Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions.
Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several operations defined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups, rings, and fields, based on the number of operations they use and the laws they follow, called axioms. Universal algebra and category theory provide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures.
Algebraic methods were first studied in the ancient period to solve specific problems in fields like geometry. Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until the 16th and 17th centuries when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond a theory of equations to cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry, topology, number theory, and calculus, and other fields of inquiry, like logic and the empirical sciences.
Definition and etymology
Algebra is the branch of mathematics that studies algebraic structures and the operations they use. An algebraic structure is a non-empty set of mathematical objects, such as the integers, together with algebraic operations defined on that set, like addition and multiplication. Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines the use of variables in equations and how to manipulate these equations.Algebra is often understood as a generalization of arithmetic. Arithmetic studies operations like addition, subtraction, multiplication, and division, in a particular domain of numbers, such as the real numbers. Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in the form of variables in addition to numbers. A higher level of abstraction is found in abstract algebra, which is not limited to a particular domain and examines algebraic structures such as groups and rings. It extends beyond typical arithmetic operations by also covering other types of operations. Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates the characteristics of algebraic structures in general.
The term algebra is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra. When used as a countable noun, an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation, a bilinear map. Depending on the context, "algebra" can also refer to other algebraic structures, like a Lie algebra or an associative algebra.
The word algebra comes from the Arabic term الجبر, which originally referred to the surgical treatment of bonesetting. In the 9th century, the term received a mathematical meaning when Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to name a method for transforming equations and used it in the title of his treatise which was translated into Latin as Liber Algebrae et Almucabola. The word entered the English language in the 16th century from Italian, Spanish, and medieval Latin. Initially, its meaning was restricted to the theory of equations, that is, to the art of manipulating polynomial equations in view of solving them. This changed in the 19th century when the scope of algebra broadened to cover the study of diverse types of algebraic operations and structures together with their underlying axioms, the laws they follow.
Major branches
Elementary algebra
Elementary algebra, also called school algebra, college algebra, and classical algebra, is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on variables and examines how mathematical statements may be transformed.Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition, subtraction, multiplication, division, exponentiation, extraction of roots, and logarithm. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in
Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the equation belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the commutative property of multiplication, which is expressed in the equation
Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention, the lowercase letters,, and represent variables. In some cases, subscripts are added to distinguish variables, as in,, and. The lowercase letters,, and are usually used for constants and coefficients. The expression is an algebraic expression created by multiplying the number 5 with the variable and adding the number 3 to the result. Other examples of algebraic expressions are and
Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is a statement formed by comparing two expressions, saying that they are equal. This can be expressed using the equals sign, as in. Inequations involve a different type of comparison, saying that the two sides are different. This can be expressed using symbols such as the less-than sign, the greater-than sign, and the inequality sign. Unlike other expressions, statements can be true or false, and their truth value usually depends on the values of the variables. For example, the statement is true if is either 2 or −2 and false otherwise. Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, such as the equation. Conditional equations are only true for some values. For example, the equation is only true if is 5.
The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as solving the equation for that variable. For example, the equation can be solved for by adding 7 to both sides, which isolates on the left side and results in the equation.
There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression can be replaced with the expression since by the distributive property. For statements with several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that then one can simplify the expression to arrive at. In a similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables.
Algebraic equations can be interpreted geometrically to describe spatial figures in the form of a graph. To do so, the different variables in the equation are understood as coordinates and the values that solve the equation are interpreted as points of a graph. For example, if is set to zero in the equation, then must be −1 for the equation to be true. This means that the -pair is part of the graph of the equation. The -pair, by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of -pairs that solve the equation.
Polynomials
A polynomial is an expression consisting of one or more terms that are added or subtracted from each other, like. Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive integer power. A monomial is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials. The degree of a polynomial is the maximal value of the sum of the exponents of the variables. Polynomials of degree one are called linear polynomials. Linear algebra studies systems of linear polynomials. A polynomial is said to be univariate or multivariate, depending on whether it uses one or more variables.Factorization is a method used to simplify polynomials, making it easier to analyze them and determine the values for which they evaluate to zero. Factorization consists of rewriting a polynomial as a product of several factors. For example, the polynomial can be factorized as. The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if is either −2 or 5. Before the 19th century, much of algebra was devoted to polynomial equations, that is equations obtained by equating a polynomial to zero. The first attempts for solving polynomial equations were to express the solutions in terms of th roots. The solution of a second-degree polynomial equation of the form is given by the quadratic formula
Solutions for the degrees 3 and 4 are given by the cubic and quartic formulas. There are no general solutions for higher degrees, as proven in the 19th century by the Abel–Ruffini theorem. Even when general solutions do not exist, approximate solutions can be found by numerical tools like the Newton–Raphson method.
The fundamental theorem of algebra asserts that every univariate polynomial equation of positive degree with real or complex coefficients has at least one complex solution. Consequently, every polynomial of a positive degree can be factorized into linear polynomials. This theorem was proved at the beginning of the 19th century, but this does not close the problem since the theorem does not provide any way for computing the solutions.