Rational Numbers between Two Rational Numbers

We will learn to insert rational numbers between two rational numbers. Let us recall integers and properties of various operations on them. We know between two non-consecutive integers x and y there are (x - y - 1) integers. However, there is no integer between two consecutive integers.

For example, between -7 and 7 there are 7 - (-7) - 1 = 7 + 7 - 1 = 14 – 1 = 13 integers. The integers are -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 and 6 but there is no integer between 2 and 3 since they are consecutive integers.

Thus, we find that between two given integers there may or may not lie any integer.

How to insert many rational numbers between two rational numbers?

We can insert infinitely many rational numbers between any two rational numbers. This property of rational numbers is known as the dense property.

How to find out some rational numbers lying between two given rational numbers, say between -4/7 and 2/7. The four rational numbers -3/7, -2/7, -1/7, 0/7 and 1/7 lies between -4/7 and 2/7.

We can apply the same procedure to insert more rational numbers between -4/7 and 2/7.

The rational numbers -4/7 and 2/7 can also be written as -40/70 and 20/70 respectively.

Clearly, -39/70, -38/70, -37/70, -36/70, -35/70, …….., 0/70, 1/70, 2/70, 3/70, 4/70, …….., 18/70, 19/70 are rational numbers between -4/7 and 2/7.

The total number of these rational numbers is same as the number of integers between -40 and 70, i.e., 70 - (-40) - 1 = 70 + 40 - 1 = 110 - 1 = 109.

Similarly, by re-writing -4/7 and 2/7 as -400/700 and 200/700, we can insert 700 - (-400) - 1 = 700 + 400 - 1 = 1100 - 1 = 1099 rational numbers between -4/7 and 2/7.

Therefore, we can apply the same procedure to insert as many rational numbers between -4/7 and 2/7.

Solved examples on rational numbers between two rational numbers:

Find out 100 rational numbers lying between -9/19 and 5/19.

Solution:

We have,

-9/19 = -9 × 10/19 × 10 = -90/190 and,

5/19 = 5 × 10/19 × 10 = 50/190

We know that

-90 < -89 < -88 < -87 < -86 < -85 < …….. < -25 < -24 < -23 < -22 < …….. < -1 < 0 < 1 < 2 < …….. < 9 < 10

⇒ -90/190 < -89/190 < -88/190 < -87/190 < -86/190 < -85/190 < …….. < -25/190 < -24/190 < -23/190 < -22/190 < …….. < -1/190 < 0/190 < 1/190 < 2/190 < …….. < 9/190 < 10/190

Hence, < -89/190 < -88/190 < -87/190 < -86/190 < -85/190 < …….. < -25/190 < -24/190 < -23/190 < -22/190 < …….. < -1/190 < 0/190 < 1/190 < 2/190 < …….. < 9/190 < 10/190 are the 100 rational numbers between -9/19 = -90/190 and 5/19 = 50/190.

● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms