Question

Find two irrational numbers between 786 and 787.

Answer 1

Hint: In order to solve this problem you need to know that the sum of a rational number and an irrational number is an irrational number. Knowing this will solve your problem.

Complete step-by-step answer:
An Irrational Number is a real number that cannot be written as a simple fraction. "Irrational" means "no ratio", so it isn't a rational number.
We know that the number 786 and 787 are rational and we also know that $\dfrac{1}{{\sqrt 2 }}$ and $\dfrac{1}{{\sqrt 3 }}$ is an irrational number. Since after decimal in the values of $\sqrt 2 $ and $\sqrt 3 $ neither the numbers are repeated nor come to an end. Then the same is with $\dfrac{1}{{\sqrt 2 }}$ and $\dfrac{1}{{\sqrt 3 }}$. So, the numbers showing this property are irrational.

As, the difference between 786 and 787 is one so to get two irrational number between them we need to add an irrational numbers less than 1 and here those numbers are $\dfrac{1}{{\sqrt 2 }}$ and $\dfrac{1}{{\sqrt 3 }}$ since $\sqrt 2 $ and $\sqrt 3 $ is greater than 1 so $\dfrac{1}{{\sqrt 2 }}$and $\dfrac{1}{{\sqrt 3 }}$ will be less than 1.
So, we can say irrational numbers between 786 & 787 are:
$
   \Rightarrow 786 + \dfrac{1}{{\sqrt 2 }} = \dfrac{{786\sqrt 2 + 1}}{{\sqrt 2 }} \\
   \Rightarrow 786 + \dfrac{1}{{\sqrt 3 }} = \dfrac{{786\sqrt 3 + 1}}{{\sqrt 3 }} \\
$
Hence, the two irrational numbers between 786 and 787 numbers are $\dfrac{{786\sqrt 2 + 1}}{{\sqrt 2 }}$ and $\dfrac{{786\sqrt 3 + 1}}{{\sqrt 3 }}$.

Note: To solve such problems we need to know that the most common irrational number is $\pi $. Here we have to find two irrational number between 786 and 787 then we need to add an irrational numbers less than 1 and here those numbers are $\dfrac{1}{{\sqrt 2 }}$ and $\dfrac{1}{{\sqrt 3 }}$ since $\sqrt 2 $ and $\sqrt 3 $ is greater than 1 so $\dfrac{1}{{\sqrt 2 }}$and $\dfrac{1}{{\sqrt 3 }}$ will be less than 1. Proceeding like this you will get the right answer.