Question

Find the 12 rational numbers between \[ - 1\] and 2?

Answer 1

Hint:
Here we have to find the 12 rational numbers between \[ - 1\] and 2. Rational number is defined as a number that can be expressed in the form \[\dfrac{p}{q}\], where \[q\] cannot be zero. For that, we will find the average between the given numbers, which will be the rational numbers between the given numbers.

Complete step by step solution:
Rational numbers include every integer, fraction, decimal. Now we will find all the required rational numbers between \[ - 1\] and 2.
First rational number between \[ - 1\] and 2 can be calculated by finding average between them, which is
1) First rational number \[ = \dfrac{{ - 1 + 2}}{2} = \dfrac{1}{2}\]
Now, we have three numbers i.e. \[ - 1\], \[\dfrac{1}{2}\] and 2. So remaining rational numbers can be calculated by taking average between \[ - 1\] and \[\dfrac{1}{2}\] , and between \[\dfrac{1}{2}\] and 2.

2) Second rational number between \[ - 1\] and \[\dfrac{1}{2}\] can be calculated by finding the average between them.
Second rational number \[ = \dfrac{{ - 1 + \dfrac{1}{2}}}{2} = - \dfrac{1}{4}\]
 The third rational number between 2 and \[\dfrac{1}{2}\] can be calculated by finding the average between them.
3) Third rational number \[ = \dfrac{{2 + \dfrac{1}{2}}}{2} = \dfrac{5}{4}\]
4) Similarly the fourth rational number between -1 and \[\dfrac{5}{4}\] can be calculated by finding the average between them.
Fourth rational number \[ = \dfrac{{ - 1 + \dfrac{5}{4}}}{2} = \dfrac{1}{8}\]
5) Similarly, the fifth rational number between 2 and \[\dfrac{5}{4}\] can be calculated by finding the average between them.
Fifth rational number\[ = \dfrac{{2 + \dfrac{5}{4}}}{2} = \dfrac{{13}}{8}\]
6) Sixth rational number between \[ - 1\] and \[ - \dfrac{1}{4}\] can be calculated by finding the average between them.
Sixth rational number\[ = \dfrac{{ - 1 - \dfrac{1}{4}}}{2} = - \dfrac{5}{8}\]
7) The seventh rational number between 2 and \[ - \dfrac{1}{4}\] can be calculated by finding the average between them.
Seventh rational number\[ = \dfrac{{2 - \dfrac{1}{4}}}{2} = \dfrac{7}{8}\]

8) Eighth rational numbers between \[ - 1\] and \[ - \dfrac{5}{8}\] can be calculated by finding the average between them.
Eighth rational number\[ = \dfrac{{ - 1 - \dfrac{5}{8}}}{2} = - \dfrac{{13}}{{16}}\]
9) The ninth rational number between 2 and \[ - \dfrac{5}{8}\] can be calculated by finding the average between them.
Ninth rational number\[ = \dfrac{{2 - \dfrac{5}{8}}}{2} = \dfrac{{11}}{{16}}\]
10) Tenth rational number between \[ - 1\] and \[\dfrac{7}{8}\] can be calculated by finding the average between them.
Tenth rational number\[ = \dfrac{{ - 1 + \dfrac{7}{8}}}{2} = - \dfrac{1}{{16}}\]
11) The eleventh rational number between 2 and \[\dfrac{7}{8}\] can be calculated by finding the average between them.
Eleventh rational number \[ = \dfrac{{2 + \dfrac{7}{8}}}{2} = \dfrac{{23}}{{16}}\]
At last for the twelfth rational number, we will find the average between the numbers \[-1\] and \[ - \dfrac{{13}}{{16}}\]
12) Twelfth rational number\[ = \dfrac{{2 - \dfrac{{13}}{{16}}}}{2} = \dfrac{{19}}{{32}}\]

Thus, the 12 rational number between \[-1\] and 2 are \[\dfrac{1}{2}\], \[\dfrac{5}{4}\], \[ - \dfrac{1}{4}\], \[\dfrac{1}{8}\], \[\dfrac{{13}}{8}\], \[ - \dfrac{5}{8}\], \[\dfrac{7}{8}\], \[ - \dfrac{{13}}{{16}}\], \[\dfrac{{11}}{{16}}\], \[ - \dfrac{1}{{16}}\], \[\dfrac{{23}}{{16}}\]and \[\dfrac{{19}}{{32}}\].

Note:
We found five rational numbers between 3 and 4. Here we have found out 5 rational numbers. We can say that the number we found is a rational number because the denominator is not equal to zero. If the denominator of a fraction is zero then they are termed as infinite numbers. We could have found the answer using a number line and placing the given numbers on the number line. And then observe which numbers come in between 3 and 4.