Density property

The density property states that between two rational numbers, there is another rational number. For example, is there a rational number between 0 and 1/2 ? Yes, there is a rational number between 0 and 1/2 and that rational number is 1/4. 

0           1/4            1/2

Are there rational numbers between 0 and 1/4 and between 1/4 and 1/2 ? Yes, there are. The rational number between 0 and 1/4 is 1/8 and the rational number between 1/4 and 1/2 is 3/8. Keep reading in order to see how you can find the rational numbers between 0 and 1/4 and between 1/4 and 1/2.

In the figure below, we illustrate the density property with a number line. 

How do we find the number between two rational numbers? Just take the average of the two rational numbers.

For example, to find the rational number between 0 and 1/4, just add 0 and 1/4 and then divide the sum by 2 as demonstrated below.

0 + 1/4 = 1/4 since anything added to zero gives you the same thing.

Now divide 1/4 by 2

1/4 ÷ 2 = 1/4 ÷ 2/1

1/4 ÷ 2 = 1/4 × 1/2

1/4 ÷ 2 = (1 × 1)/(4 × 2)

1/4 ÷ 2 = 1/8

By the same token, to find the rational number between 1/4 and 1/2, just add 1/4 and 1/2 and then divide the sum by 2 as demonstrated below.

1/4 + 1/2 = 1/4 + (1 × 2)/( 2 × 2)

1/4 + 1/2 = 1/4 + 2/4

1/4 + 1/2 = (1 + 2 )/4

1/4 + 1/2 = 3/4

Now divide 3/4 by 2

3/4 ÷ 2 = 3/4 ÷ 2/1

3/4 ÷ 2 = 3/4 × 1/2

3/4 ÷ 2 = (3 × 1)/(4 × 2)

3/4 ÷ 2 = 3/8

Here is a generic way to express the density property

If a/b < c/d, then there exists a rational number g/h such that a/b < g/h < c/d

If you want to learn more about the density property, feel free to visit mathonline.wikidot.com