In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads whenever n is any non-negative integer, the numbers are the binomial coefficients, and n! denotes the factorial of n. This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to Chinese mathematician Yang Hui in the 13th century. The Persian mathematicia ...

Including:

• Binomial theorem - Newton's generalized binomial theorem
• Binomial theorem - Binomial type
• Binomial theorem - Proof inductive
• Binomial theorem - Trivia

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Blaise Pascal (June 19, 1623–August 19, 1662) was a French mathematician, physicist, and religious philosopher. Pascal was a child prodigy, who was educated by his father. Pascal's earliest work was in the natural and applied sciences, where he made important contributions to the construction of mechanical calculators and the study of fluids, and clarified the concepts of pressure and vacuum by expanding the work of Evangelista Torricelli. Pascal also w ...

Including:

• Blaise Pascal - Early life and education
• Blaise Pascal - Contributions to mathematics
• Blaise Pascal - Philosophy of mathematics
• Blaise Pascal - Contributions to the physical sciences
• Blaise Pascal - Mature life religion philosophy and literature
• Blaise Pascal - Religious conversion
• Blaise Pascal - Upon brink of death
• Blaise Pascal - The Provincial Letters
• Blaise Pascal - Miracle
• Blaise Pascal - The Pensées
• Blaise Pascal - Last works and death
• Blaise Pascal - Legacy
• Blaise Pascal - Works
• Blaise Pascal - Notes

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Blaise Pascal - Religious conversion. Biographically, we can say that two basic influences led him to his conversion: sickness and Jansenism. As early as his eighteenth year he suffered from a nervous ailment that left him hardly a day without pain. In 1647 a paralytic attack so disabled him that he could not move without crutches. His head ached, his bowels burned, his legs and feet were continually cold, and required wearisome aids to circulation of the blood; he wore stockings steeped in brandy to warm his fee ...

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Blaise Pascal, Blaise Pascal - Early life and education, Blaise Pascal - Contributions to mathematics, Blaise Pascal - Philosophy of mathematics, Blaise Pascal - Contributions to the physical sciences, Blaise Pascal - Mature life religion philosophy and literature, Blaise Pascal - Religious conversion, Blaise Pascal - Upon brink of death, Blaise Pascal - The Provincial Letters, Blaise Pascal - Miracle, Blaise Pascal - The Pensées, Blaise Pascal - Last works and death, Blaise Pascal - Legacy, Blaise Pascal - Works, Blaise Pascal - Notes

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Magic square - The Lo Shu Square 3x3 magic square. Chinese literature dating from as early as 2800 BC tells the legend of Lo Shu or "scroll of the river Lo". In ancient China, there was a huge flood. The people tried to offer some sacrifice to the river god of one of the flooding rivers, the Lo river, to calm his anger. Then, there emerged from the water a turtle with a curious figure/pattern on its shell; there were circular dots of numbers that were arranged in a three by three nine-grid pattern such that the s ...

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Magic square, Magic square - Brief history of magic squares, Magic square - The Lo Shu Square 3x3 magic square, Magic square - The early squares of order four 4x4 magic squares, Magic square - Cultural significance of magic squares, Magic square - Albrecht Dürer's magic square, Magic square - The Sagrada Família magic square, Magic square - Types of magic squares and their construction, Magic square - A method for constructing a magic square of odd order, Magic square - A method of constructing a magic square of doubly even order, Magic square - Counting magic squares, Magic square - Generalizations, Magic square - Extra constraints, Magic square - Different constraints, Magic square - Other operations, Magic square - Other magic shapes, Magic square - Combined extensions, Magic square - Related problems, Magic square - Magic Square of Primes, Magic square - n-Queens problem

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Isaac Newton generalized the formula to other exponents by considering an infinite series: where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r − 1), etc., do not a ...

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Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - Proof inductive, Binomial theorem - Trivia

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Pascal's triangle can be used as a lookup table for the number of arbitrarily dimensioned elements within a single arbitrarily dimensioned version of a triangle (known as a simplex). For example, consider the 3rd line of the triangle, with values 1, 3, 3, 1. A 2-dimensional triangle has one 2-dimensional element (itself), 3 1-dimensional elements (lines, or edges), and 3 0-dimensional elements (vertices, or corners). The meaning of the final number (1) is more difficult to explain (but see below). Continuing with our example, a tetrah ...

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Pascal's triangle, Pascal's triangle - The triangle, Pascal's triangle - Uses of Pascal's triangle, Pascal's triangle - Properties of Pascal's triangle, Pascal's triangle - Geometric properties of Pascal's triangle, Pascal's triangle - Pascal's triangle and the matrix exponential, Pascal's triangle - History

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Isaac Newton generalized the formula to other exponents by considering an infinite series: where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r − 1), etc., do not a ...

