Class 9 › Mathematics › Ganita Manjari The World of Numbers Class 9 Mathematics · Ganita Manjari · Chapter 3
Summary Questions & answers Practise this chapter → This chapter traces the expanding universe of numbers, from counting by one-to-one correspondence to the real number line. Natural numbers \(\mathbb{N}=\{1,2,3,\dots\}\) grew from the human need to count. Brahmagupta formalised zero (shunya) and the negative numbers, giving the integers \(\mathbb{Z}\) and rules such as \((-)\times(-)=(+)\). Combining integers with fractions gives the rational numbers \(\mathbb{Q}\), any number expressible as \(\dfrac{p}{q}\) with \(p,q\) integers and \(q\ne 0\). Rationals are dense — between any two there is always another, found by averaging. Some lengths, like the diagonal of a unit square \(\sqrt{2}\), cannot be written as a ratio; these are irrational, proved using proof by contradiction. Uniting rationals and irrationals forms the continuous real number line \(\mathbb{R}\). A number is rational exactly when its decimal expansion terminates or repeats, and irrational when it neither terminates nor repeats. The chapter shows how to convert repeating decimals into fractions and to predict, from prime factors of the denominator, whether a decimal terminates.
Natural numbers
The counting numbers \(\mathbb{N}=\{1,2,3,\dots\}\).
Integers
The set \(\mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\}\) of zero, positives and negatives.
Rational number
A number of the form \(\dfrac{p}{q}\) where \(p,q\) are integers and \(q\ne 0\).
Irrational number
A number that cannot be written as a ratio of integers, such as \(\sqrt{2}\) or \(\pi\).
Density
The property that between any two rational numbers lies another rational number.
Absolute value
The distance \(|x|\) of a number \(x\) from \(0\) on the number line, always non-negative.
Q1 Why must \(q \ne 0\) in the definition of a rational number \(\dfrac{p}{q}\)? Answer Q2 Find a rational number between \(1\) and \(\dfrac{3}{2}\). Answer Q3 How do you prove \(\sqrt{2}\) is irrational? Answer Q4 Which rational numbers have terminating decimal expansions? Answer Q5 Convert \(0.\overline{45}\) into the form \(\dfrac{p}{q}\). Answer Q6 Classify \(0.123451234512345\dots\) as rational or irrational. Answer Q7 What does the absolute value \(\left|\dfrac{-5}{3}\right|\) equal? Answer Q8 Why does \(0.\overline{9}\) equal \(1\)? Answer Q9 Is the product of two negative numbers positive or negative? Answer Q10 What set of numbers do the rationals and irrationals together form? Answer Q1 1. A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with? Answer Q2 2. Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern. Answer Q3 3. We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer. Answer Q4 *4. Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems? Answer Q1 Why does a negative times a negative equal a positive? Think of it in terms of action and debt. If a negative number represents a debt, then multiplying by a negative number represents the removal of that debt. (Hint: If someone takes away (-) four of your debts that are each worth ₹3 (that is, -3), you are effectively ₹12 richer! Therefore, (-3) × (-4) = +12.) Answer Q1 1. The temperature in the high-altitude desert of Ladakh is recorded as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight temperature? Answer Q2 2. A spice trader takes a loan (debt) of ₹850. The next day, he makes a profit (fortune) of ₹1,200. The following week, he incurs a loss of ₹450. Write this sequence as an equation using integers and calculate his final financial standing. Answer Q3 3. Calculate the following using Brahmagupta's laws: (i) (-12) × 5 Answer Q4 3. Calculate the following using Brahmagupta's laws: (ii) (-8) × (-7) Answer Q5 3. Calculate the following using Brahmagupta's laws: (iii) 0 - (-14) Answer Q6 3. Calculate the following using Brahmagupta's laws: (iv) (-20) ÷ 4 Answer Q7 4. Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., 10 - (-5) = 15). Answer Q1 Can you explain why we need q ≠ 0 in the definition of a rational number? Answer Q1 1. While adding or subtracting two rational numbers having different denominators, how will you make the denominators equal? Answer Q2 2. Verify the distributive law for rational numbers. Answer Q1 1. Prove that the following rational numbers are equal: (i) \(\frac{2}{3}\) and \(\frac{4}{6}\) Answer Q2 1. Prove that the following rational numbers are equal: (ii) \(\frac{5}{4}\) and \(\frac{10}{8}\) Answer Q3 1. Prove that the following rational numbers are equal: (iii) \(\frac{-3}{5}\) and \(\frac{-6}{10}\) Answer Q4 1. Prove that the following rational numbers are equal: (iv) \(\frac{9}{3}\) and \(3\) Answer Q5 2. Find the sum: (i) \(\frac{2}{5} + \frac{3}{10}\) Answer Q6 2. Find the sum: (ii) \(\frac{7}{12} + \frac{5}{8}\) Answer Q7 2. Find the sum: (iii) \(\frac{-4}{7} + \frac{3}{14}\) Answer Q8 3. Find the difference: (i) \(\frac{5}{6} - \frac{1}{4}\) Answer Q9 3. Find the difference: (ii) \(\frac{11}{8} - \frac{3}{4}\) Answer Q10 3. Find the difference: (iii) \(\frac{-7}{9} - \left(\frac{-2}{3}\right)\) Answer Q11 4. Find the product: (i) \(\frac{2}{3} \times \frac{3}{10}\) Answer Q12 4. Find the product: (ii) \(\frac{7}{11} \times \frac{5}{8}\) Answer Q13 4. Find the product: (iii) \(\frac{-4}{7} \times \frac{5}{14}\) Answer Q14 5. Find the quotient: (i) \(\frac{2}{3} \div \frac{3}{10}\) Answer Q15 5. Find the quotient: (ii) \(\frac{7}{11} \div \frac{5}{8}\) Answer Q16 5. Find the quotient: (iii) \(\frac{-4}{7} \div \frac{5}{14}\) Answer Q17 6. Show that: \(\left(\frac{1}{2} + \frac{3}{4}\right) \times \frac{8}{3} = \frac{1}{2} \times \frac{8}{3} + \frac{3}{4} \times \frac{8}{3}\). Answer Q18 7. Simplify the following using the distributive property: \(\frac{7}{9} \times \left(\frac{6}{7} - \frac{3}{4}\right)\). Answer Q19 8. Find the rational number \(x\) such that: \(\frac{5}{6} \times \left(x + \frac{3}{5}\right) = \frac{5}{6}x + \frac{1}{2}\). Answer Q1 Try and represent \(\frac{8}{5}\) and \(-\frac{7}{4}\) on a number line. Answer Q1 1. Represent the rational numbers \(\frac{2}{3}\), \(-\frac{5}{4}\) and \(\frac{11}{2}\) on a single number line. Answer Q2 2. Find three distinct rational numbers that lie strictly between \(-\frac{1}{2}\) and \(\frac{1}{4}\). Answer Q3 3. Simplify the expression: \(\left|-\frac{1}{4}\right| + \left|\frac{5}{12}\right\). Answer Q4 4. A tailor has \(15\frac{3}{4}\) metres of fine silk. If making one kurta requires \(2\frac{1}{4}\) metres of silk, exactly how many kurtas can he make? Answer Q5 5. Find three rational numbers between 3.1415 and 3.1416. Answer Q6 *6. Can you think of other way(s) to find a rational number between any two rational numbers? Answer Q1 Can \(\sqrt{2}\) be written as a rational number \(\frac{p}{q}\) ? Answer Q1 Try to prove the irrationality of \(\sqrt{3}\) using the approach of proof by contradiction. Will the same approach work for \(\sqrt{5}\), \(\sqrt{7}\), or \(\sqrt{10}\) ? Answer Q1 We have seen how to obtain a line whose length is a rational number. How do we obtain lines whose lengths are irrational? Answer Q1 Try to extend this method for constructing line segments of lengths \(\sqrt{3}\) and \(\sqrt{5}\) using a ruler and a compass. Generalise this method to construct a line segment of any length of the form \(\sqrt{n}\), where n is a positive integer. Answer Q1 Try to find the decimal expansions of \(\frac{10}{3}\) and \(\frac{11}{12}\). What do you observe about the repetition of the digits after the decimal point? Answer Q1 The decimal expansion of \(\frac{p}{q}\) will be terminating precisely when the prime factors of q are only 2, only 5 or both 2 and 5. Can you explain why? Answer Q1 1. Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: \(\frac{7}{20}\), \(\frac{4}{15}\) and \(\frac{13}{250}\). Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals. Answer Q2 2. Perform the long division for \(\frac{1}{13}\). Identify the repeating block of digits. Does it show cyclic properties if you evaluate \(\frac{2}{13}\)? Now compute \(\frac{3}{13}\), \(\frac{4}{13}\), etc. What do you notice? Answer Q3 3. Classify the following numbers as rational or irrational: (i) \(\sqrt{81}\) Answer Q4 3. Classify the following numbers as rational or irrational: (ii) \(\sqrt{12}\) Answer Q5 3. Classify the following numbers as rational or irrational: (iii) \(0.33333\dots\) Answer Q6 3. Classify the following numbers as rational or irrational: (iv) \(0.123451234512345\dots\) Answer Q7 3. Classify the following numbers as rational or irrational: (v) \(1.01001000100001\dots\) (Notice the pattern: Is it repeating a single block?) Answer Q8 3. Classify the following numbers as rational or irrational: (vi) \(23.560185612239874790120\) Find the explicit fractions in case they are rational. Answer Q9 4. The number \(0.