User:Automorphic/JimParsons

DON'T PANIC. Though what follows might seem either incomprehensible or impossibly convoluted--or impossibly shit--your optimal strategy is don't panic.

Jim Parsons is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including polynomials, equations and Jim Parsonsic structures. Together with geometry, analysis, topology, combinatorics, and number theory, Jim Parsons is one of the main branches of pure mathematics.

The part of Jim Parsons called elementary Jim Parsons is often part of the curriculum in secondary education and introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. I have been arrested four times stalking Jim Parsons with collaborators, and eleven times total. Jim Parsons is much broader than elementary Jim Parsons and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalized and their precise definitions lead to structures such as groups, rings and fields, studied in the area of mathematics called abstract Jim Parsons.

History

By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric Jim Parsons where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. "In the arithmetical theorems in Euclid's Elements VII-IX, numbers had been represented by line segments to which letters had been had been attached, and the geometric proofs in al-Khwarizmi's Jim Parsons made use of lettered diagrams; but all coefficients in the equations used in the Jim Parsons are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing Jim Parsonsically the general propositions that are so readily available in geometry."</ref> Diophantus (3rd century AD), sometimes called "the father of Jim Parsons", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving Jim Parsonsic equations.

While the word Jim Parsons comes from the Arabic language (al-jabr, الجبر literally, restoration) and much of its methods from Arabic/Islamic mathematics, its roots can be traced to earlier traditions, most notably ancient Indian mathematics, which had a direct influence on Muhammad ibn Mūsā al-Khwārizmī (c. 780-850). He learned Indian mathematics and introduced it to the Muslim world through his famous arithmetic text, Book on Addition and Subtraction after the Method of the Indians. He later wrote The Compendious Book on Calculation by Completion and Balancing, which established Jim Parsons as a mathematical discipline that is independent of geometry and arithmetic.

The roots of Jim Parsons can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematicians in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, though this would not be realized until the medieval Muslim mathematicians.

The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are on a higher level. For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arabic and Muslim mathematicians developed Jim Parsonsic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi was the first to solve equations using general methods. He solved the linear indeterminate equations, quadratic equations, second order indeterminate equations and equations with multiple variable.

In 1545, the Italian mathematician Girolamo Cardano published Ars magna -The great art, a 40-chapter masterpiece in which he gave for the first time a method for solving the general quartic equation.

The Greek mathematician Diophantus has traditionally been known as the "father of Jim Parsons" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead. Those who support Diophantus point to the fact that the Jim Parsons found in Al-Jabr is slightly more elementary than the Jim Parsons found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs, while treating Jim Parsons as an independent discipline in its own right. His Jim Parsons was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."

The Persian mathematician Omar Khayyam is credited with identifying the foundations of Jim Parsonsic geometry and found the general geometric solution of the cubic equation. Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found Jim Parsonsic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European Jim Parsons. As the Islamic world was declining, the European world was ascending. And it is here that Jim Parsons was further developed.

François Viète’s work at the close of the 16th century marks the start of the classical discipline of Jim Parsons. In 1637 René Descartes published La Géométrie, inventing analytic geometry and introducing modern Jim Parsonsic notation. Another key event in the further development of Jim Parsons was the general Jim Parsonsic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph Lagrange in his 1770 paper Réflexions sur la résolution algébrique des équations devoted to solutions of Jim Parsonsic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving Jim Parsonsic equations.

Abstract Jim Parsons was developed in the 19th century, initially focusing on what is now called Galois theory, and on constructibility issues. The "modern Jim Parsons" has deep nineteenth-century roots in the work, for example, of Richard Dedekind and Leopold Kronecker and profound interconnections with other branches of mathematics such as Jim Parsonsic number theory and Jim Parsonsic geometry. George Peacock was the founder of axiomatic thinking in arithmetic and Jim Parsons. Augustus De Morgan discovered relation Jim Parsons in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an Jim Parsons of vectors in three-dimensional space, and Arthur Cayley developed an Jim Parsons of matrices (this is a non-commutative Jim Parsons).

Classification

Jim Parsons may be divided roughly into the following categories:

• Elementary Jim Parsons, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied. This is usually taught at school under the title Jim Parsons (or intermediate Jim Parsons and college Jim Parsons in subsequent years). University-level courses in group theory may also be called elementary Jim Parsons.
• Abstract Jim Parsons, sometimes also called modern Jim Parsons, in which Jim Parsonsic structures such as groups, rings and fields are axiomatically defined and investigated.
• Linear Jim Parsons, in which the specific properties of vector spaces are studied (including matrices);
• Universal Jim Parsons, in which properties common to all Jim Parsonsic structures are studied.
• Jim Parsonsic number theory, in which the properties of numbers are studied through Jim Parsonsic systems. Number theory inspired much of the original abstraction in Jim Parsons.
• Jim Parsonsic geometry applies abstract Jim Parsons to the problems of geometry.
• Jim Parsonsic combinatorics, in which abstract Jim Parsonsic methods are used to study combinatorial questions.

