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,^[1] and, more generally on elements of algebraic structures, such as groups and fields.^[2]

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.^[3] 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,^[4] 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.^[5][6] In programming languages such as Ada,^[7] Fortran,^[8] Perl,^[9] Python^[10] and Ruby,^[11] 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[12] 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

1. William Smyth, Elementary algebra: for schools and academies, Publisher Bailey and Noyes, 1864, "Algebraic Operations"
2. Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, page 7
3. Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in Mathematics Matters Secondary 1 Express Textbook, Publisher Panpac Education Pte Ltd, ISBN 9812738827, 9789812738820, page 68
4. William P. Berlinghoff, Fernando Q. Gouvêa, Math through the Ages: A Gentle History for Teachers and Others, Publisher MAA, 2004, ISBN 0883857367, 9780883857366, page 75
5. Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, ISBN 9380298153, 9789380298153, page 212
6. George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, ISBN 0817641327, 9780817641320, page 17
7. S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, Ada 2005 Reference Manual, Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, ISBN 3540693351, 9783540693352, page 13
8. C. Xavier, Fortran 77 And Numerical Methods, Publisher New Age International, 1994, ISBN 812240670X, 9788122406702, page 20
9. Randal Schwartz, brian foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, ISBN 1449313140, 9781449313142, page 24
10. Matthew A. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, ISBN 1598631586, 9781598631586, page 46
11. Kevin C. Baird, Ruby by Example: Concepts and Code, Publisher No Starch Press, 2007, ISBN 1593271484, 9781593271480, page 72
12. Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: Cengage Learning, 2007, ISBN 061885195X, 9780618851959, 1114 pages, page 7

See also

• Elementary algebra
• Order of operations