Consider the integers 4 and 10. We can locate five integers namely 5,6,7,8 and 9 (shown in dark dots) between them. Isn’t it?

**Rational
numbers between any two given rational numbers**

Consider
the integers 4 and 10. We can locate five integers namely 5,6,7,8 and 9 (shown in
dark dots) between them. Isn’t it?

How many
integers can you find between 3 and –2? List them.

Are there
any integers between –5 and –4? No, is the answer.

This shows that the choice of integers between two given integers
is limited. They are finite in number or may be nothing between them. Let us think
what will happen, if we consider rational numbers instead of integers? We will see
that we can have many rational numbers between any two rational numbers. There are
at least two methods to find more rational numbers between any two rational numbers.

** **

**Method
of Average**

We know that
that the average of any two numbers always lies at the middle of them.

For example,
the average of 2 and 8 is 2 +8 / 2 = 5 and this 5 lies at
the middle of 2 and 8 as shown in the following number line.

We use this
idea to find more rational numbers between any two rational numbers.

**Example 1.8**

Find a rational
number between 1/3 and 5/9.

*Solution:*

Note that
4/9 is one rational number we have found in between 1/3 and 5/9 and we can find
many such numbers in between 1/3 and 5/9.This shows that between any two rational
numbers there lie an unlimited number of rational numbers! Mathematically, we say
that there lie an infinite number of rational numbers between any two rational numbers.

**Think**

Are there any rational numbers between −7/11 and 6/−11 ?

**Solution: **

−7/11 = −70/110 ; 6/−11 = −60/110

−61/110, −62/110 …. −69/110

∴ Thre are many rational numbers between −7/11 and 6/−11

** **

**Method
of Equivalent rational numbers:**

We can use
the idea of equivalent fractions to get more rational numbers between any two rational
numbers. This is clearly explained in the following illustration.

**Illustration:**

Let us now
try to find more rational numbers say between 3/7 and 4/7 by the following visual
explanation on the number line. If we get the multiples of the denominator of the
equivalent rational numbers (the easy one will be to multiply by 10), then we can
insert as many
rational numbers as we want. We shall write 3/7 as 30/70 and 4/7 as 40/70 and see
that there are 9 rational numbers between 3/7 and 4/7 as given in the number line
below.

Now, if we
want more rational numbers between say 37/70 and 38/70 we can write 37/70 as 370/700 and 38/70 as 380/700.
Then again, we will get nine rational numbers between 37//70 and 38//70 as 371/700_{,}
372/700_{,} 373/700_{,} 374/700_{,} 375/700 _{,}
376/700_{,} 377/700 _{,} 378/700 and 379/700

The following
diagram helps us to understand this nicely with a magnifying lens used between 0
and 1 and further zoomed into the fractional parts also

Thus, we
can see that there are an unlimited number of rational numbers between any two given
rational numbers.

**Example 1.9**

Find atleast
two rational numbers between -3/4 and -2/5.

**Solution:**

The denominators
are different for the given rational numbers. The LCM of the denominators 4 and
5 is 20. Make the rational numbers such that they have common denominators as 20.
Here,

It is easy
now to find and insert rational numbers between -15/20 and -8/20 and as shown below.

We can list a few rational numbers as between

Are these
the only rational numbers between -15/20 and -8/20? Think! Try to find 10 more rational
numbers between them, if possible!

** **

**Note:**

We can find many rational numbers between – 7/11 and 5/-9 quickly
as given below:

The range of rational numbers can be got by the cross multiplication
of denominators with the numerators after writing the given fractions in standard
form. The cross multiplication here gives the range of rational numbers
from –63 to –55 with the denominator 99. This is nothing but making
the given rational numbers equivalent with the denominator 99!.

Tags : Numbers | Chapter 1 | 8th Maths , 8th Maths : Chapter 1 : Numbers

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