Question

Which of the following are irrational numbers
$$2,3.14,\pi ,\sqrt{9}$$

Answer 1

Hint: For solving this problem we have to try to convert the given numbers into general forms of rational numbers. Rational numbers are defined as the numbers that can be represented in the form of $${\displaystyle \frac{p}{q}}$$ where $$p,q$$ are integers and in simplest form and $$q\ne 0$$. So, if we can represent the numbers in the form as shown above then we can say that number is rational and if we cannot represent then we can conclude that a number is an irrational number.

Complete step-by-step solution
Now, let us consider the first number that is ‘2’
Here we can write the 2 in the form $${\displaystyle \frac{2}{1}}$$.
We can say that this number is a rational number since it has the form of $${\displaystyle \frac{p}{q}}$$ where $$p,q$$ are integers which are in simplest form and $$q\ne 0$$.
So, we can conclude that ‘2’ is a rational number, not an irrational number.
Now let us consider the second number given that is ‘3.14’
We can represent the number ‘3.14’ as $${\displaystyle \frac{314}{100}}$$ and by making it into the simplest form we will get $$\Rightarrow {\displaystyle \frac{314}{100}}={\displaystyle \frac{157}{50}}$$
Since we represented the number ‘3.14’ in the form of $${\displaystyle \frac{p}{q}}$$ where $$p,q$$ are integers which are in simplest form and $$q\ne 0$$, we can conclude that ‘3.14’ is a rational number.
Now let us consider $${}^{\prime }{\pi }^{\prime }$$
We know that the value of $${}^{\prime }{\pi }^{\prime }$$ after the decimal point has neither no end nor no repeating set of numbers.
Without knowing where the value after the decimal point ends we cannot represent it in the form of $${\displaystyle \frac{p}{q}}$$. So, we can conclude that the number $${}^{\prime }{\pi }^{\prime }$$ is an irrational number.
Now let us consider the third number that is $$\sqrt{9}$$
Here we know that ‘9’ is a perfect square. Which means that ‘9 is the square of some other integer. We also know that every integer is a rational number; we can conclude that $$\sqrt{9}$$ is a rational number, not an irrational number.
Therefore we can say that $${}^{\prime }{\pi }^{\prime }$$ is the only irrational number in the given list.

Note: Students will make mistakes in writing that 3.14 is a rational number because we take the value of $${}^{\prime }{\pi }^{\prime }$$ as ‘3.14’. Students need to keep in mind that the value of $${}^{\prime }{\pi }^{\prime }$$ is not 3.14. For calculation purposes, we are given that the value of $${}^{\prime }{\pi }^{\prime }$$ is 3.14. But it doesn’t mean that $${}^{\prime }{\pi }^{\prime }$$ is having the value 3.14. This is the only point that needs to be taken care of.