Solve:
Question 1: Which of the following numbers are irrational numbers?
Solution:
(a) \(\sqrt{3}\), \(\sqrt{7}\), \(\frac{3}{5}\), \(\frac{6}{7}\)
Irrational numbers: \(\sqrt{3}\), \(\sqrt{7}\).
(b) \(\sqrt{4}\), \(7\), \(\frac{3}{5}\), \(\sqrt{3}\)
Irrational numbers: \(\sqrt{3}\).
(c) \(\sqrt{36}\), \(-1\), \(\sqrt{3}\), \(0.8\)
Irrational numbers: \(\sqrt{3}\).
(d) \(\sqrt{4}\), \(\frac{3}{5}\), \(\sqrt{5}\), \(\frac{5}{7}\)
Irrational numbers: \(\sqrt{5}\).
(e) \(\frac{27}{5}\), \(\frac{5}{2}\), \(0.2\), \(\sqrt{3}\)
Irrational numbers: \(\sqrt{3}\).
(f) \(\frac{5}{8}\), \(\sqrt{100}\), \(\sqrt{11}\), (\(\sqrt{5} \times \sqrt{5}\))
Irrational numbers: \(\sqrt{11}\).
(g) \(\frac{22}{7}\), \(\frac{4}{5}\), \(\sqrt{8}\), \(\sqrt{9}\)
Irrational numbers: \(\sqrt{8}\).
(h) \(0\), \(\frac{4}{7}\), \(0.4\), \(\sqrt{2}\)
Irrational numbers: \(\sqrt{2}\).
(i) \(10\), \(\frac{1}{3}\), \(0.44\), \(\sqrt{11}\)
Irrational numbers: \(\sqrt{11}\).
Question 2: Identify whether each of the following numbers is rational or irrational.
Solution:
(a) 3.5
3.5 = \(\frac{35}{10} = \frac{7}{2}\), a terminating decimal, rational.
(b) -1.75
-1.75 = \(-\frac{175}{100} = -\frac{7}{4}\), a terminating decimal, rational.
(c) 0.333...
0.333... is a repeating decimal, equal to \(\frac{1}{3}\), rational.
(d) \(\sqrt{25}\)
\(\sqrt{25} = 5\), an integer, rational.
(e) \(\sqrt{3}\)
\(\sqrt{3} \approx 1.732050...\) (non-repeating, non-terminating), irrational.
(f) 0.326...
0.326... is a non-repeating, non-terminating decimal, so it is irrational.
Question 3: Identify whether each of the following numbers is rational or irrational.
Solution:
(a) \(\sqrt{16}\)
\(\sqrt{16} = 4\), an integer, rational.
(b) \(\frac{2}{3}\)
\(\frac{2}{3} \approx 0.666...\) (repeating decimal), a fraction, rational.
(c) \(\sqrt{22}\)
\(\sqrt{22} \approx 4.690415...\) (non-repeating, non-terminating), irrational.
(d) -8
-8 = \(-\frac{8}{1}\), an integer, rational.
(e) 0.666...
0.666... is a repeating decimal, equal to \(\frac{2}{3}\), rational.
(f) \(\sqrt{64}\)
\(\sqrt{64} = 8\), an integer, rational.
Question 4: Identify whether each of the following numbers is rational or irrational.
Solution:
(a) 1.5
1.5 = \(\frac{15}{10} = \frac{3}{2}\), a terminating decimal, rational.
(b) \(\frac{5}{7}\)
\(\frac{5}{7} \approx 0.714285714285...\) (repeating decimal), a fraction, rational.
(c) \(\sqrt{5}\)
\(\sqrt{5} \approx 2.236067...\) (non-repeating, non-terminating), irrational.
(d) \(-\sqrt{49}\)
\(\sqrt{49} = 7\), so \(-\sqrt{49} = -7\), an integer, rational.
(e) 0.5644217...
0.5644217... = irrational.
(f) \(\frac{1}{23}\)
\(\frac{1}{23} \approx 0.043478260869565217391304347826...\) (repeating decimal), a fraction, rational.
Question 5: Identify whether each of the following numbers is rational or irrational.
Solution:
(a) 0.65
0.65 = \(\frac{65}{100} = \frac{13}{20}\), a terminating decimal, rational.
(b) \(\frac{7}{9}\)
\(\frac{7}{9} \approx 0.777...\) (repeating decimal), a fraction, rational.
(c) \(\sqrt{7}\)
\(\sqrt{7} \approx 2.645751...\) (non-repeating, non-terminating), irrational.
(d) \(-\sqrt{36}\)
\(\sqrt{36} = 6\), so \(-\sqrt{36} = -6\), an integer, rational.
(e) \(\pi\)
\(\pi \approx 3.1415926535...\) (non-repeating, non-terminating), irrational.
(f) 0.98762
0.98762 = \(\frac{98762}{100000} = \frac{49381}{50000}\), a terminating decimal, rational.
