# Operations with Real Numbers

## Adding Real Numbers

### Properties

1. **Closure**:

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

**A + b **

+

2. **Associative** :

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

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

** + 0** =

5. **Additive inverse:**

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

**a + (−a)= 0**

**e − e = 0 **

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

−(−) =

#### 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**:

The result of multiplying two real numbers is another real number.

**a · b **

2. **Associative: **

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

**(a · b) · c = a · (b · c)**

**(e · )** · = **e · ( · ) **

3. **Commutative: **

The order of factors does not change the product.

**a · b = b · a **

4. **Multiplicative Identity**:

The **1** is the neutral element of the multiplication because any number multiplied by it gives the same number.

**a · 1 = a **

** · 1 = **

5. **Multiplicative inverse**:

A number is the reciprocal of another if when multiplied by each other, the product is the muliplicative identity.

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**

· **(e + ) **= · **e +** ·

** Removing a common factor:**

It is the reverse of the distributive property.

**a · b + a · c = a · (b + c)**

· **e +** · = · **(e + ) **

#### Dividing Real Numbers

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