Solving an Equation for a Variable. Inequalities in One Variable. Study Plan for Review of Chapter 2. Ordered Pairs and Solutions of Linear Equations. Standard Form and Graphing with Intercepts. Average Rate of Change and Slope. Slope Intercept Form. Linear Inequalities in Two Variables. Relations and Functions. Study Plan for Review of Chapter 3.
Cumulative Reviews Chapters Systems of Linear Equations. The Substitution Method. The Elimination Method. More Applications. Systems of Linear Inequalities. Study Plan for Review of Chapter 4.
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Exponential Expressions and the Exponent Rules. Scientific Notation. Introduction to Polynomials. Multiplication of Polynomials. Division of Polynomials.
Study Plan for Review of Chapter 5. Introduction to Factoring. Factoring Trinomials. Factoring Completely. The Zero Product Property. Study Plan for Review of Chapter 6. Cumulative Review Chapters Simplifying Rational Expressions. Multiplication and Division of Rational Expressions. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices this is a noncommutative algebra.
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Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory , ring theory , and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name. Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic.
In algebra, numbers are often represented by symbols called variables such as a , n , x , y or z. This is useful because:. A polynomial is an expression that is the sum of a finite number of non-zero terms , each term consisting of the product of a constant and a finite number of variables raised to whole number powers. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication.
A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.
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Two important and related problems in algebra are the factorization of polynomials , that is, expressing a given polynomial as a product of other polynomials that can not be factored any further, and the computation of polynomial greatest common divisors. A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are listed fundamental concepts in abstract algebra. Sets : Rather than just considering the different types of numbers , abstract algebra 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. Set theory is a branch of logic and not technically a branch of algebra.
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The notion of binary operation is meaningless without the set on which the operation is defined. 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. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers 1, 2, 3, Inverse elements : The negative numbers give rise to the concept of inverse elements. Associativity : Addition of integers has a property called associativity.
That is, the grouping of the numbers to be added does not affect the sum. This property is shared by most binary operations, but not subtraction or division or octonion multiplication. Commutativity : Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative. Combining the above concepts gives one of the most important structures in mathematics: a group.
For example, the set of integers under the operation of addition is a group. The non-zero rational numbers form a group under multiplication. 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. 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 and , which separates the finite simple groups into roughly 30 basic types. Semi-groups , quasi-groups , 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 semi-group has an associative binary operation, but might not have an identity element. A monoid is a semi-group which does have an identity but might not have an inverse for every element. A quasi-group satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however the binary operation might not be associative.
Groups just have one binary operation.
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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 generalises the distributive law for numbers. The integers are an example of a ring. The integers have additional properties which make it an integral domain. From Wikipedia, the free encyclopedia. For the kind of algebraic structure, see Algebra over a field. Study of mathematical symbols and the rules for manipulating them. Main articles: History of algebra and Timeline of algebra.
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