Question 6: Show the following irrational numbers in a number line:
a. \(\sqrt{3}\)
b. \(\sqrt{2}\)
c. \(\sqrt{5}\)
d. \(\sqrt{7}\)
e. -\(\sqrt{2}\)
f. 2\(\sqrt{3}\)
Question 9: Express the following decimals as fractions:
(a) 0.\(\overline{4}\)
Solution:
Let x = 0.\(\overline{4}\) = 0.4444...... (i)
Multiplying (i) by 10
10x = 4.4444...... (ii)
Subtracting (i) from (ii),
9x = 4
or, x = \(\frac{4}{9}\)
Thus, 0.\(\overline{4}\) = \(\frac{4}{9}\)
(b) 2.\(\overline{3}\)
Solution:
Let x = 2.\(\overline{3}\) = 2.3333...... (i)
Multiplying (i) by 10,
10x = 23.3333...... (ii)
Subtracting (i) from (ii):
9x = 21
ir, x = \(\frac{21}{9}\) = \(\frac{7}{3}\)
Thus, 2.\(\overline{3}\) = \(\frac{7}{3}\) = \(2\frac{1}{3}\)
(c) 12.\(\overline{5}\)
Solution:
Let x = 12.\(\overline{5}\) = 12.5555...... (i)
Multiplying (i) by 10
10x = 125.5555...... (ii)
Subtracting (i) from (ii),
9x = 113
or, x = \(\frac{113}{9}\)
Thus, 12.\(\overline{5}\) = \(\frac{113}{9}\) = \(12\frac{5}{9}\)
(d) 1.2\(\overline{7}\)
Solution:
Let x = 1.2\(\overline{7}\) = 1.27777...... (i)
10x = 12.7777...... (ii)
100x = 127.7777...... (iii)
Subtracting (ii) from (iii):
90x = 115
or, x = \(\frac{115}{90}\) = \(\frac{23}{18}\)
Thus, 1.2\(\overline{7}\) = \(\frac{23}{18}\) = \(1\frac{5}{18}\)
(e) 2.3\(\overline{4}\)
Solution:
Let x = 2.3\(\overline{4}\) = 2.34444...... (i)
10x = 23.4444...... (ii)
100x = 234.4444...... (iii)
Subtracting (ii) from (iii):
90x = 211
x = \(\frac{211}{90}\)
Thus, 2.3\(\overline{4}\) = \(\frac{211}{90}\) = \(2\frac{31}{90}\)
(f) 4.\(\overline{57}\)
Solution:
Here, we have 4.\(\overline{57}\) = 4.575757......
Let x = 4.\(\overline{57}\) = 4.575757...... (i)
Multiplying both sides of (i) by 100:
100x = 100 × 4.\(\overline{57}\) = 457.575757...... (ii)
Subtracting (i) from (ii):
100x - x = 457.575757...... - 4.575757......
or, 99x = 453
or, x = \(\frac{453}{99}\)
or, x = \(\frac{151}{33}\) = \(4\frac{19}{33}\)
Thus, the fraction is \(4\frac{19}{33}\).
(g) 13.\(\overline{34}\)
Solution:
Here, we have 13.\(\overline{34}\) = 13.343434......
Let x = 13.\(\overline{34}\) = 13.343434...... (i)
Multiplying both sides of (i) by 100:
100x = 100 × 13.\(\overline{34}\) = 1334.343434...... (ii)
Subtracting (i) from (ii):
100x - x = 1334.343434...... - 13.343434......
or, 99x = 1321
or, x = \(\frac{1321}{99}\)
Thus, the fraction is \(\frac{1321}{99}\) = \(13\frac{34}{99}\).
(h) 12.2\(\overline{75}\)
Solution:
Here, we have 12.2\(\overline{75}\) = 12.2757575......
Let x = 12.2\(\overline{75}\) = 12.2757575...... (i)
10x = 122.757575...... (ii)
1000x = 12275.757575...... (iii)
Subtracting (ii) from (iii),
1000x - 10x = 12275.757575...... - 122.757575......
or, 990x = 12153
or, x = \(\frac{12153}{990}\)
or, x = \(\frac{4051}{330}\) = \(12\frac{91}{330}\)
Thus, the fraction is \(12\frac{91}{330}\).
(i) 11.\(\overline{374}\)
Solution:
Here, we have 11.\(\overline{374}\) = 11.374374374......
Let x = 11.\(\overline{374}\) = 11.374374374...... (i)
Multiplying both sides of (i) by 1000,
1000x = 1000 × 11.\(\overline{374}\) = 11374.374374374...... (ii)
Subtracting (i) from (ii),
1000x - x = 11374.374374374...... - 11.374374374......
or, 999x = 11363
or, x = \(\frac{11363}{999}\)
or, x = \(\frac{11363}{999}\)
Thus, the fraction is \(\frac{11363}{999}\) = \(11\frac{374}{999}\).
(j) 5.\(\overline{678}\)
Solution:
Here, we have 5.\(\overline{678}\) = 5.678678678......
Let x = 5.\(\overline{678}\) = 5.678678678...... (i)
Multiplying both sides of (i) by 1000,
1000x = 1000 × 5.\(\overline{678}\) = 5678.678678678...... (ii)
Subtracting (i) from (ii),
1000x - x = 5678.678678678...... - 5.678678678......
or, 999x = 5673
or, x = \(\frac{5673}{999}\)
or, x = \(\frac{1891}{333}\)
Thus, the fraction is \(\frac{1891}{333}\) = \(5\frac{226}{333}\).
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