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Outline Of Algebra
Algebra is one of the main branches of mathematics, covering the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. In addition to working directly with numbers, algebra also covers symbols, variables, and set elements. Addition and multiplication are general operations, but their precise definitions lead to structures such as groups, rings, and fields. Branches * Pre-algebra * Elementary algebra * Boolean algebra * Abstract algebra * Linear algebra * Universal algebra Algebraic equations An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree. * Linear equation – algebraic equation of degree one. * Polynomial equation – equation in which a polynomial is set equal to another polynomial. * Transcendental equation – equation involvin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Element (mathematics)
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called containing the first four positive integers (A = \), one could say that "3 is an element of ", expressed notationally as 3 \in A . Sets Writing A = \ means that the elements of the set are the numbers 1, 2, 3 and 4. Sets of elements of , for example \, are subsets of . Sets can themselves be elements. For example, consider the set B = \. The elements of are ''not'' 1, 2, 3, and 4. Rather, there are only three elements of , namely the numbers 1 and 2, and the set \. The elements of a set can be anything. For example the elements of the set C = \ are the color red, the number 12, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Equation
In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients a_1, \ldots, a_n are required to not all be zero. Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken. The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true. In the case of just one variable, there is exactly one solution (provided that a_1\ne 0). Often, the term ''linear equation'' refers implicitly to this particular case, in which the variable is sensibly called the ''unknown''. In the case of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see Order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equation with integer coefficients and :y^4 + \frac - \frac + xy^2 + y^2 + \frac = 0 is a multivariate polynomial equation over the rationals. For many authors, the term ''algebraic equation'' refers only to the univariate case, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables (the ''multivariate'' case), in which case the term ''polynomial equation'' is usually preferred. Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of studythis is the subject of group theory and ring theory in universal algebra, the object of study is the possible types of algebraic structures and their relationships. Basic idea In universal algebra, an (or algebraic structure) is a set ''A'' together with a collection of operations on ''A''. Arity An ''n''- ary operation on ''A'' is a function that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a '' constant'', often denoted by a letter like ''a''. A 1-ary operation (or '' unary operation'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as Logical conjunction, conjunction (''and'') denoted as , disjunction (''or'') denoted as , and negation (''not'') denoted as . Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his ''An Investigation of the Laws of Thought'' (1854). According to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Algebra
Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics), variables (quantities without fixed values). This use of variables entails use of algebraic notation and an understanding of the general rules of the Operation (mathematics), operations introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real number, real and complex numbers. It is typically taught to secondary school students and at introductory college level in the United States, and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pre-algebra
Pre-algebra is a common name for a course (education), course taught in middle school mathematics in the United States, usually taught in the 6th, 7th, 8th, or 9th grade. The main objective of it is to prepare students for the study of algebra. Usually, Algebra I is taught in the Eighth grade, 8th or Ninth grade, 9th grade. As an intermediate stage after arithmetic, pre-algebra helps students pass specific conceptual barriers. Students are introduced to the idea that an equals sign, rather than just being the answer to a question as in basic arithmetic, means that two sides are Equality (mathematics), equivalent and can be manipulated together. They may also learn how numbers, Variable (mathematics), variables, and words can be used in the same ways. Subjects Subjects taught in a pre-algebra course may include: * Review of natural number arithmetic * Types of numbers such as integers, Fraction (mathematics), fractions, decimals and negative numbers * Ratios and Percentage, perce ... [...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] |
