Exercise 1.3 Practice
Real Numbers and Decimal Expansions
Q1: Decimal Expansion
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Write the following in decimal form and say what kind of decimal expansion each has:
(i) $\frac{36}{100}$
(ii) $\frac{1}{11}$
(iii) $4\frac{1}{8}$
(iv) $\frac{3}{13}$
(v) $\frac{2}{11}$
(vi) $\frac{329}{400}$
(i) $\frac{36}{100}$
(ii) $\frac{1}{11}$
(iii) $4\frac{1}{8}$
(iv) $\frac{3}{13}$
(v) $\frac{2}{11}$
(vi) $\frac{329}{400}$
(i) $\frac{36}{100} = 0.36$ (Terminating)
(ii) $\frac{1}{11} = 0.0909... = 0.\overline{09}$ (Non-terminating recurring)
(iii) $4\frac{1}{8} = 4.125$ (Terminating)
(iv) $\frac{3}{13} = 0.230769... = 0.\overline{230769}$ (Non-terminating recurring)
(v) $\frac{2}{11} = 0.1818... = 0.\overline{18}$ (Non-terminating recurring)
(vi) $\frac{329}{400} = 0.8225$ (Terminating)
Q2: Prediction
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You know that $\frac{1}{7} = 0.\overline{142857}$. Can you predict the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ without actually doing the long division? If so, how?
Yes. Multiply the value of $1/7$ by the numerator.
$\frac{2}{7} = 2 \times 0.\overline{142857} = 0.\overline{285714}$
$\frac{3}{7} = 3 \times 0.\overline{142857} = 0.\overline{428571}$
$\frac{4}{7} = 4 \times 0.\overline{142857} = 0.\overline{571428}$
$\frac{5}{7} = 5 \times 0.\overline{142857} = 0.\overline{714285}$
$\frac{6}{7} = 6 \times 0.\overline{142857} = 0.\overline{857142}$
Q3: Express as p/q
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Express the following in the form $p/q$, where $p$ and $q$ are integers and $q \neq 0$:
(i) $0.\overline{6}$
(ii) $0.4\overline{7}$
(iii) $0.\overline{001}$
(i) $0.\overline{6}$
(ii) $0.4\overline{7}$
(iii) $0.\overline{001}$
(i) $0.\overline{6}$: Let $x = 0.666...$
$10x = 6.666... \Rightarrow 9x = 6 \Rightarrow x = 2/3$.
$10x = 6.666... \Rightarrow 9x = 6 \Rightarrow x = 2/3$.
(ii) $0.4\overline{7}$: Let $x = 0.4777...$
$10x = 4.777...$ and $100x = 47.777...$
$90x = 43 \Rightarrow x = 43/90$.
$10x = 4.777...$ and $100x = 47.777...$
$90x = 43 \Rightarrow x = 43/90$.
(iii) $0.\overline{001}$: Let $x = 0.001001...$
$1000x = 1.001001... \Rightarrow 999x = 1 \Rightarrow x = 1/999$.
$1000x = 1.001001... \Rightarrow 999x = 1 \Rightarrow x = 1/999$.
(i) 2/3, (ii) 43/90, (iii) 1/999
Q4: 0.999...
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Express $0.99999...$ in the form $p/q$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Let $x = 0.99999...$
Multiply by 10: $10x = 9.99999...$
Subtract x from 10x:
$10x - x = (9.999...) - (0.999...)$
$9x = 9$
$x = 1$
$10x - x = (9.999...) - (0.999...)$
$9x = 9$
$x = 1$
Discussion: Since 0.999... goes on forever, there is no gap between 1 and 0.999..., so they are equal.
1
Q5: Repeating Block
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What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $1/17$? Perform the division to check your answer.
Maximum digits: The maximum number of digits in the repeating block is always less than the divisor. Here divisor is 17, so max digits = $17 - 1 = 16$.
Division:
$1 \div 17 = 0.0588235294117647...$
$1 \div 17 = 0.0588235294117647...$
The repeating block is $\overline{0588235294117647}$.
16 digits
Q6: Property of q
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Look at several examples of rational numbers in the form $p/q$ ($q \neq 0$) where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
Examples:
$1/2 = 0.5$ (q=2)
$1/5 = 0.2$ (q=5)
$1/4 = 0.25$ (q=2²)
$1/10 = 0.1$ (q=2×5)
$1/2 = 0.5$ (q=2)
$1/5 = 0.2$ (q=5)
$1/4 = 0.25$ (q=2²)
$1/10 = 0.1$ (q=2×5)
Property: The prime factorisation of $q$ must contain only powers of 2 or powers of 5 or both.
$q = 2^n 5^m$
Q7: Non-terminating Examples
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Write three numbers whose decimal expansions are non-terminating non-recurring.
These are irrational numbers. We can construct them by creating a pattern that never repeats.
1. $0.101001000100001...$
2. $0.202002000200002...$
3. $\pi = 3.14159...$ or $\sqrt{2} = 1.4142...$
Q8: Irrational Numbers
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Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
Convert to Decimal:
$\frac{5}{7} = 0.\overline{714285}$
$\frac{9}{11} = 0.\overline{81}$
$\frac{5}{7} = 0.\overline{714285}$
$\frac{9}{11} = 0.\overline{81}$
Find Irrationals: We need non-terminating, non-recurring numbers between 0.714... and 0.818...
Examples:
1. $0.720720072000...$
2. $0.750750075000...$
3. $0.808008000...$
1. $0.720720072000...$
2. $0.750750075000...$
3. $0.808008000...$
Q9: Classify Numbers
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Classify the following numbers as rational or irrational:
(i) $\sqrt{23}$
(ii) $\sqrt{225}$
(iii) $0.3796$
(iv) $7.478478...$
(v) $1.101001000100001...$
(i) $\sqrt{23}$
(ii) $\sqrt{225}$
(iii) $0.3796$
(iv) $7.478478...$
(v) $1.101001000100001...$
(i) $\sqrt{23}$: 23 is not a perfect square. Irrational.
(ii) $\sqrt{225}$: $\sqrt{225} = 15$. Rational.
(iii) $0.3796$: Terminating decimal. Rational.
(iv) $7.478478...$: Non-terminating but recurring ($7.\overline{478}$). Rational.
(v) $1.101001...$: Non-terminating and non-recurring. Irrational.