Algebraic operation

Algebraic operations in the solution to the quadratic equation. The radical sign, √ denoting a square root, is equivalent to exponentiation to the power of ½. The ± sign represents the equation written with either a + and with a - sign.

In mathematics, a basic algebraic operation is any one of the traditional operations of arithmetic, which are addition, subtraction, multiplication, division, raising to an integer power, and taking roots (fractional power). 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, and, more generally on elements of algebraic structures, such as groups and fields.

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 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. 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, and it must be explicitly used, for example, 3x is written as 3 * x.

Rather than using the obelus symbol, ÷, division is usual represented with a vinculum, a horizontal line, e.g. 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, e.g. x². In plain text, and in the TeX mark-up language, the caret symbol, ^, represents exponents, so x² is written as x ^ 2. In programming languages such as Ada, Fortran, Perl, Python and Ruby, 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 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	$(5 \times 5) + 5 + 5 + 3$ equivalent to: $5^2 + (2 \times 5) + 3$	$(b \times b) + b + b + a$ equivalent to: $b^2 + 2b + a$	$\begin{align} 2 \times b & \equiv 2b\\ b + b + b & \equiv 3b\\ b \times b & \equiv b^2 \end{align}$
Subtraction	$(7 \times 7) - 7 - 5$ equivalent to: $7^2 - 7 - 5$	$(b \times b) - b - a$ equivalent to: $b^2 - b - a$	$\begin{align}b^2 - b & \not\equiv b\\ 3b - b & \equiv 2b\\ b^2 - b & \equiv b(b-1)\end{align}$
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	$3^{\frac{1}{2}}$   $2^3$	$a^{\frac{1}{2}}$   $a^3$	$a^{\frac{1}{2}}$ is the same as $\sqrt a$   $a^3$ is the same as $a \times a \times a$

Note: the use of the letters $a$ and $b$ is arbitrary, and the examples would be equally valid if we had used $x$ and $y$.

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. $(a - b) \not\equiv (b - a)$ (except when $a=b$)
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

• Elementary algebra
• Order of operations