Is zero a rational number? Can you write it in the form \( \dfrac{p}{q} \) where \(p\) and \(q\) are integers and \(q \neq 0\)?
Yes. \(0 = \dfrac{0}{1} = \dfrac{0}{2} = \dfrac{0}{3}\) etc. The denominator \(q\) can also be taken as a negative integer.
Find six rational numbers between 3 and 4.
One method is to write: \(3 = \dfrac{21}{6+1}\) and \(4 = \dfrac{28}{6+1}\). The six rational numbers are \(\dfrac{22}{7}, \dfrac{23}{7}, \dfrac{24}{7}, \dfrac{25}{7}, \dfrac{26}{7}, \dfrac{27}{7}\).
Find five rational numbers between \( \dfrac{3}{5} \) and \( \dfrac{4}{5} \).
Since \( \dfrac{3}{5} = \dfrac{30}{50} \) and \( \dfrac{4}{5} = \dfrac{40}{50} \), five rational numbers between them are: \( \dfrac{31}{50}, \dfrac{32}{50}, \dfrac{33}{50}, \dfrac{34}{50}, \dfrac{35}{50} \).
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.
(i) True, since the collection of whole numbers contains all the natural numbers.
(ii) False, for example \(-2\) is not a whole number.
(iii) False, for example \(\dfrac{1}{2}\) is a rational number but not a whole number.
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.
(i) True, since the collection of real numbers is made up of rational and irrational numbers.
(ii) False, no negative number can be the square root of any natural number.
(iii) False, for example 2 is real but not irrational.
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.
No. For example, \( \sqrt{4} = 2 \) is a rational number.
Show how \( \sqrt{5} \) can be represented on the number line.
Repeat the procedure as in Fig. 1.8 several times. First obtain \( \sqrt{4} \) and then \( \sqrt{5} \).
Classroom activity (Constructing the ‘square root spiral’): Follow the steps to construct the square root spiral starting with a unit segment OP1 and repeatedly drawing perpendicular unit segments to obtain points P1, P2, P3, ... depicting \( \sqrt{2}, \sqrt{3}, \sqrt{4}, ... \).
Construction activity—no specific written answer required.
Write the following in decimal form and say what kind of decimal expansion each has:
(i) \( \dfrac{36}{100} \)
(ii) \( \dfrac{1}{11} \)
(iii) \( \dfrac{4}{8} \)
(iv) \( \dfrac{3}{13} \)
(v) \( \dfrac{2}{11} \)
(vi) \( \dfrac{329}{400} \)
(i) 0.36, terminating.
(ii) 0.\overline{09}, non-terminating repeating.
(iii) 4.125, terminating.
(iv) 0.230769\overline{,} non-terminating repeating.
(v) 0.\overline{18}, non-terminating repeating.
(vi) 0.8225, terminating.
You know that \( \dfrac{1}{7} = 0.142857\overline{} \). Can you predict the decimal expansions of \( \dfrac{2}{7} , \dfrac{3}{7} , \dfrac{4}{7} , \dfrac{5}{7} , \dfrac{6}{7} \) without actually doing the long division? If so, how?
\( \dfrac{2}{7} = 2 \times \dfrac{1}{7} = 0.285714 \)
\( \dfrac{3}{7} = 3 \times \dfrac{1}{7} = 0.428571 \)
\( \dfrac{4}{7} = 4 \times \dfrac{1}{7} = 0.571428 \)
\( \dfrac{5}{7} = 5 \times \dfrac{1}{7} = 0.714285 \)
\( \dfrac{6}{7} = 6 \times \dfrac{1}{7} = 0.857142 \)
Express the following in the form \( \dfrac{p}{q} \), where \(p\) and \(q\) are integers and \(q \neq 0\):
(i) 0.\overline{6}
(ii) 0.47\overline{7}
(iii) 0.001
(i) \( \dfrac{2}{3} \)
(ii) \( \dfrac{43}{90} \)
(iii) \( \dfrac{1}{999} \)
Express 0.99999... in the form \( \dfrac{p}{q} \). Are you surprised by your answer?
