**Associative laws for addition
and multiplication:** If and
are arbitrary real numbers then

As a consequence of equations C.3 and C.4 we usually omit
the parentheses in triple sums or products, and write or
We know that all meaningful ways of inserting parentheses yield the same
result.

**Distributive laws:** If and
are arbitrary real numbers,
and is an arbitrary non-zero real number then

(C.5) | |||

(C.6) | |||

(C.7) | |||

(C.8) | |||

(C.9) | |||

(C.10) |

**Properties of zero and one:** The rational numbers
and , have the property that for all real numbers

Moreover

(C.17) |

(C.18) |

**Additive and multiplicative inverses:** For each real number
there is a real number (called the *additive inverse of* )
and for each non-zero real number there is a real number
(called *the multiplicative inverse of *) such that

Moreover for all real numbers and all non-zero real numbers

Note that by equation C.33, the expression without parentheses is unambiguous, i.e. no matter how parentheses are put in the result remains the same.

**Cancellation laws:** Let be real numbers. Then

(C.35) | |||

(C.36) | |||

(C.37) | |||

(C.38) |

**Some miscellaneous identities:** For all real numbers

(C.39) | |||

(C.40) | |||

(C.41) | |||

(C.42) | |||

(C.43) |

Moreover, if and then

If are real numbers, then

i.e. the terms of the sum are associated from right to left. It is in fact true that all meaningful ways of introducing parentheses into a long sum yield the same result, and we will assume this. I will often make statements like

without explanation. Equation C.48 can be proved from our assumptions, as is shown below, but we will usually take such results for granted.

Proof of equation C.48. Let be real numbers. Then