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Calculators generally perform operations with the same precedence from left to right, [1] but some programming languages and calculators adopt different conventions. For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.
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, [6] and it must be explicitly used; for example, 3x is written as 3 * x.
Algebraic fractions are subject to the same laws as arithmetic fractions. A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials . Thus 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}} is a rational fraction, but not x + 2 x 2 − 3 , {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}},} because ...
The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical Papyrus, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42 , 4 × 21 = 2 × 42 = 84 , 8 × 21 = 2 × 84 = 168 .
An algebraic fraction is the indicated quotient of two algebraic expressions. As with fractions of integers, the denominator of an algebraic fraction cannot be zero. Two examples of algebraic fractions are + and + . Algebraic fractions are subject to the same field properties as arithmetic fractions.
All the above multiplication algorithms can also be expanded to multiply polynomials. Alternatively the Kronecker substitution technique may be used to convert the problem of multiplying polynomials into a single binary multiplication. [31] Long multiplication methods can be generalised to allow the multiplication of algebraic formulae:
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.