# Multiplying Rational Numbers

The product of two rational numbers is another rational number that has:

As a numerator, the multiplication of the numerators.

As a denominator, the multiplication of the denominators.

### Properties of the Multiplication of Rational Numbers

1. Closure:

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

a · b

2. Associative:

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

(a · b) · c = a · (b · c)

3. Commutative:

The order of the factors does not change the product.

a · b = b · a

4. Multiplicative Identity:

The 1 is the neutral element in the multiplication because every number multiplied by it gives the same number.

a ·1 = a

5. Multiplicative inverse:

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

6. Distributive:

The product of a number and 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

Removing a common factor:

It is the reverse of the distributive property.

a · b + a · c = a · (b + c)