Introduction to Irrational Numbers

Q.

Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$.

$0.\overline{6}$

Q. If a and b are rational numbers and $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}=a-b\sqrt{77},$ find $a-b$.

1. 4

Q.

An irrational number between $\frac{1}{7}$ and $\frac{2}{7}$ is

1. $(\frac{1}{7}+\frac{2}{7})$

2. $\frac{1}{2}(\frac{1}{7}+\frac{2}{7})$

3. $\sqrt{\frac{1}{7}\times \frac{2}{7}}$

4. None of these

Q.

$0.\overline{27}$ in $\frac{p}{q}$ form is ___________ ,

Q. If a and b are rational numbers and $\frac{2+\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$ find the values of a and b.

1. a = 7, b = 4
2. a = 11, b = 14
3. a = 7, b = 11
4. a = 4, b = 7

Q.

Express the $0.00\overline{32}$ rational numbers in the form $\frac{p}{q}$ , where $pandq$ are integers and $q\ne 0$.

Q.

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

1. $\frac{8}{25}$

2. $\frac{29}{90}$

3. $\frac{32}{99}$

4. $\frac{32}{199}$

Q.

Express $2.\overline{36}+0.\overline{23}$ in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$.

Q.

Which of the following statement is not correct regarding rational number?

1. It can have non terminating recurring decimal.

2. It cannot be represented on a number line.

3. It can be written in the form of p/q where p and q are integers and q ≠ 0.

4. It can have terminating decimal form.

Q. Question 1
Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:
$\left(i\right)x^2=5\left(ii\right)y^2=9\left(iii\right)z^2=0.04\left(iv\right)u^2=\frac{17}{4}$

Q.

The decimal expansion of an irrational number maybe _______?

Q.

The decimal representation of an irrational number is?

Q.

Question 3 (ii)
Express $0.4\overline{7}$ in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$.

Q.

Question 6 (b)

Express the following as the sum of two odd primes:
36

Q.

Check whether $10$ is a perfect number or not.

Q. Question 4
Classify the following numbers as rational or irrational with justification.
$\left(i\right)\sqrt{196}\left(ii\right)3\sqrt{18}\left(iii\right)\sqrt{\frac{9}{27}}\left(iv\right)\frac{\sqrt{28}}{343}\left(v\right)-\sqrt{0.4}\left(vi\right)\frac{\sqrt{12}}{\sqrt{75}}$
(vii) 0.5918
(viii) $(1+\sqrt{5})-(4+\sqrt{5})$
(ix) 10.124124 ....
(x) 1.010010001....
To classify, use the definition a rational number is in the form of $\frac{p}{q}$, where p and q are integers and $q\ne 0$ and otherwise it is an irrational number.

Q. Write the number $0.\overline{35}$ in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$

Q.

The number of three-digit odd numbers that can be formed with $1,2,3,4,5$ is,

1. $36$

2. $60$

3. $24$

4. $53$

Q. Question 8
A rational number between $\sqrt{2}$ and $\sqrt{3}$ is

A) $\frac{\sqrt{2}+\sqrt{3}}{2}$
B) $\frac{\sqrt{2}.\sqrt{3}}{2}$
C) 1.5
D) 1.8

Q.

Express the $32.12\overline{35}$ rational numbers in the form $\frac{p}{q}$ , where $pandq$ are integers and $q\ne 0$.

Q. Which one of the following is an irrational number?

1. π
2. $\sqrt{\frac{9}{4}}$
3. $\frac{7}{11}$
4. $\sqrt{25}$

Q. Find 3 irrational numbers between $\frac{5}{7}$ and $\frac{11}{9}$. Give solution.

Q.

Question 141

Fill in the blank to make the statement true.

2 is the only ......... number which is even.

Q. $\frac{p}{q}$ form of the number $0.123\overline{456}$, where p and q are integers and $q\ne 0$, is ____

1. $\frac{3333}{99000}$
2. $\frac{41111}{999000}$
3. $\frac{41111}{333000}$
4. $\frac{12333}{99000}$

Q.

Please answer this alone very fastly

This is from congruence of numbers

Find the last two digits of 3¹⁹⁹⁷

Q. Find the hypotenuse of a right angled triangle whose perpendicular sides are $\sqrt{6}$ cm and $\sqrt{3}$ cm.

Q. Find the $\frac{p}{q}$ form of the number $0.123\overline{456}$, where p and q are integers and $q\ne 0$.

Q.

Write the number $0.\overline{35}$ in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$, is

1. $\frac{14}{9}$

2. $\frac{35}{99}$

3. $\frac{14}{13}$

4. $\frac{14}{15}$

Q. $\frac{p}{q}$ form of the number $0.123\overline{456}$, where p and q are integers and $q\ne 0$, is ____

1. $\frac{3333}{99000}$
2. $\frac{41111}{999000}$
3. $\frac{41111}{333000}$
4. $\frac{12333}{99000}$

Q. If x = 2z and z is an integer, then x is

1. a special number
2. an odd number
3. an even number