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