Sure! Let’s go through the details and provide rational and irrational numbers as requested between the given intervals.
i) Between [tex]\(-2\)[/tex] and [tex]\(2\)[/tex]:
Rational numbers can be fractions or integers which have terminating or repeating decimal expansions. Examples include:
- [tex]\(-1.5\)[/tex]
- [tex]\(-1\)[/tex]
- [tex]\(0\)[/tex]
- [tex]\(1\)[/tex]
- [tex]\(1.5\)[/tex]
Irrational numbers, however, have non-terminating and non-repeating decimal expansions. Examples include:
- [tex]\(-\sqrt{2}\)[/tex] which is roughly [tex]\(-1.4142135623730951\)[/tex]
- [tex]\(\sqrt{2}\)[/tex] which is roughly [tex]\(1.4142135623730951\)[/tex]
ii) Between [tex]\(\frac{5}{3}\)[/tex] (approximately 1.6667) and [tex]\(\frac{7}{3}\)[/tex] (approximately 2.3333):
Rational numbers in this interval might be:
- [tex]\( \frac{5}{3} + \frac{i}{6} (\frac{7}{3} - \frac{5}{3}) \)[/tex] for [tex]\( i = 1, 2, 3, 4, 5 \)[/tex]. Specifically:
- approximately [tex]\(1.7778\)[/tex]
- approximately [tex]\(1.8889\)[/tex]
- approximately [tex]\(2.0\)[/tex]
- approximately [tex]\(2.1111\)[/tex]
- approximately [tex]\(2.2222\)[/tex]
Irrational numbers could be in forms of expressions involving irrational constants multiplied in such a way they fall between our bounds:
- [tex]\( \frac{5}{3} + \frac{1}{6} (\frac{6}{3} - \frac{5}{3})\sqrt{2} \)[/tex], which is approximately [tex]\(1.7452\)[/tex]
- [tex]\( \frac{6}{3} + \frac{1}{6} (\frac{7}{3} - \frac{6}{3})\sqrt{3} \)[/tex], which is approximately [tex]\(2.0962\)[/tex]
iii) Between [tex]\(\frac{5}{7}\)[/tex] (approximately 0.7143) and [tex]\(\frac{9}{11}\)[/tex] (approximately 0.8182):
Rational numbers can be:
- [tex]\( \frac{5}{7} + \frac{i}{6} (\frac{9}{11} - \frac{5}{7}) \)[/tex] for [tex]\( i = 1, 2, 3, 4, 5 \)[/tex]. Specifically:
- approximately [tex]\(0.7316\)[/tex]
- approximately [tex]\(0.7489\)[/tex]
- approximately [tex]\(0.7662\)[/tex]
- approximately [tex]\(0.7835\)[/tex]
- approximately [tex]\(0.8009\)[/tex]
Irrational numbers can be:
- [tex]\( \frac{5}{7} + \frac{1}{6} (\frac{8}{9} - \frac{5}{7})\sqrt{2} \)[/tex], which is approximately [tex]\(0.7554\)[/tex]
- [tex]\( \frac{8}{9} + \frac{1}{6} (\frac{9}{11} - \frac{8}{9})\sqrt{3} \)[/tex], which is approximately [tex]\(0.8685\)[/tex]
iv) Between 0.1 and 0.12:
Rational numbers could be:
- 0.102
- 0.104
- 0.106
- 0.108
- 0.110
Irrational numbers could be:
- 0.1010101
- 0.1112112
These examples provide a detailed list of rational and irrational numbers between the given intervals.
