词条 | Algebraic operation | |||||||||||||||||||||||||||||||||||
释义 |
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. NotationMultiplication 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. {{sfrac|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 a positive-or-negative term such as ±x. Arithmetic vs algebraic operationsAlgebraic operations work in the same way as arithmetic operations, as can be seen in the table below.
Note: the use of the letters and is arbitrary, and the examples would be equally valid if we had used and . Properties of arithmetic and algebraic operations
References1. ^William Smyth, Elementary algebra: for schools and academies, Publisher Bailey and Noyes, 1864, "[https://books.google.com/books?id=BqQZAAAAYAAJ&lpg=PA55&ots=ex07zH_ljg&dq=%22Algebraic%20operations%22&pg=PA55#v=onepage&q=%22Algebraic%20operations%22&f=false Algebraic Operations]" 2. ^Horatio Nelson Robinson, New elementary algebra: containing the rudiments of science for schools and academies, Ivison, Phinney, Blakeman, & Co., 1866, [https://books.google.com/books?id=dKZXAAAAYAAJ&dq=Elementary%20algebra%20notation&pg=PA7#v=onepage&q=Elementary%20algebra%20notation&f=false 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, [https://books.google.com/books?id=nL5ObMmDvPEC&lpg=PR9-IA8&ots=T_h6l40AE5&dq=%22Algebraic%20notation%22%20multiplication%20omitted&pg=PR9-IA8#v=onepage&q=%22Algebraic%20notation%22%20multiplication%20omitted&f=false 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, [https://books.google.com/books?id=JAXNVaPt7uQC&lpg=PA75&ots=-P78Lrz792&dq=calculator%20asterisk%20multiplication&pg=PA75#v=onepage&q=calculator%20asterisk%20multiplication&f=false page 75] 5. ^Ramesh Bangia, Dictionary of Information Technology, Publisher Laxmi Publications, Ltd., 2010, {{ISBN|9380298153}}, 9789380298153, [https://books.google.com/books?id=zQa5I2sHPKEC&lpg=PA212&ots=s6pWav1Z_D&dq=%22plain%20text%22%20math%20caret%20exponent&pg=PA212#v=onepage&q=exponentiation%20caret&f=false page 212] 6. ^George Grätzer, First Steps in LaTeX, Publisher Springer, 1999, {{ISBN|0817641327}}, 9780817641320, [https://books.google.com/books?id=mLdg5ZdDKToC&lpg=PP1&ots=V9DFIaAAh0&dq=tex%20math&pg=PA17#v=onepage&q=subscripts%20and%20superscripts%20caret&f=false 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, [https://books.google.com/books?id=694P3YtXh-0C&lpg=PA718&ots=O_EgQ75FeB&dq=ada%20%20asterisk&pg=PA12#v=onepage&q=double%20star%20exponentiate&f=false page 13] 8. ^C. Xavier, Fortran 77 And Numerical Methods, Publisher New Age International, 1994, {{ISBN|812240670X}}, 9788122406702, [https://books.google.com/books?id=WYMgF9WFty0C&lpg=PA20&ots=BTtzs9F-NB&dq=fortran%20asterisk%20exponentiation&pg=PA20#v=onepage&q=fortran%20asterisk%20exponentiation&f=false page 20] 9. ^Randal Schwartz, brian foy, Tom Phoenix, Learning Perl, Publisher O'Reilly Media, Inc., 2011, {{ISBN|1449313140}}, 9781449313142, [https://books.google.com/books?id=l2IwEuRjeNwC&lpg=PA24&ots=5nsYOLHxlD&dq=perl%20asterisk%20exponentiation&pg=PA24#v=onepage&q=double%20asterisk%20exponentiation&f=false page 24] 10. ^Matthew A. Telles, Python Power!: The Comprehensive Guide, Publisher Course Technology PTR, 2008, {{ISBN|1598631586}}, 9781598631586, [https://books.google.com/books?id=754knV_fyf8C&lpg=PA46&ots=8fEi1F-H8-&dq=python%20asterisk%20exponentiation&pg=PA46#v=onepage&q=double%20asterisk%20exponentiation&f=false page 46] 11. ^Kevin C. Baird, Ruby by Example: Concepts and Code, Publisher No Starch Press, 2007, {{ISBN|1593271484}}, 9781593271480, [https://books.google.com/books?id=kq2dBNdAl3IC&lpg=PA72&ots=0UU3k-Pvh8&dq=ruby%20asterisk%20exponentiation&pg=PA72#v=onepage&q=double%20asterisk%20exponentiation&f=false 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, [https://books.google.com/books?id=5iXVZHhkjAgC&lpg=PA6&ots=iwrSrCrrOb&dq=operations%20addition%2C%20subtraction%2C%20multiplication%2C%20division%20exponentiation.&pg=PA7#v=onepage&q=associative%20property&f=false page 7] See also
2 : Elementary algebra|Elementary mathematics |
|||||||||||||||||||||||||||||||||||
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。