Q1. Give two examples of natural numbers.

Ans: \(1,\;2\).

Q2. What is the smallest whole number?

Ans: \(0\).

Q3. Give two examples of integers less than zero.

Ans: \(-1,\;-2\).

Q4. Define a rational number.

Ans: A number expressible as \(\dfrac{m}{n}\) where \(m,n\in\mathbb{Z}\) and \(n\neq0\).

Q5. Give two examples of rational numbers (one positive, one negative).

Ans: \(\dfrac{3}{4},\; -\dfrac{5}{2}\).

Q6. State whether the number \(0\) is rational or irrational.

Ans: Rational (because \(0=\dfrac{0}{1}\)).

Q7. Is \(-7\) rational? If yes, write as \(\dfrac{m}{n}\).

Ans: Yes. \(-7=\dfrac{-7}{1}\).

Q8. Write a terminating decimal example of a rational number.

Ans: \(1.25=\dfrac{5}{4}\).

Q9. Write a recurring decimal example of a rational number.

Ans: \(0.333\ldots = 0.\overline{3} = \dfrac{1}{3}\).

Q10. Give two examples of irrational numbers.

Ans: \(\sqrt{2},\; \pi\) (decimal non-terminating non-repeating).

Q11. Is \(\dfrac{22}{7}\) rational or irrational?

Ans: Rational (it's a fraction; numerical value \(\approx3.142857\)).

Q12. Is \(\sqrt{9}\) rational or irrational?

Ans: Rational because \(\sqrt{9}=3\).

Q13. What is the decimal form of \(\dfrac{7}{4}\)?

Ans: \(\dfrac{7}{4}=1.75\).

Q14. What type of decimal is \(\dfrac{1}{8}\)?

Ans: Terminating decimal: \(\dfrac{1}{8}=0.125\).

Q15. Is the number \( -0.5\) rational? Give fraction form.

Ans: Yes. \(-0.5 = -\dfrac{1}{2}\).

Q16. Are rational numbers dense on the number line?

Ans: Yes — between any two rationals there are infinitely many rationals.

Q17. Convert \(0.\overline{23}\) to a fraction.

Ans: \(0.\overline{23}=\dfrac{23}{99}\).

Q18. Is \( \sqrt{4}\) rational? Give value.

Ans: Yes. \(\sqrt{4}=2\).

Q19. Write \(\dfrac{5}{10}\) in simplest form.

Ans: \(\dfrac{5}{10}=\dfrac{1}{2}\).

Q20. Is \( \sqrt{3}\) rational or irrational?

Ans: Irrational (non-terminating, non-repeating decimal).

Q1. Show on a number line how to place \(\dfrac{7}{3}\).

Ans: Divide each unit into 3 equal parts. Count 7 parts to the right of 0. \(\dfrac{7}{3}=2+\dfrac{1}{3}\) lies one-third after 2.

Q2. Show on a number line how to place \(-\dfrac{2}{3}\).

Ans: Place \(\dfrac{2}{3}\) to the right; then reflect to left of 0 — \(-\dfrac{2}{3}\) is two-thirds left of 0.

Q3. Compare \(\dfrac{5}{4}\) and \(\dfrac{2}{3}\).

Ans: Convert to common denominator: \(\dfrac{5}{4}=\dfrac{15}{12}\), \(\dfrac{2}{3}=\dfrac{8}{12}\). Since \(15>8\), \(\dfrac{5}{4}>\dfrac{2}{3}\).

Q4. Compare \(-\dfrac{7}{9}\) and \(\dfrac{4}{5}\).

Ans: Any negative number \(<\) any positive number, so \(-\dfrac{7}{9}<\dfrac{4}{5}\).

Q5. Compare \(-\dfrac{7}{3}\) and \(-\dfrac{5}{2}\).

Ans: Compare positive parts: \(\dfrac{7}{3}\approx2.333\), \(\dfrac{5}{2}=2.5\). Since \(2.333<2.5\), we get \(-\dfrac{7}{3}>-\dfrac{5}{2}\).

Q6. Convert \(\dfrac{7}{4}\) to decimal.

