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

If you do not quite understand why 0 is a rational number check the lesson about rational numbers.

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. 

Density property

As you can see in the figure above, no matter how densely packed the number line is, you can always find more rational numbers to put in between other rationals numbers.

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

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