See also:

Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - A proof, Binomial theorem - Trivia

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Blaise Pascal - Religious conversion. Biographically, we can say that two basic influences led him to his conversion: sickness and Jansenism. As early as his eighteenth year he suffered from a nervous ailment that left him hardly a day without pain. In 1647 a paralytic attack so disabled him that he could not move without crutches. His head ached, his bowels burned, his legs and feet were continually cold, and required wearisome aids to circulation of the blood; he wore stockings steeped in brandy to warm his fee ...

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Blaise Pascal, Blaise Pascal - Early life and education, Blaise Pascal - Contributions to mathematics, Blaise Pascal - Philosophy of mathematics, Blaise Pascal - Contributions to the physical sciences, Blaise Pascal - Mature life, religion, philosophy, and literature, Blaise Pascal - Religious conversion, Blaise Pascal - Upon brink of death, Blaise Pascal - The Provincial Letters, Blaise Pascal - Miracle, Blaise Pascal - The Pensées, Blaise Pascal - Last works and death, Blaise Pascal - Legacy, Blaise Pascal - Works, Blaise Pascal - Notes

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Some simple patterns are immediately apparent in Pascal's triangle: The diagonals going along the left and right edges contain only 1s. The diagonals next to the edge diagonals contain the natural numbers in order. Moving inwards, the next pair of diagonals contain the triangle numbers in order. The next pair of diagonals contain the tetrahedral numbers in order, and the next pair give pentatope numbers. In general, each next pair of diagonals contains the next higher dimensional "d-triangle" numbers, whic ...

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Pascal's triangle, Pascal's triangle - The triangle, Pascal's triangle - Uses of Pascal's triangle, Pascal's triangle - Properties of Pascal's triangle, Pascal's triangle - Geometric properties of Pascal's triangle, Pascal's triangle - Pascal's triangle and the matrix exponential, Pascal's triangle - History

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We use mathematical induction. When n = 0, we have For the inductive step, assume the theorem holds when the exponent is m. Then for n = m + 1, (a + b)m + 1 = a(a + b)m + b(a + b)m by the inductive hypothesis

The binomial theorem can be stated by saying that the polynomial sequence is of binomial type. ...

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Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - A proof, Binomial theorem - Trivia

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Pascal's triangle has many uses in binomial expansions. For example (x + 1)2 = 1x2 + 2x + 12. Notice the coefficients are the third row of Pascal's triangle: 1, 2, 1. In general, when a binomial is raised to a positive integer power we have: (x + y)n = a0xn + a1xn−1y + a2xn−2y< ...

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Pascal's triangle, Pascal's triangle - The triangle, Pascal's triangle - Uses of Pascal's triangle, Pascal's triangle - Properties of Pascal's triangle, Pascal's triangle - Geometric properties of Pascal's triangle, Pascal's triangle - Pascal's triangle and the matrix exponential, Pascal's triangle - History

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In honor of his scientific contributions, the name Pascal has been given to the SI unit of pressure, to a programming language, and Pascal's law (an important principle of hydrostatics), and as mentioned above, Pascal's triangle and Pascal's wager still bear his name. In Canada, there is an annual math contest named in his honour. The Pascal Contest is open to any student in Canada that is 14 ye ...

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Blaise Pascal, Blaise Pascal - Early life and education, Blaise Pascal - Contributions to mathematics, Blaise Pascal - Philosophy of mathematics, Blaise Pascal - Contributions to the physical sciences, Blaise Pascal - Mature life religion philosophy and literature, Blaise Pascal - Religious conversion, Blaise Pascal - Upon brink of death, Blaise Pascal - The Provincial Letters, Blaise Pascal - Miracle, Blaise Pascal - The Pensées, Blaise Pascal - Last works and death, Blaise Pascal - Legacy, Blaise Pascal - Works, Blaise Pascal - Notes

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Born in Clermont, in the Auvergne region of France, Blaise Pascal lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal (1588–1651), was a local judge and member of the petite noblesse, who also had an interest in science and mathematics. Blaise Pascal was brother to Jacqueline Pascal and two other sisters, only one of whom, Gilberte, survived past childhood. In 1631, Étienne moved with his children to Paris. Étienne decided that he would educate his son, who showed extraordinary mental and int ...

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Blaise Pascal, Blaise Pascal - Early life and education, Blaise Pascal - Contributions to mathematics, Blaise Pascal - Philosophy of mathematics, Blaise Pascal - Contributions to the physical sciences, Blaise Pascal - Mature life religion philosophy and literature, Blaise Pascal - Religious conversion, Blaise Pascal - Upon brink of death, Blaise Pascal - The Provincial Letters, Blaise Pascal - Miracle, Blaise Pascal - The Pensées, Blaise Pascal - Last works and death, Blaise Pascal - Legacy, Blaise Pascal - Works, Blaise Pascal - Notes

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When n = 1, . For the inductive step, assume it holds for m. Then for n = m + 1, (a + b)m + 1 = a(a + b)m + b(a + b)m = by the inductive hypot ...

