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# Products

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In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.

Multiplication is one of the four elementary mathematical operations of arithmetic, with the others being addition, subtraction and division.

Mathematics is the study of topics such as quantity, structure, space, and change.

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Thus, for instance, 30 is the product of 6 and 5, and

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{\displaystyle x\cdot }

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The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication.

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When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors.

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field.

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Matrix multiplication, for example, and multiplication in other algebras is in general non-commutative.

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There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structures.

In mathematics, and more specifically in abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms.

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Placing several stones into a rectangular pattern with

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{\displaystyle r\cdot s=\sum _{i=1}^{s}r=\underbrace {r+r+\cdots +r} _{s{\text{ times}}}=\sum _{j=1}^{r}s=\underbrace {s+s+\cdots +s} _{r{\text{ times}}}}

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Another approach to multiplication that applies also to real numbers is continuously stretching the number line from 0, so that the 1 is stretched to the one factor, and looking up the product, where the other factor is stretched to.

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Integers allow positive and negative numbers.

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Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule, which is a necessary consequence of demanding distributivity of the multiplication over addition, but is no additional rule.

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{\displaystyle {\begin{array}{|c|c c|}\hline \cdot &-&+\\\hline -&+&-\\+&-&+\\\hline \end{array}}}

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Minus times Minus gives Plus

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Minus times Plus gives Minus

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Plus times Minus gives Minus

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Plus times Plus gives Plus

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Two fractions can be multiplied by multiplying their numerators and denominators:

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{\displaystyle {\frac {z}{n}}\cdot {\frac {z'}{n'}}={\frac {z\cdot z'}{n\cdot n'}}}

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For a rigorous definition of the product of two real numbers see Construction of the real numbers.

In mathematics, there are several ways of defining the real number system as an ordered field.