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.
Definition 7 A constant is a number or symbol with a fixed value.
Definition 8 An expression is any group of variables, constants, and numerals combined with arithmetic operations such as addition, subtraction, multiplication, division, or exponentiation.

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 a and b, a+b = b+a and ab = ba.
2. The Associative Property of Addition and Multiplication - For any numbers a, b, and c, a + (b +c) = (a +b) +c and a(bc) = (ab)c.
3. The Distributive Property of Multiplication Over Addition - For any numbers a, b, and c, a(b +c) = ab +ac.
Note: The distributive property will be addressed in more detail in when we discuss the Order of Operations.

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 which a + (-a) = 0.
6.2 The Additive Property of Zero - For and number a, a + 0 = a.
6.3 0 x a = 0 (anything times zero is zero).
6.4 0/a = 0 ÷ a = 0 if a ≠ 0
6.5 a/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.