In mathematics, a basic algebraic operation is a mathematical operation similar to 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). The operations of elementary algebra 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,William Smyth, Elementary algebra: for schools and academies, Publisher Bailey and Noyes, 1864, "Algebraic Operations" and more generally, on elements of algebraic structures, such as groups and fields.Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, page 7
An algebraic operation on a set may be defined more formally as a function that maps to the tuples of a given length of elements of . The length of the tuples is called the arity of the operation, and each member of the tuple is called an operand. The most common case is the case of arity two, where the operation is called a binary operation and the operands form an ordered pair. A unary operation is an operation of arity one that has only one operand; for example, the square root. An example of a ternary operation (arity three) is the triple product.
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.Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in Mathematics Matters Secondary 1 Express Textbook, Publisher Panpac Education Pte Ltd, , 9789812738820, page 68 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,William P. Berlinghoff, Fernando Q. Gouvêa, Math through the Ages: A Gentle History for Teachers and Others, Publisher MAA, 2004, , 9780883857366, page 75 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 . 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.Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, , 9789380298153, page 212George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, , 9780817641320, page 17 In programming languages such as Ada,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, , 9783540693352, page 13 Fortran,C. Xavier, Fortran 77 And Numerical Methods, Publisher New Age International, 1994, , 9788122406702, page 20 Perl,Randal Schwartz, brian foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, , 9781449313142, page 24 PythonMatthew A. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, , 9781598631586, page 46 and Ruby,Kevin C. Baird, Ruby by Example: Concepts and Code, Publisher No Starch Press, 2007, , 9781593271480, page 72 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.
OperationArithmeticAlgebraComments Additionequivalent to:equivalent to: Subtractionequivalent 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 PropertyArithmeticAlgebraComments CommutativityAddition and multiplication arecommutative and associative.Ron Larson, Robert Hostetler, Bruce H. Edwards, Algebra And Trigonometry: A Graphing Approach, Publisher: Cengage Learning, 2007, , 9780618851959, 1114 pages, page 7 Subtraction and division are not: e.g. Associativity
See also Algebraic expression Algebraic function Elementary algebra Factoring a quadratic expression Order of operations
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