Ans: \(\dfrac{7}{4}=1.75\) (terminating).

Q7. Express \(0.08333\ldots\) as a fraction.

Ans: \(0.08333\ldots=0.08\overline{3}=\dfrac{5}{60}+\dfrac{1}{120}?\) — better: write \(x=0.08333\ldots\). Then \(100x=8.333\ldots\), so \(100x- x =8.25\Rightarrow99x=8.25\Rightarrow x=\dfrac{8.25}{99}=\dfrac{825}{9900}=\dfrac{11}{132}=\dfrac{1}{12}\). (Simplify carefully): actually \(0.08333\ldots = \dfrac{1}{12}\).

Q8. Convert \(0.\overline{243}\) to a fraction.

Ans: \(0.\overline{243}=\dfrac{243}{999}=\dfrac{81}{333}=\dfrac{27}{111}=\dfrac{9}{37}\).

Q9. Convert \(\dfrac{23}{99}\) to decimal form.

Ans: \(\dfrac{23}{99}=0.\overline{23}\) (recurring).

Q10. Is \(\dfrac{22}{7}\) exactly equal to \(\pi\)?

Ans: No. \(\dfrac{22}{7}\) is a rational approximation of \(\pi\) but \(\pi\) is irrational.

Q11. Show that \(\dfrac{3}{5} = \dfrac{6}{10}\) (brief explanation).

Ans: Multiply numerator and denominator of \(\dfrac{3}{5}\) by \(2\): \(\dfrac{3\times2}{5\times2}=\dfrac{6}{10}\).

Q12. Write decimal form of \(\dfrac{9}{14}\) (state if terminating/recurring).

Ans: \(\dfrac{9}{14}=0.642857142857\ldots\) = \(0.\overline{642857}\) (recurring block of 6 digits).

Q13. If \(a

Ans: \(-a> -b\) (sign flips when negative).

Q14. Compare \(\dfrac{3}{5}\) and \(\dfrac{6}{10}\).

Ans: \(\dfrac{3}{5}=\dfrac{6}{10}\), so they are equal.

Q15. Convert \(0.2323\ldots\) to fraction.

Ans: \(0.\overline{23}=\dfrac{23}{99}\).

Q16. Which is larger: \(\dfrac{15}{12}\) or \(\dfrac{7}{16}\)?

Ans: Convert to decimals or common denom: \(\dfrac{15}{12}=1.25,\;\dfrac{7}{16}=0.4375\). So \(\dfrac{15}{12}>\dfrac{7}{16}\).

Q17. Compare \(-\dfrac{25}{8}\) and \(-\dfrac{9}{4}\).

Ans: \(-\dfrac{25}{8}=-3.125,\; -\dfrac{9}{4}=-2.25.\) Since \(-3.125 < -2.25\), we have \(-\dfrac{25}{8} < -\dfrac{9}{4}\).

Q18. Simplify and state rational/irrational: \(3 - \sqrt{9}\).

Ans: \(\sqrt{9}=3\) so \(3-3=0\). \(0\) is rational.

Q19. Convert \( \dfrac{12}{15}\) into simplest form.

Ans: \(\dfrac{12}{15}=\dfrac{4}{5}\) after dividing by 3.

Q20. Write decimal representation of \(\dfrac{5}{6}\).

Ans: \(\dfrac{5}{6}=0.83333\ldots=0.8\overline{3}\) (recurring).

Q1. Compare \(-\dfrac{7}{3}\) and \(-\dfrac{5}{2}\) with full working.

Ans: Compare positive counterparts: \(\dfrac{7}{3}=\dfrac{14}{6},\;\dfrac{5}{2}=\dfrac{15}{6}\). Since \(\dfrac{14}{6}<\dfrac{15}{6}\), for negatives \(-\dfrac{7}{3}>-\dfrac{5}{2}\).

Q2. Compare \(\dfrac{3}{5}\) and \(\dfrac{6}{10}\) showing cross-multiplication.

Ans: Cross-multiply: \(3\times10=30\) and \(6\times5=30\). Since equal, \(\dfrac{3}{5}=\dfrac{6}{10}\).

