Algebraic operation

In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power).^[1] These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions,^[2] and more generally, on elements of algebraic structures, such as groups and fields.^[3] An algebraic operation may also be defined more generally as a function from a Cartesian power of a given set to the same set.^[4]

Algebraic operations in the solution to the quadratic equation. The radical sign √, denoting a square root, is equivalent to exponentiation to the power of ⁠1/2⁠. The ± sign means the equation can be written with either a + or a – sign.

The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation on numerical functions and algebraic expressions that is not algebraic.

Notation

Multiplication symbols are usually omitted, and implied, when there is no operator between two variables or terms, or when a coefficient is used. For example, 3 × x² is written as 3x², and 2 × x × y is written as 2xy.^[5] Sometimes, multiplication symbols are replaced with either a dot or center-dot, so that x × y is written as either x . y or x · y. Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol,^[6] and it must be explicitly used; for example, 3x is written as 3 * x.

Rather than using the ambiguous division sign (÷),^[a] division is usually represented with a vinculum, a horizontal line, as in ⁠3/x + 1⁠. In plain text and programming languages, a slash (also called a solidus) is used, e.g. 3 / (x + 1).

Exponents are usually formatted using superscripts, as in x². In plain text, the TeX mark-up language, and some programming languages such as MATLAB and Julia, the caret symbol, ^, represents exponents, so x² is written as x ^ 2.^[8][9] In programming languages such as Ada,^[10] Fortran,^[11] Perl,^[12] Python^[13] and Ruby,^[14] a double asterisk is used, so x² is written as x ** 2.

The plus–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 equations y = x + 1 and y = 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 as arithmetic operations, as can be seen in the table below.

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

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

Properties of arithmetic and algebraic operations

Property	Arithmetic Example	Algebra Example	Comments ≡ means "equivalent to" ≢ means "not equivalent to"
Commutativity	${\displaystyle 3+5=5+3}$ ${\displaystyle 3\times 5=5\times 3}$	${\displaystyle a+b=b+a}$ ${\displaystyle a\times b=b\times a}$	Addition and multiplication are commutative and associative.[15] Subtraction and division are not: e.g. ${\displaystyle a-b\not \equiv b-a}$
Associativity	${\displaystyle (3+5)+7=3+(5+7)}$ ${\displaystyle (3\times 5)\times 7=3\times (5\times 7)}$	${\displaystyle (a+b)+c=a+(b+c)}$ ${\displaystyle (a\times b)\times c=a\times (b\times c)}$

See also

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

Notes

1. In some countries, this symbol indicates subtraction or a wrong answer. ISO 80000-2 advises that it not be used.^[7] For more information, see Obelus.