Question

Classify the following as rational or irrational numbers? $3+\sqrt{2}$

Answer 1

• Rational number:

A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero.

Example:

$\frac{1}{4},\frac{3}{5},....$are all rational numbers.

• Irrational number:

An irrational number is a type of real number which cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio.

Example:

$\pi ,\sqrt{2},....$are the irrational numbers.

Given Data:

Given number $3+\sqrt{2}$

Explanation:

1. Let, $3+\sqrt{2}$is a rational number.
2. Then, It can be written in the form, $3+\sqrt{2}=\frac{p}{q}$where p and q are integers and q≠0
3. $$\begin{gathered} 3+\sqrt{2}=\frac{p}{q} \\ or,\sqrt{2}=\frac{p}{q}-3 \\ or,\sqrt{2}=\frac{(p-3b)}{q}, \end{gathered}$$

is rational as p ,q, and 3 are integers.

4. It means that, $\sqrt{2}$is rational.

5. But this contradicts the fact that $\sqrt{2}$is irrational. So, Our assumption is wrong.

6. So, $3+\sqrt{2}$is an irrational number.

Final Answer:
Hence, $3+\sqrt{2}$is an irrational number.