# Properties of the Real Numbers

**Additive**

**Multiplicative**

**Distributive**

These properties of the real numbers form the basis of algebra. There are five properties each for addition and multiplication: **closure**, **associativity**, **commutativity**, **identity**, and **inverses**. Furthermore, there is one property that combines addition and multiplication: **distributivity**.

### Explanations

**Closure**

The closure property tells us that the real numbers are closed under addition and multiplication. That is, if we add or multiply two real numbers, the result is a real number. For an example of a set that is not closed, consider this set of integers ** A = {1, 2, 3}**. Now if we add

**and**

*3***, which are in**

*2***, we get the number**

*A***, which is not in**

*5***. So,**

*A***is not closed under addition.**

*A***Associative**

The associative property tells us that when we add (or multiply) three numbers, it does not matter whether we perform the first or second operation first. So, if we have 3 + 4 + 9, we can sum it as (3 + 4) + 9 = 7 + 9 = 16 or 3 + (4 + 9) = 3 + 13 = 16 and the answer is the same either way.

**Commutative**

The commutative property tells us that when we add (or multiply) two numbers, it does not which number comes first. So, if we add 3 and 4, we can write the sum as 3 + 4 = 7 or 4 + 3 = 7 and the answer is the same either way.

**Identity**

The identity property tells us that there are real numbers 0 and 1 such that if we add or multiply them respectively, with any other real number, it does not change the value. So, 3 + 0 = 3 and 3*1 = 3. Note that, by commutativity, this is true in the other direction too: 0 + 3 = 3 and 1*3 = 3.

**Inverse**

The inverse property tells us that for any nonzero real number ** x**, there exist real numbers

**and**

*-x***such that**

*1/x***and**

*x + (-x) = 0***. For the real number**

*x(1/x) = 1***, it is its own additive inverse, but it has no multiplicative inverse. By commutivity, the sums and products are true in both directions:**

*0***and**

*x + (-x) = (-x) + x = 0***.**

*x(1/x) = (1/x)x = 1*Many sets do not contain inverses. The natural numbers do not contain additive inverses, for example. The set of integers does not have multiplicative inverses.

**Distributive**

The distributive property defines how multiplication and addition interact. Namely, for real numbers ** x**,

**, and**

*y***,**

*z***; this is called the left-distributive property. By commutativity and a few more steps, we also have**

*x(y + z) = xy + xz***; this is called the right-distributive property.**

*(x + y)z = xz + yz*To see that the right-distributive property follows from the left-distributive property, we write ** (x + y)z = z(x + y) = zx + zy = xz + yz**, where the first and third equalities follow from commutativity and the second follows from the left-distributive property.

To see a numerical example of the distributive property, note that 3(4 + 2) = 3*6 = 18. On the other hand, 3*4 + 3*2 = 12 + 6 = 18. So, we see that the answer is 18 either way.