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# associativity of multiplication

It’s important to know the following interpretation of the associative law

$\displaystyle a\cdot(b\cdot c)=(a\cdot b)\cdot c$ | (1) |

of multiplication in arithmetics and elementary algebra:

A product ($b\cdot c$) is multiplied by a number ($a$) so that only one factor ($b$) of the product is multiplied by that number.

This rule is sometimes violated even in high school level. A pupil may calculate e.g. like

$10\cdot 2.5\cdot 0.3=25\cdot 3=75,$ |

which is wrong. Or when solving an equation like

$x\cdot\frac{2x-1}{3}=1$ |

one would like to multiply both sides by 3 for removing the denominator, getting perhaps

$3x(2x-1)=3;$ |

then the both factors of left side have incorrectly been multiplied by 3.

The reason of such mistakes is very likely that one confuses the associative law with the distributive law; cf. (1) with this latter

$\displaystyle a\cdot(b+c)=a\cdot b+a\cdot c,$ | (2) |

which contains two different operations, multiplication and addition; both addends must be multiplied separately.

## Mathematics Subject Classification

12D99*no label found*00A35

*no label found*

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