Q3. Show why \(\sqrt{2}\) cannot be written as a terminating or repeating decimal (brief idea).

Ans: \(\sqrt{2}\) is irrational (proof by contradiction will be taught later). Its decimal expansion is non-terminating and non-repeating — e.g., \(1.41421356\ldots\).

Q4. Convert \(0.142857142857\ldots\) to fraction.

Ans: \(0.\overline{142857}=\dfrac{142857}{999999}=\dfrac{1}{7}\).

Q5. Write \(\dfrac{9}{37}\) in recurring decimal form (brief result).

Ans: \(\dfrac{9}{37}=0.\overline{243}\) (recurring period 3).

Q6. Show that \(\dfrac{3}{5} < \dfrac{4}{5}\) using numerators.

Ans: Same denominator \(5\); compare numerators \(3<4\Rightarrow\dfrac{3}{5}<\dfrac{4}{5}\).

Q7. Express \(1.1666\ldots\) as a fraction.

Ans: \(1.1666\ldots=1.1\overline{6}=\dfrac{7}{6}\). (Method: \(x=1.1666\ldots\Rightarrow10x=11.666\ldots; 100x=116.666\ldots; 100x-10x=105\Rightarrow90x=105\Rightarrow x=\dfrac{105}{90}=\dfrac{7}{6}\)).

Q8. Convert \(-\dfrac{103}{5}\) to mixed number and decimal.

Ans: \(-\dfrac{103}{5}=-20\dfrac{3}{5}=-20.6\).

Q9. Using Pythagoras, show how to mark \(2\) on number line (short steps).

Ans: Build right triangle with legs 1 and 1; hypotenuse \(=\sqrt{1^2+1^2}=\sqrt{2}\). With centre at origin and radius \(\sqrt{2}\), intersection with line gives \(\sqrt{2}\). (This is method outline; for exact 2 use distance construction as in book.)

Q10. Why is \(\pi\) irrational (short statement)?

Ans: \(\pi\) is proven to be non-terminating, non-repeating (transcendental); therefore irrational — cannot be expressed as \(\dfrac{m}{n}\).

Q11. Show why \(\dfrac{5}{6}=0.8\overline{3}\) by division.

Ans: \(5\div6=0.83333\ldots\) giving repeating 3. So \(0.8\overline{3}\).

Q12. Prove \(\dfrac{3}{5} = 0.6\) using decimal conversion.

Ans: \(\dfrac{3}{5}= \dfrac{3\times2}{5\times2}=\dfrac{6}{10}=0.6\).

Q13. Write \(\dfrac{11}{13}\) approximately up to 4 decimals and state recurring/terminating.

Ans: \(\dfrac{11}{13}\approx0.8461\ldots\) (repeating, non-terminating).

Q14. Compare \(\dfrac{40}{29}\) and \(\dfrac{141}{29}\).

Ans: Same denom — compare numerators: \(40<141\Rightarrow\dfrac{40}{29}<\dfrac{141}{29}\).

Q15. Compare \(-\dfrac{17}{20}\) and \(-\dfrac{13}{20}\).

Ans: Same denom, \( -\dfrac{17}{20} < -\dfrac{13}{20}\) because \(-17<-13\).

Q16. Convert \(0.75\) to fraction and simplify.

Ans: \(0.75=\dfrac{75}{100}=\dfrac{3}{4}\).

Q17. Show that \(3\) can be marked using previous marking of \(2\) (Pythagorean step).

Ans: If OQ=2 and QR=1 perpendicular at Q, then OR\(^2\)=OQ\(^2\)+QR\(^2\)=4+1=5, so OR=\(\sqrt{5}\). Then repeat to construct 3 via nested steps (book demonstrates full geometric steps).

Q18. If decimal terminates, is number rational? Explain.

Ans: Yes. Every terminating decimal can be written as fraction with denominator \(10^k\), hence rational.

Q19. Convert \(0.2323\ldots\) into simplest fraction and show steps.

Ans: Let \(x=0.\overline{23}\). Then \(100x=23.\overline{23}\). Subtract: \(99x=23\Rightarrow x=\dfrac{23}{99}\).

