Question

Find two rational and two irrational number between $$\sqrt{2}$$ and $$\sqrt{3}$$

Answer 1

Hint: As we know that $$\sqrt{2}$$ and $$\sqrt{\text{3}}$$ are irrational numbers so their approximate values are $$1.414$$ and $$\text{1}\text{.732}$$. Now , we need to calculate rational an irrational number between $$1.414$$ and $$\text{1}\text{.732}$$

Complete step by step answer:

Given Irrational Numbers $$\sqrt{2}$$ and $$\sqrt{\text{3}}$$
Firstly find its rational and so we need to consider rational points for that,
Calculating rational numbers between $$1.4$$and $$1.7$$.
So, they are $$\Rightarrow {\displaystyle \frac{1.4+1.7}{2}}=1.55$$ and can also be integers as $$1.5,1.6...$$
And now calculating the irrational terms,
$$\Rightarrow {\displaystyle \frac{\sqrt{2}+\sqrt{3}}{2}}=1.572$$
So the numbers between $$\sqrt{2}$$ and $$\sqrt{\text{3}}$$ which are non-terminating and cannot be expressed in $${\displaystyle \frac{\text{p}}{q}}$$ form, so it can be $$1.665\bar{7},1.543\bar{9},....$$
Hence, $$1.5,1.6$$ are 2 rational numbers and $$1.665\bar{7},1.543\bar{9}$$ are irrational terms between $$\sqrt{2}$$ and $$\sqrt{\text{3}}$$.

Note: An Irrational Number is a real number that cannot be written as a simple fraction. Irrational means, not Rational number.
A number that can be made by dividing two integers (an integer is a number with no fractional part). The word comes from "ratio".
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.