# Real Numbers

The set formed by rational and irrational numbers is the set of real numbers.

The real number is denoted by .

With real numbers** all operations can be performed, except for the root of an even index and negative radicand and division by zero. **

## Adding Real Numbers

### Properties

1. **Closure**:

**A + b **

2. **Associative** :

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

3. **Commutative** :

**a + b = b + a **

4. **Additive Identity:**

**a + 0 = a**

5. **Additive Inverse:**

**a + (−a)= 0**

−(−a) = a

#### Subtracting Real Numbers

The difference of two real numbers is defined as
** the sum of the minuend plus the opposite of the subtrahend**.

**a − b = a + (−b)**

## Multipying Real Numbers

The rule of signs for the product of integers and rational numbers is still maintained with the real numbers.

### Properties

1. **Closure**:

**a · b **

2. **Associative: **

**(a · b) · c = a · (b · c)**

3. **Commutative: **

**a · b = b · a **

4. **Multiplicative Identity**:

**a · 1 = a **

5. **Multiplicative Inverse**:

6. **Distributive**:

The product of a number for a sum is equal to the sum of the products of this number for each of the addends.

**a · (b + c) = a · b + a · c**

#### Dividing Real Numbers

The division of two real numbers is defined as the product of the dividend by the reciprocal of the divisor.