Q20. Explain in one line why \(-2 < -1\) on number line.

Ans: On number line, numbers increase to the right; \(-2\) is left of \(-1\), so \(-2< -1\).

PS1.1 Q1 (1): Show \(\dfrac{3}{2},\; \dfrac{5}{2},\; -\dfrac{3}{2}\) on separate number lines.

Ans: For each number line:
— Divide each unit into 2 equal parts (half units).
— \(\dfrac{3}{2}=1+\dfrac{1}{2}\) lies at halfway between \(1\) and \(2\).
— \(\dfrac{5}{2}=2+\dfrac{1}{2}\) lies halfway between \(2\) and \(3\).
— \(-\dfrac{3}{2}\) is halfway between \(-2\) and \(-1\) on the left side.

PS1.1 Q1 (2): Show \(\dfrac{7}{5},\; -\dfrac{2}{5},\; -\dfrac{4}{5}\).

Ans:
— Divide each unit into 5 equal parts.
— \(\dfrac{7}{5}=1+\dfrac{2}{5}\) (two fifths past 1).
— \(-\dfrac{2}{5}\) is two-fifths to left of 0.
— \(-\dfrac{4}{5}\) is four-fifths to left of 0.

PS1.1 Q1 (3): Show \(-\dfrac{5}{8},\; \dfrac{11}{8}\).

Ans:
— Divide each unit into 8 equal parts.
— \(\dfrac{11}{8}=1+\dfrac{3}{8}\) (three-eighths after 1).
— \(-\dfrac{5}{8}\) lies five-eighths to the left of 0.

PS1.1 Q1 (4): Show \(\dfrac{13}{10},\; -\dfrac{17}{10}\).

Ans:
— Divide each unit into 10 equal parts (tenths).
— \(\dfrac{13}{10}=1+\dfrac{3}{10}\).
— \(-\dfrac{17}{10}= -1 - \dfrac{7}{10}\) (one and seven-tenths to left of 0).

PS1.1 Q2: Observe the number line (A,B,C,D,E,F, etc.) and answer: (1) Which number is indicated by point B? (2) Which point indicates \(1\dfrac{3}{4}\)? (3) Is 'point D denotes \(\dfrac{5}{2}\)' true or false?

Ans:
— (1) Without the picture: typically B would be labelled; follow the book's given number line. (If on provided figure B corresponds to \(\dfrac{3}{2}\) etc.) (If you paste the exact image into your page I will annotate exact labels.)
— (2) \(1\dfrac{3}{4}=1.75\) — the point three-fourths after \(1\).
— (3) Check D's position: if D is at \(2.5\) then statement true; otherwise false. (From typical figure if D at 2.5 then true.)

PS1.2 Q1 (1): Compare \(-7\) and \(-2\).

Ans: On number line \(-7\) is left of \(-2\), so \(-7< -2\).

PS1.2 Q1 (2): Compare \(0\) and \(-\dfrac{9}{5}\).

Ans: \(0> -\dfrac{9}{5}\) since any positive (including 0) is greater than a negative.

PS1.2 Q1 (3): Compare \(\dfrac{8}{7}\) and \(0\).

Ans: \(\dfrac{8}{7}\approx1.142>0\).

PS1.2 Q1 (4): Compare \(-\dfrac{5}{4}\) and \(\dfrac{1}{4}\).

Ans: \(-\dfrac{5}{4}<\dfrac{1}{4}\) since negative < positive.

PS1.2 Q1 (5): Compare \(\dfrac{40}{29}\) and \(\dfrac{141}{29}\).

Ans: Same denominators \(29\). Compare numerators: \(40<141\Rightarrow\dfrac{40}{29}<\dfrac{141}{29}\).

PS1.2 Q1 (6): Compare \(-\dfrac{17}{20}\) and \(-\dfrac{13}{20}\).

Ans: Same denom: \(-17<-13\Rightarrow -\dfrac{17}{20}< -\dfrac{13}{20}\).

PS1.2 Q1 (7): Compare \(\dfrac{15}{12}\) and \(\dfrac{7}{16}\).

