Write the following in decimal form and say what kind of decimal it is. (i) 100/360 (ii) 1/11 (iii) 1/7 (iv) 1/13. Express the following in the form p/q, where p and q are integers... Write the following in decimal form and say what kind of decimal it is. (i) 100/360 (ii) 1/11 (iii) 1/7 (iv) 1/13. Express the following in the form p/q, where p and q are integers: (a) 0.06 (b) 0.47. Express 0.999... in the form p/q. Find three different irrational numbers between the rational numbers 5/7 and 1/2. Classify the following numbers as rational or irrational: (i) √23 (ii) √22 (iii) 1.101010001000... (iv) 0.376.
Understand the Problem
The question is asking for various mathematical concepts related to decimal representations, particularly for rational and irrational numbers. It requires converting fractions into decimal form and understanding their characteristics.
Answer
(i) $0.\overline{27}$, (ii) $0.\overline{09}$, (iii) $0.\overline{142857}$, (iv) $0.\overline{076923}$, (a) $\frac{3}{50}$, (b) $\frac{47}{100}$, $0.999... = 1$, Irrational: $\frac{\sqrt{2}}{2}, \frac{3}{5}, \frac{\sqrt{3}}{2}$; Classifications: (i) Irrational, (ii) Irrational, (iii) Irrational, (iv) Rational.
Answer for screen readers
(i) $\frac{100}{360} \to 0.\overline{27}$ (repeating)
(ii) $\frac{1}{11} \to 0.\overline{09}$ (repeating)
(iii) $\frac{1}{7} \to 0.\overline{142857}$ (repeating)
(iv) $\frac{1}{13} \to 0.\overline{076923}$ (repeating)
(a) $0.06 \to \frac{3}{50}$
(b) $0.47 \to \frac{47}{100}$
$0.999... = 1$
Three irrational numbers: $\frac{\sqrt{2}}{2}, \frac{3}{5}, \frac{\sqrt{3}}{2}$
Classifications: (i) Irrational, (ii) Irrational, (iii) Irrational, (iv) Rational.
Steps to Solve
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Convert Fractions to Decimal Form
To convert the fractions to decimal form, we perform the division for each fraction:
(i) For $\frac{100}{360}$: [ 100 \div 360 = 0.2777... \text{ (repeating) or } 0.\overline{27} ]
(ii) For $\frac{1}{11}$: [ 1 \div 11 = 0.090909... \text{ (repeating) or } 0.\overline{09} ]
(iii) For $\frac{1}{7}$: [ 1 \div 7 = 0.142857142857... \text{ (repeating) or } 0.\overline{142857} ]
(iv) For $\frac{1}{13}$: [ 1 \div 13 = 0.076923076923... \text{ (repeating) or } 0.\overline{076923} ]
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Convert Decimal to Fraction Form
Now, we express the decimals in the form $\frac{p}{q}$, where ( p ) and ( q ) are integers:
(a) For $0.06$: [ 0.06 = \frac{6}{100} = \frac{3}{50} ]
(b) For $0.47$: [ 0.47 = \frac{47}{100} ]
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Convert Repeating Decimal to Fraction
For $0.999...$: Let ( x = 0.999... )
Then, ( 10x = 9.999... )
Subtracting gives: [ 10x - x = 9.999... - 0.999... \implies 9x = 9 \implies x = 1 ] Thus, ( 0.999... = 1 ).
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Finding Irrational Numbers Between Two Rationals
We need to find three irrational numbers between $\frac{5}{7}$ and $\frac{1}{2}$. Some examples include: [ \frac{\sqrt{2}}{2}, \frac{3}{5}, \frac{\sqrt{3}}{2} ] (All are approximations within the range.)
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Classify Given Numbers as Rational or Irrational
(i) $\sqrt{23}$: Irrational
(ii) $\sqrt{22}$: Irrational
(iii) $1.101010001000...$: Irrational (non-repeating)
(iv) $0.376$: Rational (terminating)
(i) $\frac{100}{360} \to 0.\overline{27}$ (repeating)
(ii) $\frac{1}{11} \to 0.\overline{09}$ (repeating)
(iii) $\frac{1}{7} \to 0.\overline{142857}$ (repeating)
(iv) $\frac{1}{13} \to 0.\overline{076923}$ (repeating)
(a) $0.06 \to \frac{3}{50}$
(b) $0.47 \to \frac{47}{100}$
$0.999... = 1$
Three irrational numbers: $\frac{\sqrt{2}}{2}, \frac{3}{5}, \frac{\sqrt{3}}{2}$
Classifications: (i) Irrational, (ii) Irrational, (iii) Irrational, (iv) Rational.
More Information
The decimal forms discussed here demonstrate the difference between terminating and repeating decimals. Rational numbers can be expressed as fractions, while irrational numbers cannot be expressed as simple fractions.
Tips
- Confusing terminating decimals with repeating decimals.
- Miscalculating or incorrectly simplifying fractions when converting decimals.
- Misidentifying irrational numbers; remember that irrational numbers cannot be expressed as $\frac{p}{q}$ where both are integers.
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