Question

List five irrational numbers between
(a) -1 and 0
(b) -2 and -1

Answer 1

Hint:In this question we have been asked to find 5 rational numbers between -1 and 0 and -2 and -1. Therefore, we should first understand what are rational numbers and then use it to find the required answer.

Complete step-by-step answer:
In this question, we are asked to find rational numbers in a given interval. Therefore, we should understand what rational numbers are first.
A number is said to be a rational number if it can be expressed in the form of ${\displaystyle \frac{p}{q}}$ where p and q are integers and $q\ne 0$……………………(1.1)

 i) Considering equation (1.1), we note that, to find rational numbers between -1 and 0, we can find out five fractions of the interval between -1 and 0 which will lie between -1 and 0. As, we have to find 5 numbers, we can divide the interval between -1 and 0 into 6 parts and each part will correspond to one rational number between -1 and 0. Thus, each division should be equal to ${\displaystyle \frac{\text{total length of interval}}{\text{number of divisions}}}={\displaystyle \frac{0-(-1)}{6}}={\displaystyle \frac{1}{6}}$ and the fractions of the interval at which the points lie would be
$\begin{array}{rl}& r_1={\displaystyle \frac{1}{6}}\times 1={\displaystyle \frac{-1}{6}}\\ & r_2={\displaystyle \frac{1}{6}}\times 2={\displaystyle \frac{-2}{6}}\\ & r_3={\displaystyle \frac{-1-0}{6}}\times 3={\displaystyle \frac{-3}{6}}\\ & r_4={\displaystyle \frac{1}{6}}\times 4={\displaystyle \frac{-4}{6}}\\ & r_5={\displaystyle \frac{1}{6}}\times 5={\displaystyle \frac{-5}{6}}\end{array}$
Now, these numbers are just fractions of the interval, therefore to obtain the actual numbers which lie in the interval, we should add these numbers to the starting point of the interval.
Thus, the five rational numbers lying between -1 and 0 will be $-1+{\displaystyle \frac{1}{6}},-1+{\displaystyle \frac{2}{6}},-1+{\displaystyle \frac{3}{6}},-1+{\displaystyle \frac{4}{6}},-1+{\displaystyle \frac{5}{6}}$ which are the same as ${\displaystyle \frac{-5}{6}},{\displaystyle \frac{-4}{6}},{\displaystyle \frac{-3}{6}},{\displaystyle \frac{-2}{6}},{\displaystyle \frac{-5}{6}}$
ii) Considering equation (1.1), we note that, to find rational numbers between -2 and -1, we can find out five fractions of the interval between -2 and -1 which will lie between -2 and -1. As, we have to find 5 numbers, we can divide the interval between -2 and -1 into 6 parts and each part will correspond to one rational number between -2 and -1. Thus, each division should be equal to ${\displaystyle \frac{\text{total length of interval}}{\text{number of divisions}}}={\displaystyle \frac{-1-(-2)}{6}}={\displaystyle \frac{1}{6}}$ and the fractions of the interval at which the points lie would be
$\begin{array}{rl}& r_1={\displaystyle \frac{1}{6}}\times 1={\displaystyle \frac{-1}{6}}\\ & r_2={\displaystyle \frac{1}{6}}\times 2={\displaystyle \frac{-2}{6}}\\ & r_3={\displaystyle \frac{-1-0}{6}}\times 3={\displaystyle \frac{-3}{6}}\\ & r_4={\displaystyle \frac{1}{6}}\times 4={\displaystyle \frac{-4}{6}}\\ & r_5={\displaystyle \frac{1}{6}}\times 5={\displaystyle \frac{-5}{6}}\end{array}$
Now, these numbers are just fractions of the interval, therefore to obtain the actual numbers which lie in the interval, we should add these numbers to the starting point of the interval.
Thus, the five rational numbers lying between -2 and -1 will be $-2+{\displaystyle \frac{1}{6}},-2+{\displaystyle \frac{2}{6}},-2+{\displaystyle \frac{3}{6}},-2+{\displaystyle \frac{4}{6}},-2+{\displaystyle \frac{5}{6}}$ which are the same as ${\displaystyle \frac{-11}{6}},{\displaystyle \frac{-10}{6}},{\displaystyle \frac{-9}{6}},{\displaystyle \frac{-8}{6}},{\displaystyle \frac{-7}{6}}$

Note: We should note that to find five rational numbers, we had to divide the interval into 6 parts because the sixth part will be equal to $r_6={\displaystyle \frac{1}{6}}\times 6=1$ and adding it with the starting point will give a value equal to the ending point of the interval and thus will not lie strictly inside the interval. Also, one other method to find five rational numbers is to use the property that any number having non-recurring decimal representation is a rational number. Thus, the numbers $p_1=-1+0.1=-0.9$, $p_2=-1+0.01=-0.99$, $p_3=-1+0.001=-0.999$, $p_4=-1+0.0001=-0.9999$ and $p_1=-1+0.00001=-0.99999$ will also be valid answers to part a of the question. Also, as subtraction of two rational numbers is also a rational number, we could have obtained the five rational numbers between -2 and -1 by subtracting the obtained numbers in part (i) by 1.