\overline{9}\) (which means 0.99999...) is a rational number. Using algebra (let \(x = 0.\overline{9}\), multiply by 10, and subtract), explain why \(0.\overline{9}\) is exactly equal to 1. Answer Q10 *5. We have seen that the repeating block of \(\frac{1}{7}\) is a cyclic number. Try to find more numbers (n) whose reciprocals (\(\frac{1}{n}\)) produce decimals with repeating blocks that are cyclic. Answer Q1 Consider this puzzle: What is the square root of -1? We know that \(1 \times 1 = 1\). We also know that \((-1) \times (-1) = 1\). There is no Real Number that, when multiplied by itself, results in a negative number. Thus, \(\sqrt{-1}\) cannot exist on number line. Answer Q1 1. Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division: (i) \(\frac{3}{50}\) Answer Q2 1. Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division: (ii) \(\frac{2}{9}\) Answer Q3 2. Prove that \(\sqrt{5}\) is an irrational number. Answer Q4 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (i) \(12.\overline{6}\) Answer Q5 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (ii) \(0.0\overline{120}\) Answer Q6 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (iii) \(3.0\overline{52}\) Answer Q7 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (iv) \(1.\overline{235}\) Answer Q8 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (v) \(0.\overline{23}\) Answer Q9 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (vi) \(2.0\overline{5}\) Answer Q10 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (vii) \(2.1\overline{25}\) Answer Q11 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (viii) \(3.\overline{125}\) Answer Q12 3. Convert the following decimal numbers in the form of \(\frac{p}{q}\). (ix) \(2.16\overline{25}\) Answer Q13 4. Locate the following rational numbers on the number line. (i) \(0.5\overline{32}\) Answer Q14 4. Locate the following rational numbers on the number line. (ii) \(1.\overline{15}\) Answer Q15 5. Find 6 rational numbers between 3 and 4. Answer Q16 6. Find 5 rational numbers between \(\frac{2}{5}\) and \(\frac{3}{5}\). Answer Q17 7. Find 5 rational numbers between \(\frac{1}{6}\) and \(\frac{2}{5}\). Answer Q18 8. If \(\frac{x}{3} + \frac{x}{5} = \frac{16}{15}\), find the rational number \(x\). Answer Q19 9. Let a and b be two non-zero rational numbers such that \(a + b = 0\). Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer. Answer Q20 10. A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form \(\frac{p}{10^{4}}\), where p is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by \(2^{4}\) or \(5^{4}\)? Give reasons. Answer Q21 11. Without performing division, determine whether the decimal expansion of \(\frac{18}{125}\) is terminating or non-terminating. If it terminates, state the number of decimal places. Answer Q22 12. A rational number in its lowest form has denominator \(2^{3} \times 5\). How many decimal places will its decimal expansion have? Explain your answer. Answer Q23 *13. Let \(a = \frac{7}{12}\) and \(b = \frac{5}{6}\). Express both a and b in the form \(\frac{k_{1}}{m}\) and \(\frac{k_{2}}{m}\) where \(k_{1}\), \(k_{2}\) and m are integers and \(k_{2} - k_{1} > 6\). Using the same denominator m, write exactly five distinct rational numbers lying between a and b keeping an integer numerator. Explain why the condition \(k_{2} - k_{1} > n + 1\) is necessary to find n such rational numbers between the two rational numbers a and b using this method. Answer Q24 *14. Three rational numbers x, y, z satisfy \(x + y + z = 0\) and \(xy + yz + zx = 0\). Show that all the rational numbers x, y, z must be simultaneously zero. Answer Q25 *15. Show that the rational number \(\frac{a + b}{2}\) lies between the rational numbers a and b. Answer Q26 16. Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14 which is referred to as the square root spiral. Answer
A rational number is any number that can be written as:
A \(\dfrac{p}{q}\) with \(p, q\) integers and \(q \ne 0\) B \(\dfrac{p}{q}\) with \(q = 0\) C only a whole number D a non-terminating non-repeating decimal
Flashcards & interactive practice 
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