Ans: Convert to decimals: \(\dfrac{15}{12}=1.25,\; \dfrac{7}{16}=0.4375\). So \(\dfrac{15}{12}>\dfrac{7}{16}\).

PS1.2 Q1 (8): Compare \(-\dfrac{25}{8}\) and \(-\dfrac{9}{4}\).

Ans: \(-\dfrac{25}{8}=-3.125,\; -\dfrac{9}{4}=-2.25\). So \(-\dfrac{25}{8}< -\dfrac{9}{4}\).

PS1.2 Q1 (9): Compare \(\dfrac{12}{15}\) and \(\dfrac{3}{5}\).

Ans: \(\dfrac{12}{15}=\dfrac{4}{5}=0.8,\; \dfrac{3}{5}=0.6\). So \(\dfrac{12}{15}>\dfrac{3}{5}\).

PS1.2 Q1 (10): Compare \(-\dfrac{7}{11}\) and \(-\dfrac{3}{4}\).

Ans: Convert to decimals: \(-\dfrac{7}{11}\approx-0.63636,\; -\dfrac{3}{4}=-0.75\). Since \(-0.63636 > -0.75\), therefore \(-\dfrac{7}{11} > -\dfrac{3}{4}\).

PS1.3 Q1 (1): Write \(\dfrac{9}{37}\) in decimal form.

Ans: \(\dfrac{9}{37}=0.\overline{243}\) (recurring). We can confirm by long division: \(9\div37=0.243243\ldots\).

PS1.3 Q1 (2): Write \(\dfrac{18}{42}\) in decimal form (simplify first).

Ans: Simplify \(\dfrac{18}{42}=\dfrac{3}{7}\). \(\dfrac{3}{7}=0.\overline{428571}\) (repeating block of 6 digits).

PS1.3 Q1 (3): Write \(\dfrac{9}{14}\) in decimal form.

Ans: \(\dfrac{9}{14}=0.642857142857\ldots = 0.\overline{642857}\) (period 6).

PS1.3 Q1 (4): Write \( -\dfrac{103}{5}\) in decimal form.

Ans: \(-\dfrac{103}{5}=-20.6\) (since \(103\div5=20.6\)).

PS1.4 Q1: Fill the boxes to obtain 3 on number line using 2 (steps provided in book).

Ans (completed steps):
— The point Q shows the number \(2\).
— Draw a perpendicular at Q. Take R on this perpendicular such that \(QR=1\).
— Triangle \(OQR\) has \(OQ=2,\;QR=1\).
— By Pythagoras: \(|OR|^2 = |OQ|^2 + |QR|^2 = 2^2 + 1^2 = 4+1=5\).
— So \(|OR|=\sqrt{5}\).
— Now, with centre O and radius \(\sqrt{5}\) draw an arc meeting number line at C. C shows \(\sqrt{5}\).
— To get 3: repeat construction using \(\sqrt{5}\) and another step or use Pythagorean chain such that final distance becomes 3 (book's activity concludes that the point C shows 3 using the correct sequence).

PS1.4 Q2: Show the number 5 on the number line (outline).

Ans: Repeatedly use right triangles: build \(OQ=3, QR=4\) (say) so that hypotenuse \(=\sqrt{3^2+4^2}=5\). Place arc with radius 5 from origin to mark 5 on the line (construct 3 and 4 using prior steps or direct measurements as allowed).

PS1.4 Q3: Show the number 7 on the number line (outline).

Ans: Combine constructions (for example, create a 3–4–5 triangle then use another triangle to add lengths to reach 7) or measure 7 units directly by marking unit segments. The book's geometric steps show a systematic Pythagorean construction.

Example: Compare \(5/4\) and \(2/3\) (from chapter example).

Ans: \(5/4=\dfrac{15}{12},\;2/3=\dfrac{8}{12}\Rightarrow \dfrac{5}{4}>\dfrac{2}{3}\).

Example: Compare \(-7/9\) and \(4/5\).

Ans: Negative < positive, so \(-7/9<4/5\).

Example: Compare \(-7/3\) and \(-5/2\) (detailed earlier).

Ans: We found \(-7/3> -5/2\) because \(7/3 < 5/2\).