Question

Find two irrational numbers between 786 and 787.

Answer 1

Hint: Consider two numbers as $a=786$ and the other as $b=787$. Now find the irrational numbers between them using the formula $\sqrt{ab}$ and then find the next irrational number by taking $a=786$ and $b=787$.

Complete step-by-step answer:
We have to find two irrational numbers between 786 and 787. Before proceeding with this question, let us find out what rational and irrational numbers are. Rational numbers are the numbers that can be expressed in the form of $\dfrac{p}{q}$, where $p,q$ are integers and $q\ne 0$. Examples are $1,0,\dfrac{2}{3},-10$ etc. Real numbers that are not rational numbers are called irrational numbers or we can say that it is a number that cannot be expressed in the form of a fraction are irrational numbers. Examples are $\sqrt{2},\sqrt{3}$ etc.
Now, we know that if we have two numbers, say $a,b$, then the irrational number between them is $\sqrt{ab}$ given that $ab$ is not a perfect square. So, let us take $a=786$ and $b=787$. So, we get the irrational number between 786 and 787 as,
$\begin{align}
  & =\sqrt{ab} \\
 & =\sqrt{786\times 787} \\
 & =\sqrt{618582} \\
 & =786.499841\ldots \ldots \\
\end{align}$
Now, we know that 786.5 is between 786 and 787, so any irrational number between 786 and 786.5 would also be between 786 and 787. So, we get, irrational numbers between 786 and 786.5 as, $\begin{align}
  & =\sqrt{\left( 786 \right)\times \left( 786.5 \right)} \\
 & =\sqrt{618189} \\
 & =786.24996\ldots \ldots \\
\end{align}$
Hence, we get the two irrational numbers between 786 and 787 as 786.499841… and 786.24996….

Note: The students must note that there are infinite rational and irrational numbers between any two numbers. The irrational numbers are those numbers that are non-terminating and non-repeating, that is the numbers of the digits in any irrational number is infinite and there is no specific order in which the digits appear. We can also cross check this in our answer. We have got two irrational numbers as 786.499841… and 786.24996… Here we can see that there is no specific order in the digits of these numbers and the digits are appearing infinitely.