Algebraic operation

Algebraic operations in the solution to thequadratic equation.The radical sign √, denoting asquare root,is equivalent toexponentiationto the power of⁠1/2⁠.The± signmeans theequationcan be written with either a + or a – sign.

Inmathematics,a basicalgebraic operationis any one of the commonoperationsofelementary algebra,which includeaddition,subtraction,multiplication,division,raising to a whole numberpower,and takingroots(fractionalpower).These operations may be performed onnumbers,in which case they are often calledarithmetic operations.They may also be performed, in a similar way, onvariables,algebraic expressions,and more generally, on elements ofalgebraic structures,such asgroupsandfields.An algebraic operation may also be defined more generally as afunctionfrom aCartesian powerfrom a givensetto the same set.

The termalgebraic operationmay also be used for operations that may be defined by compounding basic algebraic operations, such as thedot product.Incalculusandmathematical analysis,algebraic operationis also used for the operations that may be defined by purelyalgebraic methods.For example,exponentiationwith anintegerorrationalexponent is an algebraic operation, but not the general exponentiation with arealorcomplexexponent. Also, thederivativeis an operation that is not algebraic.

Notation

Multiplication symbols are usually omitted, and implied, when there is no operator between two variables or terms, or when acoefficientis used. For example, 3 ×x²is written as 3x²,and 2 ×x×yis written as 2xy.Sometimes, multiplication symbols are replaced with either a dot or center-dot, so thatx×yis written as eitherx.yorx·y.Plain text,programming languages,andcalculatorsalso use a single asterisk to represent the multiplication symbol,and it must be explicitly used; for example, 3xis written as 3 *x.

Rather than using the ambiguousdivision sign(÷),division is usually represented with avinculum,a horizontal line, as in⁠3/x+ 1⁠.In plain text and programming languages, a slash (also called asolidus) is used, e.g. 3 / (x+ 1).

Exponents are usually formatted using superscripts, as inx².Inplain text,theTeXmark-up language, and some programming languages such asMATLABandJulia,thecaretsymbol, ^, represents exponents, sox²is written asx^ 2.In programming languages such asAda,Fortran,Perl,PythonandRuby,a double asterisk is used, sox²is written asx** 2.

Theplus–minus sign,±, is used as a shorthand notation for two expressions written as one, representing one expression with a plus sign, the other with a minus sign. For example,y=x± 1 represents the two equationsy=x+ 1 andy=x− 1. Sometimes, it is used for denoting a positive-or-negative term such as ±x.

Arithmetic vs algebraic operations

Algebraic operations work in the same way asarithmetic operations,as can be seen in the table below.

Operation	Arithmetic Example	Algebra Example	Comments ≡ means "equivalent to" ≢ means "not equivalent to"
Addition	$(5\times 5)+5+5+3$ equivalent to: $5^{2}+(2\times 5)+3$	$(b\times b)+b+b+a$ equivalent to: $b^{2}+2b+a$	${\begin{aligned}2\times b&\equiv 2b\\b+b+b&\equiv 3b\\b\times b&\equiv b^{2}\end{aligned}}$
Subtraction	$(7\times 7)-7-5$ equivalent to: $7^{2}-7-5$	$(b\times b)-b-a$ equivalent to: $b^{2}-b-a$	${\begin{aligned}b^{2}-b&\not \equiv b\\3b-b&\equiv 2b\\b^{2}-b&\equiv b(b-1)\end{aligned}}$
Multiplication	$3\times 5$or $3\.\ 5$or$3\cdot 5$ or$(3)(5)$	$a\times b$or $a.b$or$a\cdot b$ or$ab$	$a\times a\times a$is the same as$a^{3}$
Division	$12\div 4$or $12/4$or ${\frac {12}{4}}$	$b\div a$or $b/a$or ${\frac {b}{a}}$	${\frac {a+b}{3}}\equiv {\tfrac {1}{3}}\times (a+b)$
Exponentiation	$3^{\frac {1}{2}}$ $2^{3}$	$a^{\frac {1}{2}}$ $b^{3}$	$a^{\frac {1}{2}}$is the same as${\sqrt {a}}$ $b^{3}$is the same as$b\times b\times b$

Note: the use of the letters$a$and$b$is arbitrary, and the examples would have been equally valid if$x$and$y$were used.

Properties of arithmetic and algebraic operations

Property	Arithmetic Example	Algebra Example	Comments ≡ means "equivalent to" ≢ means "not equivalent to"
Commutativity	$3+5=5+3$ $3\times 5=5\times 3$	$a+b=b+a$ $a\times b=b\times a$	Addition and multiplication are commutative and associative. Subtraction and division are not: e.g.$a-b\not \equiv b-a$
Associativity	$(3+5)+7=3+(5+7)$ $(3\times 5)\times 7=3\times (5\times 7)$	$(a+b)+c=a+(b+c)$ $(a\times b)\times c=a\times (b\times c)$

See also

• Algebraic expression
• Algebraic function
• Elementary algebra
• Factoring a quadratic expression
• Order of operations