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Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - Proof inductive, Binomial theorem - Trivia

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In addition to the childhood marvels recorded above, Pascal continued to influence mathematics throughout his life. In 1653 Pascal wrote his Traité du triangle arithmétique in which he described a convenient tabular presentation for binomial coefficients, the "arithmetical triangle", now called Pascal's triangle. (It should be noted, however, that Yang Hui, a Chinese mathematician of the Qin dynasty, had independently worked out a ...

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Blaise Pascal, Blaise Pascal - Early life and education, Blaise Pascal - Contributions to mathematics, Blaise Pascal - Philosophy of mathematics, Blaise Pascal - Contributions to the physical sciences, Blaise Pascal - Mature life religion philosophy and literature, Blaise Pascal - Religious conversion, Blaise Pascal - Upon brink of death, Blaise Pascal - The Provincial Letters, Blaise Pascal - Miracle, Blaise Pascal - The Pensées, Blaise Pascal - Last works and death, Blaise Pascal - Legacy, Blaise Pascal - Works, Blaise Pascal - Notes

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Magic square - Magic Square of Primes. Rudolf Ondrejka discovered the following 3x3 magic square of primes, in this case nine Chen primes: Magic square - n-Queens problem. In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into N-queens solutions, and vice versa. ...

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Magic square, Magic square - Brief history of magic squares, Magic square - The Lo Shu Square 3x3 magic square, Magic square - The early squares of order four 4x4 magic squares, Magic square - Cultural significance of magic squares, Magic square - Albrecht Dürer's magic square, Magic square - The Sagrada Família magic square, Magic square - Types of magic squares and their construction, Magic square - A method for constructing a magic square of odd order, Magic square - A method of constructing a magic square of doubly even order, Magic square - Counting magic squares, Magic square - Generalizations, Magic square - Extra constraints, Magic square - Different constraints, Magic square - Other operations, Magic square - Other magic shapes, Magic square - Combined extensions, Magic square - Related problems, Magic square - Magic Square of Primes, Magic square - n-Queens problem

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Magic square - Extra constraints. Certain extra restrictions can be imposed on magical squares. If not only the main diagonals but also the broken diagonals sum to the magic constant, the result is a panmagic square. If raising each number to certain powers yields another magic square, the result is a bimagic, a trimagic, or, in general, a multimagic square. ...

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Magic square, Magic square - Brief history of magic squares, Magic square - The Lo Shu Square 3x3 magic square, Magic square - The early squares of order four 4x4 magic squares, Magic square - Cultural significance of magic squares, Magic square - Albrecht Dürer's magic square, Magic square - The Sagrada Família magic square, Magic square - Types of magic squares and their construction, Magic square - A method for constructing a magic square of odd order, Magic square - A method of constructing a magic square of doubly even order, Magic square - Counting magic squares, Magic square - Generalizations, Magic square - Extra constraints, Magic square - Different constraints, Magic square - Other operations, Magic square - Other magic shapes, Magic square - Combined extensions, Magic square - Related problems, Magic square - Magic Square of Primes, Magic square - n-Queens problem

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There are many ways to construct magic squares, but the standard (and most simple) way is to follow certain configurations / formulas which generate regular patterns. Magic squares exist for all values of n, with only one exception - it is impossible to construct a magic square of order 2. Magic squares can be classified into three types: odd, doubly even (n divisible by four) and singly even (n even, but not divisible by four). Odd and doubly even magic squares are easy to generate; the construction of singly even magic ...

See also:

Magic square, Magic square - Brief history of magic squares, Magic square - The Lo Shu Square 3x3 magic square, Magic square - The early squares of order four 4x4 magic squares, Magic square - Cultural significance of magic squares, Magic square - Albrecht Dürer's magic square, Magic square - The Sagrada Família magic square, Magic square - Types of magic squares and their construction, Magic square - A method for constructing a magic square of odd order, Magic square - A method of constructing a magic square of doubly even order, Magic square - Counting magic squares, Magic square - Generalizations, Magic square - Extra constraints, Magic square - Different constraints, Magic square - Other operations, Magic square - Other magic shapes, Magic square - Combined extensions, Magic square - Related problems, Magic square - Magic Square of Primes, Magic square - n-Queens problem

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The binomial theorem can be stated by saying that the polynomial sequence is of binomial type. ...

See also:

Binomial theorem, Binomial theorem - Newton's generalized binomial theorem, Binomial theorem - Binomial type, Binomial theorem - Proof inductive, Binomial theorem - Trivia

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