# Adding Rational Numbers

### With the Same Denominator

**Add** the numerators together and the** denominator** is maintained.

### With a Different Denominator

First, reduce the denominators to a common denominator, and add or subtract the numerators of the equivalent fractions obtained.

### Properties of the Addition of Rational Numbers

1. Closure:

The result of** adding** two** rational numbers** is another** rational number**.

a + b

2. Associative:

The way in which the summands are grouped does not change the result.

(a + b) + c = a + (b + c)

3. Commutative:

The order of the addends does not change the sum.

a + b = b + a

4. Additive identity:

The 0 is the neutral element in the addition because every number added to it gives the same number.

a + 0 = a

5. Additive inverse:

Two numbers are opposites if they are added together and the result is zero.

a + (−a) = 0

The opposite of the opposite of a number is equal to the same number.

As a result of these properties, *the subtraction*

**of two rational numbers**is defined as the addition of the minuend plus the opposite of the subtrahend.

a − b = a + (−b)