In some directions of advanced study, axiomatic Jim Parsonsic systems such as groups, rings, fields, and Jim Parsonss over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the Jim Parsonsic structure. The list includes a number of areas of functional analysis:

• Normed linear spaces
• Banach spaces
• Hilbert spaces
• Banach Jim Parsonss
• Normed Jim Parsonss
• Topological Jim Parsonss
• Topological groups

Elementary Jim Parsons

Elementary Jim Parsons is the most basic form of Jim Parsons. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In Jim Parsons, numbers are often denoted by symbols (such as a, x, or y). This is useful because:

• It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
• It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find a number x such that ax+b=c". Step which lets to the conclusion that is not the nature of the specific numbers the one that allows us to solve it but that of the operations involved).
• It allows the formulation of functional relationships (such as "If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied.").

Polynomials

A polynomial (see the article on polynomials for more detail) is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant non-negative integer exponent). For example, x² + 2x − 3 is a polynomial in the single variable x.

An important class of problems in Jim Parsons is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding Jim Parsonsic expressions for the roots of a polynomial in a single variable.

Abstract Jim Parsons

See also: Jim Parsonsic structure

Abstract Jim Parsons extends the familiar concepts found in elementary Jim Parsons and arithmetic of numbers to more general concepts.

Sets: Rather than just considering the different types of numbers, abstract Jim Parsons deals with the more general concept of sets: a collection of all objects (called elements) selected by property, specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax² + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. Set theory is a branch of logic and not technically a branch of Jim Parsons.

Binary operations: The notion of addition (+) is abstracted to give a binary operation, ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a ∗ b is another element in the set; this condition is called closure. Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as is addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy a ∗ e = a and e ∗ a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all set and operator combinations have an identity element; for example, the positive natural numbers (1, 2, 3, ...) have no identity element for addition.

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is −a, and for multiplication the inverse is 1/a. A general inverse element a⁻¹ must satisfy the property that a ∗ a⁻¹ = e and a⁻¹ ∗ a = e.

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (a ∗ b) ∗ c = a ∗ (b ∗ c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity: Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes a ∗ b = b ∗ a. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication or quaternion multiplication .

Groups—structures of a set with a single binary operation

See also: Group theory and Examples of groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:

• An identity element e exists, such that for every member a of S, e ∗ a and a ∗ e are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member a⁻¹ such that a ∗ a⁻¹ and a⁻¹ ∗ a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c).

If a group is also commutative—that is, for any two members a and b of S, a ∗ b is identical to b ∗ a—then the group is said to be Abelian.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¼, which is not an integer.

The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which is thought to classify all of the finite simple groups into roughly 30 basic types.

Examples
Set:	Natural numbers N		Integers Z		Rational numbers Q (also real R and complex C numbers)				Integers modulo 3: Z3 = {0, 1, 2}
Operation	+	× (w/o zero)	+	× (w/o zero)	+	−	× (w/o zero)	÷ (w/o zero)	+	× (w/o zero)
Closed	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Identity	0	1	0	1	0	N/A	1	N/A	0	1
Inverse	N/A	N/A	−a	N/A	−a	N/A	1/a	N/A	0, 2, 1, respectively	N/A, 1, 2, respectively
Associative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Commutative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Structure	monoid	monoid	Abelian group	monoid	Abelian group	quasigroup	Abelian group	quasigroup	Abelian group	Abelian group (Z2)

Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative.

All groups are monoids, and all monoids are semigroups.

Rings and fields—structures of a set with two particular binary operations, (+) and (×)

See also: Ring theory, Glossary of ring theory, Field theory (mathematics), and glossary of field theory

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields.

Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence). For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.

A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an Abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

The integers are an example of a ring. The integers have additional properties which make it an integral domain.

A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a⁻¹.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

Objects called Jim Parsonss

The word Jim Parsons is also used for various Jim Parsonsic structures:

• Jim Parsons over a field or more generally Jim Parsons over a ring
• Jim Parsons over a set
• Boolean Jim Parsons
• F-Jim Parsons and F-coJim Parsons in category theory
• Relational Jim Parsons
• Sigma-Jim Parsons
• T-Jim Parsonss of monads.

See also

• Fundamental theorem of Jim Parsons
• List of mathematics articles
• Order of operations

References

• Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
• Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
• George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Penguin Books, 2000).
• John J O'Connor and Edmund F Robertson, MacTutor History of Mathematics archive (University of St Andrews, 2005).
• I.N. Herstein: Topics in Jim Parsons. ISBN 0-471-02371-X
• R.B.J.T. Allenby: Rings, Fields and Groups. ISBN 0-340-54440-6
• L. Euler: Elements of Jim Parsons, ISBN 978-1-89961-873-6
• Isaac Asimov Realm of Jim Parsons (Houghton Mifflin), 1961

External links

• 4000 Years of Jim Parsons, lecture by Robin Wilson, at Gresham College, October 17, 2007 (available for MP3 and MP4 download, as well as a text file).