Elementary algebra, also known as high school algebra or college algebra,Pierce, R., College Algebra, Maths is Fun, accessed 28 August 2023 encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers,H.E. Slaught and N.J. Lennes, Elementary algebra, Publ. Allyn and Bacon, 1915, page 1 (republished by Forgotten Books) whilst algebra introduces numerical variables (quantities without fixed values).Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, , 9780534999728, 654 pages, page 2 In arithmetic, operations (such as addition and subtraction) can only be performed on numbers. In algebra, operations can be performed on numbers, variables, and terms. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of 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 problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations.
Algebraic operations
Algebraic notation
Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression has the following components:
A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.Richard N. Aufmann, Joanne Lockwood, Introductory Algebra: An Applied Approach, Publisher Cengage Learning, 2010, , 9781439046043, page 78 Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. ) are typically used to represent constants, and those toward the end of the alphabet (e.g. and ) are used to represent variables.William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, The Rosen Publishing Group, 2010, , 9781615302192, page 71 They are usually printed in italics.James E. Gentle, Numerical Linear Algebra for Applications in Statistics, Publisher: Springer, 1998, , 9780387985428, 221 pages, [James E. Gentle page 184]
Algebraic operations work in the same way as arithmetic operations,Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, page 7 such as addition, subtraction, multiplication, division and exponentiation,Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: Cengage Learning, 2007, , 9780618851959, 1114 pages, page 6 and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, is written as , and may be written .Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in Mathematics Matters Secondary 1 Express Textbook, Publisher Panpac Education Pte Ltd, , 9789812738820, page 68
Usually terms with the highest power (exponent), are written on the left, for example, is written to the left of . When a coefficient is one, it is usually omitted (e.g. is written ).David Alan Herzog, Teach Yourself Visually Algebra, Publisher John Wiley & Sons, 2008, , 9780470185599, 304 pages, page 72 Likewise when the exponent (power) is one, (e.g. is written ).John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, , 9780766861893, 1613 pages, page 31 When the exponent is zero, the result is always 1 (e.g. is always rewritten to ).Jerome E. Kaufmann, Karen L. Schwitters, Algebra for College Students, Publisher Cengage Learning, 2010, , 9780538733540, 803 pages, page 222 However , being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.
Alternative notation Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g., , in plain text, and in the TeX mark-up language, the caret symbol represents exponentiation, so is written as "x^2".Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, , 9789380298153, page 212George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, , 9780817641320, page 17 This also applies to some programming languages such as Lua. In programming languages such as Ada,S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual, Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, , 9783540693352, page 13 Fortran,C. Xavier, Fortran 77 And Numerical Methods, Publisher New Age International, 1994, , 9788122406702, page 20 Perl,Randal Schwartz, Brian Foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, , 9781449313142, page 24 PythonMatthew A. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, , 9781598631586, page 46 and Ruby,Kevin C. Baird, Ruby by Example: Concepts and Code, Publisher No Starch Press, 2007, , 9781593271480, page 72 a double asterisk is used, so is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,William P. Berlinghoff, Fernando Q. Gouvêa, Math through the Ages: A Gentle History for Teachers and Others, Publisher MAA, 2004, , 9780883857366, page 75 and it must be explicitly used, for example, is written "3*x".
Concepts
Variables
Elementary algebra builds on and extends arithmeticThomas Sonnabend, Mathematics for Teachers: An Interactive Approach for Grades K-8, Publisher: Cengage Learning, 2009, , 9780495561668, 759 pages, page xvii by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.
Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as .Lewis Hirsch, Arthur Goodman, Understanding Elementary Algebra With Geometry: A Course for College Students, Publisher: Cengage Learning, 2005, , 9780534999728, 654 pages, page 48 Variables allow one to describe general problems,Lawrence S. Leff, College Algebra: Barron's Ez-101 Study Keys, Publisher: Barron's Educational Series, 2005, , 9780764129148, 230 pages, page 2 without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to seconds. A more general (algebraic) description may state that the number of seconds, , where m is the number of minutes. Variables allow one to describe mathematical relationships between quantities that may vary.Ron Larson, Kimberly Nolting, Elementary Algebra, Publisher: Cengage Learning, 2009, , 9780547102276, 622 pages, page 210 For example, the relationship between the circumference, c, and diameter, d, of a circle is described by . Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as .Charles P. McKeague, Elementary Algebra, Publisher: Cengage Learning, 2011, , 9780840064219, 571 pages, page 49
Simplifying expressions
Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example, Added terms are simplified using coefficients. For example, can be simplified as (where 3 is a numerical coefficient). Multiplied terms are simplified using exponents. For example, is represented as Like terms are added together,Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, Publisher Kaplan Publishing, 2007, , 9781419552885, 288 pages, page 51 for example, is written as , because the terms containing are added together, and the terms containing are added together. Brackets can be "multiplied out", using the distributive property. For example, can be written as which can be written as Expressions can be factored. For example, , by dividing both terms by the common factor, can be written as
Equations
An equation states that two expressions are equal using the symbol for equality, (the equals sign).Mark Clark, Cynthia Anfinson, Beginning Algebra: Connecting Concepts Through Applications, Publisher Cengage Learning, 2011, , 9780534419387, 793 pages, page 134 One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:Alan S. Tussy, R. David Gustafson, Elementary and Intermediate Algebra, Publisher Cengage Learning, 2012, , 9781111567682, 1163 pages, page 493
This equation states that , representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by and .
An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as ); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. is true only for and . The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.
Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: where represents 'greater than', and where represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.
Properties of equality
By definition, equality is an equivalence relation, meaning it is reflexive (i.e. ), symmetric (i.e. if then ), and transitive (i.e. if and then ).Douglas Downing, Algebra the Easy Way, Publisher Barron's Educational Series, 2003, , 9780764119729, 392 pages, page 20 It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:
if and then and ; if then and ; more generally, for any function , if then .
Properties of inequality
The relations less than and greater than have the property of transitivity:Ron Larson, Robert Hostetler, Intermediate Algebra, Publisher Cengage Learning, 2008, , 9780618753529, 857 pages, page 96 If and then ; If and then ; If and then ; If and then . By reversing the inequation, and can be swapped,Chris Carter, Physics: Facts and Practice for A Level, Publisher Oxford University Press, 2001, , 9780199147687, 144 pages, page 50 for example: is equivalent to
Substitution