Question

Write any two irrational numbers lying between 3 and 4.

Answer 1

Hint: In this question, we first need to look into some of the basic definitions of the number system. Then from the definition of the irrational numbers we have a formula for writing irrational numbers that lie between two rational numbers. We can get those irrational numbers between 3 and 4 by using the formula:
\[{{a}_{i}}=a+\dfrac{b-a}{2\left( n+1 \right)}\sqrt{2}\times i,where\text{ }i=1,2,3,...,n\]

Complete step-by-step answer:

Now, let us look at some of the basic definitions first.

RATIONAL NUMBERS: A number which can be written in the form of \[\dfrac{p}{q}\], where \[p,q\in Z\]and \[q\ne 0\] , is called rational number. A rational number can be expressed as decimal.
IRRATIONAL NUMBER: An irrational number is a non-terminating, non-recurring decimal, which cannot be written in the form \[\dfrac{p}{q}\], is called an irrational number.
If a and b are two distinct rational numbers, then for a < b, n irrational numbers between a and b may be
\[{{a}_{i}}=a+\dfrac{b-a}{2\left( n+1 \right)}\sqrt{2}\times i,where\text{ }i=1,2,3,...,n\]
Now, we have that
\[a=3,b=4,n=2\]
\[\begin{align}
  & {{a}_{i}}=a+\dfrac{b-a}{2\left( n+1 \right)}\sqrt{2}\times i \\
 & \Rightarrow {{a}_{i}}=3+\dfrac{4-3}{2\left( 2+1 \right)}\sqrt{2}\times i \\
\end{align}\]
\[\begin{align}
  & \Rightarrow {{a}_{i}}=3+\dfrac{1}{2\left( 3 \right)}\sqrt{2}\times i \\
 & \Rightarrow {{a}_{i}}=3+\dfrac{1}{6}\sqrt{2}\times i \\
\end{align}\]
\[\left[ \because i=1,2 \right]\]
\[\begin{align}
  & \Rightarrow {{a}_{1}}=3+\dfrac{1}{6}\sqrt{2} \\
 & \therefore {{a}_{1}}=3+\dfrac{1}{3\sqrt{2}} \\
 & \Rightarrow {{a}_{2}}=3+\dfrac{1}{6}\sqrt{2}\times 2 \\
 & \therefore {{a}_{2}}=3+\dfrac{\sqrt{2}}{3} \\
\end{align}\]

Note: The number \[\sqrt{x}\], x is not a perfect square, is an irrational number.
0 is not an irrational number.
Sum, difference, product and quotient of two irrational numbers may be rational or irrational. But, the sum, difference, product and quotient of one rational and other irrational number is always irrational.
In this question, we used the same law that the sum of an irrational number and a rational number is always an irrational number to get two irrational numbers that are lying between 3 and 4.