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). 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. An algebraic operation may also be defined simply as a function from a Cartesian power of a set to the same set.
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 × x^{2} is written as 3x^{2}, and 2 × x × y is written as 2xy. Sometimes, multiplication symbols are replaced with either a dot or centerdot, 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 ambiguous division sign (÷), 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 x^{2}. In plain text, the TeX markup language, and some programming languages such as MATLAB and Julia, the caret symbol, ^, represents exponents, so x^{2} is written as x ^ 2. In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so x^{2} 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 positiveornegative 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 
equivalent to:

equivalent to:


Subtraction 
equivalent to:

equivalent to:


Multiplication  or
or or 
or
or or 
is the same as 
Division  or
or

or
or


Exponentiation  

is the same as is the same as 
Note: the use of the letters and is arbitrary, and the examples would have been equally valid if and were used.
Properties of arithmetic and algebraic operations
Property  Arithmetic Example 
Algebra Example 
Comments ≡ means "equivalent to" ≢ means "not equivalent to" 

Commutativity  Addition and multiplication are commutative and associative. Subtraction and division are not: e.g.  
Associativity 
See also
 Algebraic expression
 Algebraic function
 Elementary algebra
 Factoring a quadratic expression
 Order of operations
Notes
 ^ In some countries, this symbol indicates subtraction or a wrong answer. ISO 800002 advises that it not be used. For more information, see Obelus.