Absolute value

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, |x| = −x if x is negative, |0| = 0. For example, the absolute value of 3 is 3, the absolute value of −3 is 3; the absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is defined for the complex numbers, the quaternions, ordered rings and vector spaces; the absolute value is related to the notions of magnitude and norm in various mathematical and physical contexts. In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French for the complex absolute value, it was borrowed into English in 1866 as the Latin equivalent modulus; the term absolute value has been used in this sense from at least 1806 in French and 1857 in English. The notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.

Other names for absolute value include numerical magnitude. In programming languages and computational software packages, the absolute value of x is represented by abs, or a similar expression; the vertical bar notation appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality. Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra, for example a real number, a complex number, or a quaternion. A related but distinct notation is the use of vertical bars for either the euclidean norm or sup norm of a vector in R n, although double vertical bars with subscripts are a more common and less ambiguous notation. For any real number x, the absolute value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥ 0 − x, if x < 0. {\displaystyle |x|=\left\ The absolute value of x is thus always either positive or zero, but never negative: when x itself is negative its absolute value is positive.

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, more the absolute value of the difference of two real numbers is the distance between them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root symbol represents the unique positive square root, it follows that | x | = x 2 is equivalent to the definition above, may be used as an alternative definition of the absolute value of real numbers; the absolute value has the following four fundamental properties, that are used for generalization of this notion to other domains: Non-negativity, positive definiteness, multiplicativity are apparent from the definition. To see that subadditivity holds, first note that one of the two alternatives of taking s as either –1 or +1 guarantees that s ⋅ = | a + b | ≥ 0. Now, since − 1 ⋅ x ≤ | x | and + 1 ⋅ x ≤ | x |, it follows that, whichever is the value of s, one has s ⋅ x ≤ | x | for all real x.

| a + b | = s ⋅ = s ⋅ a + s ⋅ b ≤ | a | + | b |, as desired. Some additional useful properties are given below; these are either immediate consequences of the definition or implied by the four fundamental properties above. Two other useful properties concerning inequalities are: | a | ≤ b ⟺ − b ≤ a ≤ b | a | ≥ b ⟺ a ≤

Paramount Theater (Springfield, Massachusetts)

The Paramount Theater is an historic theater located at 1676-1708 Main Street in Springfield, Massachusetts. Built in 1926 out of part of the grand Massasoit Hotel at a cost of over $1 million, the Paramount Theater was the most ornate picture palace in Western Massachusetts; as of 2011, The Paramount is in the midst of a $1.725 million renovation to once again become a theater after decade as a disco and concert hall, when it was the center of Springfield's club scene. In 2018 the building's owners, the New England Farm Workers Council, announced plans to redevelop it in tandem with a new adjacent hotel building. In a push to renovate the Paramount along with Holyoke's Victory Theater, in October 2018, the administration of Massachusetts Governor Charlie Baker announced a $2.5 million grant to assist the project, on top of a $4 million federal loan guarantee. Pending finalizing funding for the combined restoration and new hotel, no construction timeline has been presented as of 2020. From 1926 until the 1960s, The Paramount changed names several times - including a brief stint as the Julia Sanderson Theater, honoring a famous actress from Springfield - however, it remained a movie theater until the 1960s, when it began to find use as a mixed use venue for movies, rock concerts, other live performances.

The building was added to the National Register of Historic Places in 1979. In 1999, the venue was purchased and restored by Steven Stein and Michael Barrasso of Paramount Realty Investment LLC/Creative Theater Concepts. At that time, it was turned into a lavish performance space; the venue's main floor seats were removed. The theater underwent a $1.3 million renovation in 1999, was reopened as the Hippodrome. The original organ was restored and the marquee was changed to reflect the theater's new name; the Hippodrome became a popular concert venue during the 2000s. In 2011, the theater was purchased by the New England Farm Worker's Council; as of 2011, the Paramount Theater is in the midst of a $1.725 million renovation to once again become a theater and performance space. National Register of Historic Places listings in Springfield, Massachusetts National Register of Historic Places listings in Hampden County, Massachusetts

Théophile-Jules Pelouze

Théophile-Jules Pelouze was a French chemist. He was born at Valognes, died in Paris, his father, Edmond Pelouze, was the author of several technical handbooks. The son, after spending some time in a pharmacy at La Fère acted as laboratory assistant to Gay-Lussac and Jean Louis Lassaigne at Paris from 1827 to 1829. In 1830 he was appointed associate professor of chemistry at Lille, but returning to Paris next year became repetiteur, subsequently professor at the École polytechnique, he held the chair of chemistry at the Collège de France, in 1833 became assayer to the mint and in 1848 president of the Commission des Monnaies. He resigned all his public positions in 1852. After the coup d'état in 1851 he resigned his appointments, but continued to conduct an experimental laboratory-school he had started in 1846. There he worked with other nitrosulphates, his student Ascanio Sobrero was the discoverer of nitroglycerin, another student, Alfred Nobel, was to take that discovery on to great heights in the form of commercial explosives including dynamite.

He was a major inspiration for both students. Though Pelouze made no discovery of outstanding importance, he was a busy investigator, his work including researches on salicin, on beetroot sugar, on various organic acids, on oenanthic ether, on the nitrosulphates, on guncotton, on the composition and manufacture of glass, he carried out determinations of the atomic weights of several elements, with E. Fremy, published Traité de chimie générale, his son Eugène-Philippe Pelouze married Marguerite Wilson, a rich heiress, in 1857. The couple purchased the Château de Chenonceau in 1864. Marguerite continued to live there until 1888, when she was forced to sell, his name is one of the 72 names inscribed on the Eiffel Tower. This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed.. "Pelouze, Théophile Jules". Encyclopædia Britannica. 21. Cambridge University Press. Works by Théophile-Jules Pelouze at Open Library