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List Of Things Named After Pythagoras
This is a list of things named after Pythagoras, the ancient Greek philosopher, mystic, mathematician, and music theorist. Philosophy and mysticism * Pythagoreanism – the system of philosophy of Pythagoras and his followers * Neopythagoreanism – a later philosophical system * Pythagorean cup – a drinking cup that forces its user to imbibe only in moderation * Pythagorean letter – the Greek letter upsilon, used as a symbol by the Pythagoreans * Pythagorean diet – vegetarianism * Pythagorean symbol – the tetractys * Pythagorean system – the distinctive system of numerology used by the Pythagoreans Mathematics * Pythagorean theorem – the statement that the sum of the areas of the squares on the sides of a right triangle is the area of the square on the hypotenuse * Pythagorean triple – a set of three positive integers that can occur in the Pythagorean theorem * Pythagorean quadruple - a set of four positive integers that describes the space diagonal of a cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagoras
Pythagoras of Samos (; BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of Plato, Aristotle, and, through them, Western philosophy. Modern scholars disagree regarding Pythagoras's education and influences, but most agree that he travelled to Croton in southern Italy around 530 BC, where he founded a school in which initiates were allegedly sworn to secrecy and lived a communal, ascetic lifestyle. In antiquity, Pythagoras was credited with mathematical and scientific discoveries, such as the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions, the sphericity of the Earth, the identity of the morning and evening stars as the planet Venus, and the division of the globe into five climatic zones. He was reputedly the first man to call himself a philosopher ("lover of wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagoras Tree (fractal)
The Pythagoras tree is a Plane (geometry), plane fractal constructed from Square (geometry), squares. Invented by the Netherlands, Dutch mathematics teacher Albert E. Bosman in 1942, it is named after the Ancient Greece, ancient Greek mathematician Pythagoras because each triple of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. If the largest square has a size of ''L'' × ''L'', the entire Pythagoras tree fits snugly inside a box of size 6''L'' × 4''L''.Wisfaq.nl The finer details of the tree resemble the Lévy C curve. Construction The construction of the Pythagoras tree begins with a square (geometry), square. Upon this square are constructed two squares, each scaled down by a linear ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lute Of Pythagoras
The lute of Pythagoras is a self-similar geometric figure made from a sequence of pentagrams. Constructions The lute may be drawn from a sequence of pentagrams. The centers of the pentagrams lie on a line and (except for the first and largest of them) each shares two vertices with the next larger one in the sequence... An alternative construction is based on the golden triangle, an isosceles triangle with base angles of 72° and apex angle 36°. Two smaller copies of the same triangle may be drawn inside the given triangle, having the base of the triangle as one of their sides. The two new edges of these two smaller triangles, together with the base of the original golden triangle, form three of the five edges of the polygon. Adding a segment between the endpoints of these two new edges cuts off a smaller golden triangle, within which the construction can be repeated.. Some sources add another pentagram, inscribed within the inner pentagon of the largest pentagram of the fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Technically, it should be 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 Unit square, square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational number, irrational. The fraction (≈ 1.4142857) is sometimes used as a good Diophantine approximation, rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 60 decimal places: : History The Babylonian clay tablet YBC 7289 (–1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagoras's Constant
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Technically, it should be 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 fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 60 decimal places: : History The Babylonian clay tablet YBC 7289 (–1600 BC) gives an approximation of \sqrt in four sexagesi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Addition
In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. Like the more familiar addition and multiplication operations of arithmetic, it is both associative and commutative. This operation can be used in the conversion of Cartesian coordinates to polar coordinates, and in the calculation of Euclidean distance. It also provides a simple notation and terminology for the diameter of a cuboid, the energy-momentum relation in physics, and the overall noise from independent sources of noise. In its applications to signal processing and propagation of measurement uncertainty, the same operation is also called addition in quadrature. A scaled version of this operation gives the quadratic mean or ''root mean square''. It is implemented in many programming libraries as the hypot function, in a way designed to avoid errors arising due to limited-precision calculations performed o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Mean
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean is the multiplicative inverse, reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with f(x) = \frac. For example, the harmonic mean of 1, 4, and 4 is :\left(\frac\right)^ = \frac = \frac = 2\,. Definition The harmonic mean ''H'' of the positive real numbers x_1, x_2, \ldots, x_n is :H(x_1, x_2, \ldots, x_n) = \frac = \frac. It is the reciprocal of the arithmetic mean of the reciprocals, and vice versa: :\begin H(x_1, x_2, \ldots, x_n) &= \frac, \\ A(x_1, x_2, \ldots, x_n) &= \frac, \end where the arithmetic mean is A(x_1, x_2, \ldots, x_n) = \tfrac1n \sum_^n x_i. The harmonic mean is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for positive a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean of numbers is the Nth root, th root of their product (mathematics), product, i.e., for a collection of numbers , the geometric mean is defined as : \sqrt[n]. When the collection of numbers and their geometric mean are plotted in logarithmic scale, the geometric mean is transformed into an arithmetic mean, so the geometric mean can equivalently be calculated by taking the natural logarithm of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale using the exponential function , :\sqrt[n] = \exp \left( \frac \right). The geometric mean of two numbers is the square root of their product, for example with numbers and the geometric mean is \textstyle \sqrt = The geometric mean o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a Survey (statistics), survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such as geometric mean, geometric and harmonic mean, harmonic. Arithmetic means are also frequently used in economics, anthropology, history, and almost every other academic field to some extent. For example, per capita income is the arithmetic average of the income of a nation's Human population, population. While the arithmetic mean is often used to report central tendency, central tendencies, it is not a robust statistic: it is greatly influenced by outliers (Value (mathematics), values much larger or smaller than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Means
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. Definition The three Pythagorean means are defined by the equations \begin \operatorname \left( x_1,\; \ldots,\; x_n \right) &= \frac, \\ pt \operatorname \left( x_1,\; \ldots,\; x_n \right) &= \sqrt \text \\ pt \operatorname \left( x_1,\; \ldots,\; x_n \right) &= \frac . \end Properties Each mean, \operatorname, has the following properties for positive real inputs: ; First-order homogeneity: \operatorname(bx_1, \ldots, bx_n) = b \operatorname(x_1, \ldots, x_n) ; Invariance under exchange: \operatorname(\ldots, x_i, \ldots, x_j, \ldots) = \operatorname(\ldots, x_j, \ldots, x_i, \ldots) for any i and j. ; Monotonicity: if a \leq b then \operatorname(a,x_1,x_2,\ldots x_n) \leq \operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Identity
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. Pythagorean identities The basic relationship between the sine and cosine is given by the Pythagorean identity: \sin^2\theta + \cos^2\theta = 1, where \sin^2 \theta means ^2 and \cos^2 \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
