State whether the following statements are true or false. Justify your answers.(i) Every irrational number is a real number.(ii) Every point on the number line is of the form $\sqrt{m}$, where 'm' is a natural number. (iii) Every real number is an irrational number.

To do:

We have to find whether the given statements are true or false.

Solution:

Rational Numbers:

A number that can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q$ is not equal to zero is a rational number.

For example,

$\frac{2}{3}, \frac{4}{5}, \frac{23}{6}, 8$

Note: Any number divided by zero is undefined but not infinity.

Irrational Numbers:

A number that cannot be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q$ is not equal to zero is an irrational number.

For example,

$ \sqrt{3}, \sqrt{7}, \pi$

Note: The value of Pi($\pi$) is not exactly equal to $\frac{22}{7}$. We use it for calculation purposes. Therefore, $\pi$ is an irrational number.

Real Numbers:

Rational numbers and irrational numbers together are called real numbers.

For example 

$\frac{3}{4}, \sqrt{5}, \pi, 11$

Therefore,

(i) Every irrational number is a real number. 

The given statement is true.

(ii) There are many points on the number line like $\sqrt{2.2}, \sqrt{3.3}$ which are not of the form $\sqrt{m}$ where $m$ is a natural number. 

The given statement is false.

(iii) All irrational numbers are real numbers but all real numbers are not irrational numbers. Real numbers include both rational numbers as well as irrational numbers. 

The given statement is false.

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Updated on: 10-Oct-2022

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