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! That rational number is 1/4.

0           1/4            1/2

If you do not 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

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 1/4 and 1/2, just add 1/4 and 1/2 and then divide by 2.

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, visit mathonline.wikidot.com




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