# associativity of multiplication

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

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

of multiplication^{} in arithmetics^{} and elementary algebra:

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

This rule is sometimes violated even in high school 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 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

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

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

Title | associativity of multiplication |
---|---|

Canonical name | AssociativityOfMultiplication |

Date of creation | 2013-03-22 15:09:22 |

Last modified on | 2013-03-22 15:09:22 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Application |

Classification | msc 12D99 |

Classification | msc 00A35 |