Q. Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.

Q. Write the following in decimal form and say what kind of decimal expansion each has:
(i) $\frac{36}{100}$ (ii) $\frac{1}{11}$ (iii) $4\frac{1}{8}$ (iv) $\frac{3}{13}$ (v) $\frac{2}{11}$ (vi) $\frac{329}{400}$

Q. Classroom activity(Constructing the 'square root spiral'): Take a large sheet of paper and construct the 'square root spiral' in the following fashion. Start with a point O and draw a line segment $O{P}_1$ of unit length. Draw a line segment $P_1{P}_2$ perpendicular to $O{P}_1$ of unit length. Now draw a line segment $P_2{P}_3$ perpendicular to $O{P}_2$. Then draw a line segment $P_3{P}_4$ perpendicular to $O{P}_3$. Continuing in this manner, you can get the line segment $P_{n-1}{P}_n$ by drawing a line segment of unit length perpendicular to $O{P}_{n-1}$. In this manner, you will have created the points $P_2,P_3,.......,P_n,.....,$ and joined them to create a beautiful spiral depicting $\sqrt{2},\sqrt{3},\sqrt{4},.......$

Q. Find five rational numbers between $3$ and $4$.

Q. Look at several examples of rational numbers in the form $\frac{p}{q}(q\ne 0)$, where $p$ and $q$ are integers with no common factors other than $1$ and having terminating decimal representaions (expansions). Can you guess what property $q$ must satisfy?

Q. Classify the following numbers as rational or irrational:
(i) $\sqrt{23}$
(ii) $\sqrt{225}$
(iii) $0.3796$
(iv) $\mathrm{7.478478...}$
(v) $\mathrm{1.101001000100001...}$

Q. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

Q. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.

Q. What can be the maximum number of digits in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check your answer.

Q. Express the following in the form $\frac{p}{q}$, wher $p$ and $q$ are integers and $q\ne 0.$
(i) $0.\overline{6}$
(ii) $0.4\overline{7}$
(iii) $0.\overline{001}$

Q. Show how $\sqrt{5}$ can be represented on the number line.

Q. Recall, $\pi$ is defined as the ratio of the circumference(say c) of a circle to its diameter(say d). That is, $\pi =\frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?

Q. Rationalise the denominators of the following:
(i) $\frac{1}{\sqrt{7}}$
(ii) $\frac{1}{\sqrt{7}-\sqrt{6}}$
(iii) $\frac{1}{\sqrt{5}+\sqrt{2}}$
(iv) $\frac{1}{\sqrt{7}-2}$

Q. Classify the following numbers as rational or irrational:
(i) $2-\sqrt{5}$ (ii) $(3+\sqrt{23})-\sqrt{23}$ (iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$ (iv) $\frac{1}{\sqrt{2}}$ (v) $2\pi$.

Q. Represent $\sqrt{9.3}$ on the number line.

Q. Simplify each of the following expressions:
(i) $(3+\sqrt{3})(2+\sqrt{2})$
(ii) $(3+\sqrt{3})(3-\sqrt{3})$
(iii) $(\sqrt{5}+\sqrt{2}{)}^2$
(iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$

Q. You know that $\frac{1}{7}=0.\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ are, without actually doing the long division? If so, how? (Hint: Study the remainders while finding the value of $\frac{1}{7}$ carefully.).

Q. Express $0.99999$..... in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Q. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Q. 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.