Mixed model

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, an algebraic operation is any one of the operations addition, subtraction, multiplication, division, raising to an integer power, and taking roots (fractional power). Algebraic operations are performed on an algebraic variable, term or expression, and work in the same way as arithmetic operations.

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. Electrical Engineer Stephan from Merrickville-Wolford, likes to spend time body building, property developers ec in singapore singapore and storytelling. Gets inspiration through travel and just spent 6 days at Würzburg Residence with the Court Gardens.. 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 Template:Nobold	Algebra Template:Nobold	Comments Template:Nobold
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	$3^{\frac {1}{2}}$   $2^{3}$	$a^{\frac {1}{2}}$   $a^{3}$	$a^{\frac {1}{2}}$ 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 Template:Nobold	Algebra Template:Nobold	Comments Template:Nobold
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	$(3+5)+7=3+(5+7)$ $(3\times 5)\times 7=3\times (5\times 7)$	$(a+b)+c=a+(b+c)$ $(a\times b)\times c=a\times (b\times c)$

References

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See also

• Elementary algebra
• Order of operations