1
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \( \dfrac{1}{17} \)? Perform the division to check your answer.
0.0588235294117647
Look at several examples of rational numbers in the form \( \dfrac{p}{q} \) (\(q \neq 0\)), where \(p\) and \(q\) are integers with no common factors other than 1 and having terminating decimal expansions. Can you guess what property \(q\) must satisfy?
The prime factorisation of \(q\) has only powers of 2 or powers of 5 or both.
Write three numbers whose decimal expansions are non-terminating non-recurring.
0.010001000100001..., 0.02002000200020002..., 0.003000300003...
Find three different irrational numbers between the rational numbers \( \dfrac{5}{7} \) and \( \dfrac{9}{11} \).
0.7507500750007500075..., 0.767076700767000767..., 0.808008000800080008...
Classify the following numbers as rational or irrational:
(i) \( \sqrt{23} \)
(ii) \( \sqrt{225} \)
(iii) 0.3796
(iv) 7.478478...
(v) 1.101001000100001...
(i) irrational
(ii) rational
(iii) rational
(iv) rational
(v) irrational
Classify the following numbers as rational or irrational:
(i) \(2 - \sqrt{5}\)
(ii) \((3 + \sqrt{23}) - \sqrt{23}\)
(iii) \(\dfrac{2\sqrt{7}}{7\sqrt{7}}\)
(iv) \(\dfrac{1}{\sqrt{2}}\)
(v) \(2\pi\)
(i) Irrational
(ii) Rational
(iii) Rational
(iv) Irrational
(v) Irrational
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) \((\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})\)
(i) \(6 + 3\sqrt{2} + 2\sqrt{3} + \sqrt{6}\)
(ii) 6
(iii) \(7 + 2\sqrt{10}\)
(iv) 3
Recall that \(\pi\) is defined as the ratio of the circumference (say \(c\)) of a circle to its diameter (say \(d\)). That is, \( \pi = \dfrac{c}{d} \). This seems to contradict the fact that \(\pi\) is irrational. How will you resolve this contradiction?
There is no contradiction. Measurement using a scale only gives approximate rational values, so you may not realise that either \(c\) or \(d\) is irrational.
Represent \(\sqrt{9.3}\) on the number line.
Refer Fig. 1.17.
Rationalise the denominators of the following:
(i) \(\dfrac{1}{\sqrt{7}}\)
(ii) \(\dfrac{1}{\sqrt{7} - \sqrt{6}}\)
(iii) \(\dfrac{1}{\sqrt{5} + \sqrt{2}}\)
(iv) \(\dfrac{1}{\sqrt{7} - 2}\)
(i) \(\dfrac{\sqrt{7}}{7}\)
(ii) \(\sqrt{7} + \sqrt{6}\)
(iii) \(\dfrac{\sqrt{5} - \sqrt{2}}{3}\)
(iv) \(\dfrac{\sqrt{7} + 2}{3}\)
Find the values of the following:
(i) \(64^{1/2}\)
(ii) \(32^{1/5}\)
(iii) \(125^{1/3}\)
(i) 8
(ii) 2
(iii) 5
Find the values of the following:
(i) \(9^{3/2}\)
(ii) \(32^{2/5}\)
(iii) \(16^{3/4}\)
(iv) \(125^{-1/3}\)
(i) 27
(ii) 4
(iii) 8
(iv) \(\dfrac{1}{5}\) [\((125)^{-1/3} = (5^{3})^{-1/3} = 5^{-1}\)]
Simplify the following:
(i) \(2^{2/3} \cdot 2^{1/5}\)
(ii) \(\left(\dfrac{1}{3^{3}}\right)^{7}\)
(iii) \(\dfrac{11^{1/2}}{11^{1/4}}\)
(iv) \(7^{1/2} \cdot 8^{1/2}\)
(i) \(2^{13/15}\)
(ii) \(3^{-21}\)
(iii) \(11^{1/4}\)
(iv) \(56^{1/2}\)