - Three types of infinities Infinity - Infinity in science fiction Infinity - Note

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Quantum mechanics is a fundamental physical theory that replaces Newtonian mechanics and classical electromagnetism at the atomic and subatomic levels and is the underlying framework of many fields of physics and chemistry, including condensed matter physics, quantum chemistry, and particle physics. Along with general relativity, it is one of the pillars of modern physics. Quantum mechanics - Introduction. The term quantum (Latin, "how much") refers to the discrete units that the theory assign ...

Including:

Quantum mechanics - Introduction Quantum mechanics - Description of the theory
Quantum mechanics - Quantum mechanical effects Quantum mechanics - Mathematical formulation Quantum mechanics - Interactions with other scientific theories
Quantum mechanics - Applications of quantum theory Quantum mechanics - Philosophical consequences Quantum mechanics - History
Quantum mechanics - Founding experiments
Quantum mechanics - Notes

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It might be asked what meaning more than n multiple knots in a knot vector have, since this would lead to continuities like at the location of this high multiplicity. By convention, any such situation indicates a simple discontinuity between the two adjacent polynomial pieces. This means that if a knot ti appears more than n + 1 times in an extended knot vector, all instances of it in exces ...

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Spline mathematics, Spline mathematics - Introduction, Spline mathematics - Definition, Spline mathematics - Examples, Spline mathematics - Notes, Spline mathematics - Representations and names, Spline mathematics - History

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In the mathematical subfield of numerical analysis a Bézier curve is a parametric curve important in computer graphics. A numerically stable method to evaluate Bézier curves is de Casteljau's algorithm. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces; the Bézier triangle is a special case. Bézier curves are also formed by many common forms of string art, where strings are looped across a frame of nails. Bézier curve - History. Bézier curves were widely ...

Including:

Bézier curve - History Bézier curve - Examination of cases
Bézier curve - Linear Bézier curves Bézier curve - Quadratic Bézier curves Bézier curve - Cubic Bézier curves
Bézier curve - Generalization
Bézier curve - Terminology Bézier curve - Notes
Bézier curve - Application in computer graphics Bézier curve - Rational Bézier curves

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For a given interval [a,b] and a given extended knot vector on that interval, the splines of degree n form a vector space. Briefly this means that adding any two splines of a given type produces spline of that given type, and multiplying a spline of a given type by any constant produces a spline of that given type. The dimension of the space containing all splines of a certain type can be counted from the extended knot vector: The dimension is equal to th ...

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Spline mathematics, Spline mathematics - Introduction, Spline mathematics - Definition, Spline mathematics - Examples, Spline mathematics - Notes, Spline mathematics - Representations and names, Spline mathematics - History

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Mathematical notation - Counting. It is believed that a mathematical notation was first developed at least 50,000 years ago in order to assist with counting. Early mathematical ideas for counting were represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient Sumer. The Census Quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts. Mathemat ...

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Mathematical notation, Mathematical notation - Definition, Mathematical notation - Expressions, Mathematical notation - Precise semantic meaning, Mathematical notation - History, Mathematical notation - Counting, Mathematical notation - Geometry becomes analytic, Mathematical notation - Counting is mechanized, Mathematical notation - Computerized notation, Mathematical notation - Ideographic notation, Mathematical notation - Notes

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Precision is necessary so that we can know what we are investigating. Suppose that we have statements, denoted by some formal sequence of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the denoted symbols refer to those objects, perhaps in a model. The semantics of that object has a heuristic side and a d ...

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Mathematical notation, Mathematical notation - Definition, Mathematical notation - Expressions, Mathematical notation - Precise semantic meaning, Mathematical notation - History, Mathematical notation - Counting, Mathematical notation - Geometry becomes analytic, Mathematical notation - Counting is mechanized, Mathematical notation - Computerized notation, Mathematical notation - Ideographic notation, Mathematical notation - Notes

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The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of natural logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The first indication of e as a constant was discovered by Jacob Bernoulli, trying to find the valu ...

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E mathematical constant, E mathematical constant - Definitions, E mathematical constant - Properties, E mathematical constant - History, E mathematical constant - Non-mathematical uses of e, E mathematical constant - Notes

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The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive: and , where C is a constant. It is known that e is both irrational (proof) and transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. Liouville number); the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's formula, one of the most imp ...

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E mathematical constant, E mathematical constant - Definitions, E mathematical constant - Properties, E mathematical constant - History, E mathematical constant - Non-mathematical uses of e, E mathematical constant - Notes

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The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive: and , where C is the arbitrary constant of integration. It is known that e is irrational (proof) and even more, transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. Liouville number); the proof w ...

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E mathematical constant, E mathematical constant - Definitions, E mathematical constant - Properties, E mathematical constant - History, E mathematical constant - Non-mathematical uses of e, E mathematical constant - Notes

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and tw ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and tw ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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The TeX software incorporates several aspects that were not available, or were of lower quality, in other typesetting programs at the time when TeX was released. While some of these discoveries have now been incorporated into other typesetting programs, others, such as the rules for mathematical spacing, are still relatively unique. Some of the innovations are based on interesting algorithms, and have led to a number of theses for Knuth's students. ...

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TeX, TeX - History, TeX - Motivation and early history, TeX - TeX82, TeX - Version 3, TeX - Current version and the future of TeX, TeX - The typesetting system, TeX - Novel aspects of TeX, TeX - Mathematical spacing, TeX - Hyphenation and justification, TeX - METAFONT, TeX - Examples of TeX, TeX - Mathematical examples, TeX - LaTeX examples, TeX - Software, TeX - Formats, TeX - Derived works, TeX - Compatible tools, TeX - Quality, TeX - License, TeX - Notes

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Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows beyond any assigned value, and means x is eventually less than any assigned value. Points labeled and can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat and as the same, leading to the one-point compact ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

Read more here: » Infinity: Encyclopedia II - Mathematical infinity