Question

CLASSIFY THE NUMBERS AS RATIONAL OR IRRATIONAL

[A] ROOT 23 [B] ROOT 225 [C] 0.3796 [D] 7.478478

Answer 1

Answer:

First two are irrational and rest two are rational

Answer 2

Question :-

Classify the Numbers as Rational or Irrational:

[a] $\mathtt{\sqrt{23}}$

[b] $\mathtt{\sqrt{225}}$

[c] $\mathtt{0.3796}$

[d] $\mathtt{7.478478...}$

$\begin{gathered} \end{gathered}$

Solution :-

[a] $\mathtt{\sqrt{23}}$

23 is a Prime Number.

Hence, Square Root of a Prime Number is an Irrational Number.

$\begin{gathered} \end{gathered}$

[b] $\mathtt{\sqrt{225}}$

$\mathtt{\sqrt{225}}$ is a Perfect Square.      $\boxed{\because\: \mathtt{\sqrt{225}=15}}$

Hence, $\mathtt{\sqrt{225}}$ is a Rational Number.

$\begin{gathered} \end{gathered}$

[c] $\mathtt{0.3796}$

It is a Terminating Decimal.

[Terminating Decimal :-

A Decimal that ends after a finite number of Digits is called a Terminating Decimal.

E.g., $\mathtt{\dfrac{1}{4}=0.25}$]

$\begin{gathered} \end{gathered}$

Hence, it is a Rational Number.

$\begin{gathered} \end{gathered}$

[d] $\mathtt{7.478478... = 7.\overline{478}}$

It is a Non-Terminating, Recurring Decimal.

[Non-Terminating, Recurring Decimal :-

A Decimal in which a Digit or a set of Digits is repeated periodically, is called a Repeating, or a Recurring Decimal.

E.g., $\mathtt{\dfrac{2}{3}=0.\bar6}$]

$\begin{gathered} \end{gathered}$

Hence, it is a Rational Number.

$\begin{gathered} \end{gathered}$

Additional Information :-

Rational Numbers :-

The Numbers of the form $\mathtt{\dfrac{p}{q}}$, where p and q are Integers and $\mathtt{q\neq0}$, are known as Rational Numbers.

E.g., $\mathtt{\dfrac{1}{4},\dfrac{3}{2},\dfrac{11}{79},-\dfrac{2001}{2002},etc...}$

$\begin{gathered} \end{gathered}$

Irrational Numbers :-

A Number which can neither be expressed as a Terminating Decimal nor as a Repeating (Recurring) Decimal, is called an Irrational Number.

E.g., $\mathtt{0.01001000100001...,\sqrt2, \sqrt[3]{2},\: \pi, etc... }$