Irrational Numbers

Q.

Prove that $\surd 3+\surd 5$ is irrational.

Q.

How we can express $0.\overline{001}$ in the form of $\frac{p}{q}$ , where $q\ne 0$

Q.

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

Q.

Find two irrational number between 0.5 and 0.55.

Q.

Give an example of two irrational numbers whose sum is a rational number.

Q.

Every integer is a rational number. true or false?

Q.

Examine, whether the following numbers are rational or irrational:

(i) $\sqrt{7}$ (ii) $\sqrt{4}$

(iii) $2+\sqrt{3}$ (iv) $\sqrt{3}+\sqrt{2}$

(v) $\sqrt{3}+\sqrt{5}$ (vi) $(\sqrt{2}-2{)}^2$

(vii) $(2-\sqrt{2})(2+\sqrt{2})$ (viii) $\sqrt{2}+\sqrt{3}{)}^2$

(ix) $\sqrt{5}-2$ (x) $\sqrt{23}$

(xi) $\sqrt{225}$ (xii) 0.3796

(xiii) 7.478478..... (xiv) 1.101001000100001.....

Q.

Prove that $\frac{1}{\sqrt{2}}$ is irrational:

Q.

Give an example of two irrational numbers whose sum as well as product is rational.

Q.

(a) Both Assertion (A) and Reason (R) are true and Reason (A) and Reason (R). For selecting the correct answer, use the following code :

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

$\begin{array}{cc}\text{Assertion (A)}& \text{Reason (R)}\\ \text{e is an irrational number.}& \pi isanirrationalnumber\end{array}\text{The correct answer is: (a)/(b)/(c)/(d).}$

Q.

Find 2 irrational numbers between 0.12 and 0.13

Q.

The value of $3\sqrt{54}÷2\sqrt{2}$ is

1. 
2. 
3. 
4. 

Q. Find two irrational numbers between 1/2 and 1/3

Q.

Square of an irrational number is always a rational number.

1. True

2. False

Q.

Express 0.00323232.... in the form of p/q where p , q are integers and q ≠0 .

Q.

Find two irrational numbers lying between 0.1 and 0.12.

Q. The $\frac{p}{q}$ form of $0.\overline{18}$ is (where p and q are integers and $q\ne 0$) .

1. $\frac{2}{11}$
2. $\frac{2}{22}$
3. $\frac{2}{33}$

Q.

Find an irrational number between $\frac{5}{7}$ and $\frac{7}{9}$. How many more there may be?

Q. Prove that the following are irrational.
$\frac{1}{\sqrt{2}}$

Q. Question 3
State whether the following statements are true or false? Justify your answer.
(i) $\frac{\sqrt{2}}{3}$ is a rational number.
(ii) There are infinitely many integers between any two integers.
(iii) Number of rational numbers between 15 and 18 is finite.
(iv) There are numbers which cannot be written in the form $\frac{p}{q},q\ne 0$ p and q both are integers.
(v) The square of an irrational number is always rational.
(vi) $\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as $\sqrt{1}2$ and $\sqrt{3}$ are not integers.
(vii) $\frac{\sqrt{15}}{\sqrt{3}}$ is written in the form $\frac{p}{q}$, so it is a rational number.

Q.

Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers :

(i) $\sqrt{4}$ (ii) $3\sqrt{18}$

(iii) $\sqrt{1.44}$ (iv) $\sqrt{\frac{9}{27}}$

(v) $-\sqrt{64}$ (vi) $\sqrt{100}$

Q. $0.3\overline{2}$ when expressed in the form $\frac{p}{q}$ (p, q are integers q ≠ 0), is

(a) $\frac{8}{25}$

(b) $\frac{29}{90}$

(c) $\frac{32}{99}$

(d) $\frac{32}{199}$

Q.

Find two rational numbers between √ 2 and √ 5

Q.

Which of the following is an irrational number ?

(a) 3.14

(b) 3.141414......

(c) 3.14444....

(d) 3.141141114....

Q.

Which of the following numbers is irrational?

(a) $\sqrt{\frac{4}{8}}$

(b) $\frac{\sqrt{1250}}{\sqrt{8}}$

(c) $\sqrt{8}$

(d) $\frac{\sqrt{24}}{\sqrt{6}}$

Q.

Insert a rational and an irrational number between 2 and 2.5.

Q.

Find two irrational numbers between 0.16 and 0.17.

Q. Represent 1.2353535 in p/q form

Q.

Find three different irrational numbers between the rational numbers 5/7 and 9/11

Q.

decimal expansion of 22/7