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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] |
Hippasus
Hippasus of Metapontum (; , ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods for divulging this and crediting it to himself instead of Pythagoras, which was the norm in Pythagorean society. The few ancient sources who describe this story, however, either do not mention Hippasus by name (e.g., Pappus) or alternatively tell that Hippasus drowned because he revealed how to construct a dodecahedron inside a sphere. The discovery of irrationality is not specifically ascribed to Hippasus by any ancient writer. Life Little is known about the life of Hippasus. He may have lived in the late 5th century BC, about a century after the ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QM-AM-GM-HM Inequalities
In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean (HM), geometric mean (GM), arithmetic mean (AM), and Root mean square, quadratic mean (QM; also known as root mean square). Suppose that x_1, x_2, \ldots, x_n are positive real numbers. Then : 0 0 implies the inequality: : \frac \leq \sqrt[n]. The ''n'' = 2 case When ''n'' = 2, the inequalities become :\frac \leq \sqrt \leq \frac\leq\sqrt for all x_1, x_2 > 0, which can be visualized in a semi-circle whose diameter is [''AB''] and center ''D''. Suppose ''AC'' = ''x''1 and ''BC'' = ''x''2. Construct perpendiculars to [''AB''] at ''D'' and ''C'' respectively. Join [''CE''] and [''DF''] and further construct a perpendicular [''CG''] to [''DF''] at ''G''. Then the length of ''GF'' can be calculated to be the harmonic mean, ''CF'' to be the geometric mean, ''DE'' to be the arithmetic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler Triangle
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares on the edges of this triangle have areas in another geometric progression, 1:\varphi:\varphi^2. Alternative definitions of the same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles. This triangle is named after Johannes Kepler, but can be found in earlier sources. Although some sources claim that ancient Egyptian pyramids had proportions based on a Kepler triangle, most scholars believe that the golden ratio was not known to Egyptian mathematics and architecture. History The Kepler triangle is named after the German mathematician and astronomer Johannes Kepler (1571–1630), who wrote about this shape in a 1597 letter. Two concepts that can be used to analyze th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \frac = \frac = \varphi, where the Greek letter Phi (letter), phi ( or ) denotes the golden ratio. The constant satisfies the quadratic equation and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the Straightedge and compass construction, construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average
In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by how many numbers are in the list. For example, the mean or average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, the most representative statistics, statistic to be taken as the average might be another measure of central tendency, such as the mid-range, median, Mode (statistics), mode or geometric mean. For example, the average income, personal income is often given as the median the number below which are 50% of personal incomes and above which are 50% of personal incomes because the mean would be higher by including personal incomes from a few billionaires. General properties If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic–geometric Mean
In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some mathematical constants, in particular, computing . The AGM is defined as the limit of the interdependent sequences a_i and g_i. Assuming x \geq y \geq 0, we write:\begin a_0 &= x,\\ g_0 &= y\\ a_ &= \tfrac12(a_n + g_n),\\ g_ &= \sqrt\, . \endThese two sequences converge to the same number, the arithmetic–geometric mean of and ; it is denoted by , or sometimes by or . The arithmetic–geometric mean can be extended to complex numbers and, when the branches of the square root are allowed to be taken inconsistently, it is a multivalued function. Example To find the arithmetic–geometric mean of and , iterate as follows:\begin a_1 & = & \tfr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Means Nomograms
Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Neopythagoreanism, a school of philosophy reviving Pythagorean doctrines that became prominent in the 1st and 2nd centuries AD * Pythagorean diet, the name for vegetarianism before the nineteenth century Mathematics * Pythagorean theorem * Pythagorean triple * Pythagorean prime * Pythagorean trigonometric identity * Table of Pythagoras, another name for the multiplication table Music * Pythagorean comma * Pythagorean hammers * Pythagorean tuning Other uses * Pythagorean cup * Pythagorean expectation, a baseball statistical term * Pythagorean letter See also * 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 mys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neopythagoreanism
Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic and Roman philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st century BC and flourished during the 1st and 2nd centuries AD. The ''Encyclopædia Britannica'' Eleventh Edition describes Neopythagoreanism as "a link in the chain between the old and the new" within Hellenistic philosophy. Central to Neopythagorean thought was the concept of a soul and its inherent desire for a ''unio mystica'' with the divine. The word ''Neopythagoreanism'' is a modern (19th century) term, coined as a parallel of "Neoplatonism". History In the 1st century BC Cicero's friend Nigidius Figulus made an attempt to revive Pythagorean doctrines, but the most important members of the school were Apollonius of Tyana and Moderatus of Gades in the 1st century AD. Other important Neopythagoreans include the mathematician Nicoma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eudemus Of Rhodes
Eudemus of Rhodes (; ) was an ancient Greek philosopher, considered the first historian of science. He was one of Aristotle's most important pupils, editing his teacher's work and making it more easily accessible. Eudemus' nephew, Pasicles, was also credited with editing Aristotle's works. Life Eudemus was born on the isle of Rhodes, but spent a large part of his life in Athens, where he studied philosophy at Aristotle's Peripatetic School. Eudemus's collaboration with Aristotle was long-lasting and close, and he was generally considered to be one of Aristotle's most brilliant pupils: he and Theophrastus of Lesbos were regularly called not Aristotle's "disciples", but his "companions" (ἑταῖροι). It seems that Theophrastus was the greater genius of the two, continuing Aristotle's studies in a wide range of areas. Although Eudemus too conducted original research, his ''forte'' lay in systematizing Aristotle's philosophical legacy, and in a clever didactical presentation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theaetetus (mathematician)
Theaetetus of Athens (; ''Theaítētos''; c. 417 – c. 369 BCE), possibly the son of Euphronius of the Athenian deme Sunium, was a Greece, Greek mathematician. His principal contributions were on irrational number, irrational lengths, which was included in Book X of Euclid's Elements, Euclid's ''Elements'' and proving that there are precisely five Platonic solid, regular convex polyhedra. A friend of Socrates and Plato, he is the central character in Plato's Theaetetus (dialogue), eponymous Socratic dialogue. Theaetetus, like Plato, was a student of the Greek mathematician Theodorus of Cyrene. Cyrene was a prosperous Greek colony on the coast of North Africa, in what is now Libya, on the eastern end of the Gulf of Sidra. Theodorus had explored the theory of incommensurable quantities, and Theaetetus continued those studies with great enthusiasm; specifically, he classified various forms of irrational numbers according to the way they are expressed as square roots. This theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Of Alexandria
Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known about his life except for what can be found in his own writings, many of which are lost. Pappus apparently lived in Alexandria, where he worked as a Mathematics education, mathematics teacher to higher level students, one of whom was named Hermodorus.Pierre Dedron, J. Itard (1959) ''Mathematics And Mathematicians'', Vol. 1, p. 149 (trans. Judith V. Field) (Transworld Student Library, 1974) The ''Collection'', his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics that were part of the ancient mathematics curriculum, including geometry, astronomy, and mechanics. Pappus was active in a period generally considered one of stagnation in mathematical studies, where, to s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
