There are a set of rules which can be applied to parts of an equation which will not affect the truth of the equation.

rule | name | description |
---|---|---|

x + y = y + x | commutative - additive | An operation is commutative if the order of its operands can be reversed without effecting the result |

x * y = y * x | commutative - multiplicative | |

x + (y + z) = (x + y) + z | associative - additive | An operation is associative if the order of doing multiple operations is not important |

x * (y * z) = (x * y) * z | associative - multiplicative | |

x * (y + z) = (x * y) + (x * z) | distributive | When there are two operations in this algebra say, + and *, then * is said to be distributive over + |

x + 0 = x | identity operator - additive (right) | adding 0 does not change result |

0 + x = x | identity operator - additive (left) | |

x * 1 = x | identity operator - multiplicative (right) | multiplying by 1 does not change result |

1 * x = x | identity operator - multiplicative (left) | |

if z = x + y then x = z - y |
subtracting is the inverse of adding | subtract y from both sides of the equation (this is like adding -y) |

if z = x * y then x = z / y |
dividing is the inverse of multiplying | divide both both sides of the equation by y (this is like multiplying by 1/y) |

where: x,y and z can be numbers, variables, any expression enclosed in brackets, or any expression that could be enclosed in brackets without changing the value of the equation.

These rules all apply to real number algebra, some of these rules don't apply in some other algebras. For example in matrix and quaternion algera the commutative (multiplicative) rule does not apply.

## Algebras

* distributive over + | Multiplication commutative | Multiplication associative | |

Real Numbers | yes | yes | yes |

Complex Numbers | yes | yes | yes |

Quaternions | yes | no | yes |

Octonions | yes | no | no |

Vectors | yes | no | no |

Matricies | yes | no | yes |

Multi Vectors | yes | no | no |

For further information see group theory.