**Section 2: Real Number Axioms and Properties of Zero**

To do any mathematics with the real numbers, we need a set of ground rules. Before we list the ground rules, however, we need to establish a few definitions:

**Definition 6** *A variable is a symbol (usually a letter of the English or Greek alphabet) that represents any number within a set.*

Finally we get to the ground rules. These are called **axioms**. That means that we accept them at an elemental or definitional level. Then we add some rules of logic and we can prove other properties called **theorems**

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**Axioms for Real Numbers
**1. The Commutative Property of Addition and Multiplication - For any numbers

2. The Associative Property of Addition and Multiplication - For any numbers

3. The Distributive Property of Multiplication Over Addition - For any numbers

4. The Multiplicative Property of One - For any number *a*, *a*(1)* = a*.

5. The Property of Multiplicative Inverses (Reciprocals) - For each nonzero number *a*, there is one and only one reciprocal 1/*a*, for which *a*(1/*a*)* = *1

6. **Properties of Zero **- zero is a somewhat odd number with some special rules.

6.1 The Additive Inverse Property - For each number

a, there is one and only one additive inverse,-a, for whicha+ (-a) = 0.

6.2 The Additive Property of Zero - For and numbera,a+ 0 =a.

6.3 0 xa= 0 (anything times zero is zero).

6.4 0/a= 0 ÷a= 0 if a ≠ 0

6.5a/0 =a÷ 0 = is not defined.

Although you may not have known all the proper words, you probably already knew the first three zero properties. However, some people get confused when zero is involved in a division problem. The last two rules address this. They simply state that when anything is divided into zero, the result is zero. However, when zero is divided into anything, the result is undefined.