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 2 \times 3 = 3 \times 2 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 z represent variables. In some cases, subscripts are added to distinguish variables, as in , , and . The lowercase letters , , and c are usually used for
constants and
coefficients. The expression 5x + 3 is an algebraic expression created by multiplying the number 5 with the variable x and adding the number 3 to the result. Other examples of algebraic expressions are 32xyz and {{nowrap|64x_1{}^2 + 7x_2 - c.}} 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 x^2 = 4 is true if x 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 x + 4 = 9 is only true if x 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 x - 7 = 4 can be solved for x by adding 7 to both sides, which isolates x 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 7x - 3x can be replaced with the expression 4x since 7x - 3x = (7-3)x = 4x 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 y = 3x then one can simplify the expression 7xy 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 x is set to zero in the equation , then y must be −1 for the equation to be true. This means that the (x, y)-pair (0, -1) is part of the graph of the equation. The (x, y)-pair , by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of (x, y)-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 (among its terms) of the sum of the exponents of the variables (4 in the above example). 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 x^2 - 3x - 10 can be factorized as . The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if x 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 ax^2 + bx + c = 0 is given by the
quadratic formula x = \frac{-b \pm \sqrt {b^2-4ac\ }}{2a}. 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. 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.
Linear algebra Linear algebra starts with the study of
systems of linear equations. An
equation is linear if it can be expressed in the form , where , , ..., a_n and b are constants. Examples are x_1 - 7x_2 + 3x_3 = 0 and {{tmath|1= \textstyle \frac{1}{4} x - y = 4 }}. A
system of linear equations is a set of linear equations for which one is interested in common solutions.
Matrices are rectangular arrays of values that have been originally introduced for having a compact and synthetic notation for systems of linear equations. For example, the system of equations \begin{align} 9x_1 + 3x_2 - 13x_3 &= 0 \\ 2.3x_1 + 7x_3 &= 9 \\ -5x_1 - 17x_2 &= -3 \end{align} can be written as AX=B, where , X and B are the matrices A=\begin{bmatrix}9 & 3 & -13 \\ 2.3 & 0 & 7 \\ -5 & -17 & 0 \end{bmatrix}, \quad X= \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix},\quad B = \begin{bmatrix}0 \\ 9 \\ -3 \end{bmatrix}. Under some conditions on the number of rows and columns, matrices can be
added,
multiplied, and sometimes
inverted. All methods for solving linear systems may be expressed as matrix manipulations using these operations. For example, solving the above system consists of computing an inverted matrix A^{-1} such that A^{-1}A = I, where I is the
identity matrix. Then, multiplying on the left both members of the above matrix equation by A^{-1}, one gets the solution of the system of linear equations as X=A^{-1}B. Methods of solving systems of linear equations range from the introductory, like substitution and elimination, to more advanced techniques using matrices, such as
Cramer's rule, the
Gaussian elimination, and
LU decomposition. Some systems of equations are
inconsistent, meaning that no solutions exist because the equations contradict each other. Consistent systems have either one unique solution or an infinite number of solutions. The study of
vector spaces and
linear maps form a large part of linear algebra. A vector space is an algebraic structure formed by a set with an addition that makes it an
abelian group and a
scalar multiplication that is compatible with addition (see
vector space for details). A linear map is a function between vector spaces that is compatible with addition and scalar multiplication. In the case of
finite-dimensional vector spaces, vectors and linear maps can be represented by matrices. It follows that the theories of matrices and finite-dimensional vector spaces are essentially the same. In particular, vector spaces provide a third way for expressing and manipulating systems of linear equations. From this perspective, a matrix is a representation of a linear map: if one chooses a particular
basis to describe the vectors being transformed, then the entries in the matrix give the results of applying the linear map to the basis vectors. Systems of equations can be interpreted as geometric figures. For systems with two variables, each equation represents a
line in
two-dimensional space. The point where the two lines intersect is the solution of the full system because this is the only point that solves both the first and the second equation. For inconsistent systems, the two lines run parallel, meaning that there is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to seek solutions graphically by plotting the equations and determining where they intersect. The same principles also apply to systems of equations with more variables, with the difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to
planes in
three-dimensional space, and the points where all planes intersect solve the system of equations.
Abstract algebra Abstract algebra, also called modern algebra, is the study of
algebraic structures. An algebraic structure is a framework for understanding
operations on
mathematical objects, like the addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as
groups,
rings, and
fields. The key difference between these types of algebraic structures lies in the number of operations they use and the laws they obey. On a formal level, an algebraic structure is a
set of mathematical objects, called the underlying set, together with one or several operations. Abstract algebra is primarily interested in
binary operations, which take any two objects from the underlying set as inputs and maps them to another object from this set as output. For example, the algebraic structure \langle \N, + \rangle has the
natural numbers () as the underlying set and addition () as its binary operation. For instance, the underlying set of the
symmetry group of a geometric object is made up of
geometric transformations, such as
rotations, under which the object remains
unchanged. Its binary operation is
function composition, which takes two transformations as input and has the transformation resulting from applying the first transformation followed by the second as its output.
