# factors with minus sign

The sign (cf. plus sign, opposite number) rule

$(+a)(-b)=-(ab),$ | (1) |

derived in the parent entry (http://planetmath.org/productofnegativenumbers) and concerning numbers and elements $a$, $b$ of an arbitrary ring, may be generalised to the following

Theorem. If the sign of one factor (http://planetmath.org/Product) in a ring product is changed, the sign of the product changes.

Corollary 1. The product of real numbers is equal to the product of their absolute values^{} equipped with the “$-$” sign if the number of negative factors is odd and with “$+$” sign if it is even. Especially, any odd power of a negative real number is negative and any even power of it is positive.

Corollary 2. Let us consider natural powers of a ring element. If one changes the sign of the base, then an odd power changes its sign but an even power remains unchanged:

$${(-a)}^{2n+1}=-{a}^{2n+1},{(-a)}^{2n}={a}^{2n}\mathit{\hspace{1em}\hspace{1em}}(n\in \mathbb{N})$$ |

Title | factors with minus sign |
---|---|

Canonical name | FactorsWithMinusSign |

Date of creation | 2015-02-04 12:30:11 |

Last modified on | 2015-02-04 12:30:11 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 97D40 |

Classification | msc 13A99 |

Synonym | sign rules for products |

Related topic | GeneralAssociativity |

Related topic | Multiplication |

Related topic | DoublyEvenNumber |