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 3 and 2, which are in A, we get the number 5, which is not in A. So, A is not closed under addition.

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 -x and 1/x such that x + (-x) = 0 and x(1/x) = 1. For the real number 0, it is its own additive inverse, but it has no multiplicative inverse. By commutivity, the sums and products are true in both directions: x + (-x) = (-x) + x = 0 and 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, y, and z, x(y + z) = xy + xz; this is called the left-distributive property. By commutativity and a few more steps, we also have (x + y)z = xz + yz; this is called the right-distributive property.

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.