Group theory Abstract algebra classifies algebraic structures based on the laws or
axioms that its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation is
associative and has an
identity element and
inverse elements. An operation is associative if the order of several applications does not matter, i.e., if is the same as a \circ (b \circ c) for all elements. An operation has an identity element or a neutral element if one element
e exists that does not change the value of any other element, i.e., if . An operation has inverse elements if for any element a there exists a reciprocal element a^{-1} that undoes . If an element operates on its inverse then the result is the neutral element
e, expressed formally as {{tmath|1= a \circ a^{-1} = a^{-1} \circ a = e }}. Every algebraic structure that fulfills these requirements is a group. For example, \langle \Z, + \rangle is a group formed by the set of
integers together with the operation of addition. The neutral element is 0 and the inverse element of any number a is The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements.
Group theory examines the nature of groups, with basic theorems such as the
fundamental theorem of finite abelian groups and the
Feit–Thompson theorem. The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete
classification of finite simple groups.
Ring theory and field theory A ring is an algebraic structure with two operations that work similarly to the addition and multiplication of numbers and are named and generally denoted similarly. A ring is a
commutative group under addition: the addition of the ring is associative, commutative, and has an identity element and inverse elements. The multiplication is associative and
distributive with respect to addition; that is, a (b + c) = a b + a c and (b + c) a = b a + c a. Moreover, multiplication is associative and has an
identity element generally denoted as . Multiplication needs not to be commutative; if it is commutative, one has a
commutative ring. The
ring of integers () is one of the simplest commutative rings. A
field is a commutative ring such that and each nonzero element has a
multiplicative inverse. The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of 7 is {{tmath|1= \tfrac{1}{7} }}, which is not an integer. The
rational numbers, the
real numbers, and the
complex numbers each form a field with the operations of addition and multiplication.
Ring theory is the study of rings, exploring concepts such as
subrings,
quotient rings,
polynomial rings, and
ideals as well as theorems such as
Hilbert's basis theorem. Field theory is concerned with fields, examining
field extensions,
algebraic closures, and
finite fields.
Galois theory explores the relation between field theory and group theory, relying on the
fundamental theorem of Galois theory.
Theories of interrelations among structures becomes a
semigroup if its operation is associative. Besides groups, rings, and fields, there are many other algebraic structures studied by algebra. They include
magmas,
semigroups,
monoids,
abelian groups,
commutative rings,
modules,
lattices,
vector spaces,
algebras over a field, and
associative and
non-associative algebras. They differ from each other regarding the types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more specialized structure by adding constraints. For example, a magma becomes a semigroup if its operation is associative.
Homomorphisms are tools to examine structural features by comparing two algebraic structures. A homomorphism is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form \langle A, \circ \rangle and \langle B, \star \rangle then the function h: A \to B is a homomorphism if it fulfills the following requirement: . The existence of a homomorphism reveals that the operation \star in the second algebraic structure plays the same role as the operation \circ does in the first algebraic structure.
Isomorphisms are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a
bijective homomorphism, meaning that it establishes a one-to-one relationship between the elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure. s restrict their operations to a subset of the underlying set of the original algebraic structure. Another tool of comparison is the relation between an algebraic structure and its
subalgebra. The algebraic structure and its subalgebra use the same operations, which follow the same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure. All operations in the subalgebra are required to be
closed in its underlying set, meaning that they only produce elements that belong to this set. One of those structural features concerns the
identities that are true in different algebraic structures. In this context, an identity is a
universal equation or an equation that is true for all elements of the underlying set. For example, commutativity is a universal equation that states that a \circ b is identical to b \circ a for all elements. A
variety is a class of all algebraic structures that satisfy certain identities. For example, if two algebraic structures satisfy commutativity then they are both part of the corresponding variety.
Category theory examines how mathematical objects are related to each other using the concept of
categories. A category is a collection of objects together with a collection of
morphisms or "arrows" between those objects. These two collections must satisfy certain conditions. For example, morphisms can be joined, or
composed: if there exists a morphism from object a to object , and another morphism from object b to object , then there must also exist one from object a to object . Composition of morphisms is required to be associative, and there must be an "identity morphism" for every object. Categories are widely used in contemporary mathematics since they provide a unifying framework to describe and analyze many fundamental mathematical concepts. For example, sets can be described with the
category of sets, and any group can be regarded as the morphisms of a category with just one object. == History ==