# 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

a + 0 = a

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.