Decimal numbers
You already know how to tell an exact decimal from a recurring one and how to find its generating fraction. We review the mechanics at top speed and take it a step further: using the generating fraction within calculations and looking for patterns in long periods.
The four types of decimal
Classify the following decimal numbers, indicating whether they are terminating, purely recurring, mixed recurring or irrational: $A = 3,14159$ $B = 2,\widehat{7}$…
Classify the following decimal numbers, indicating whether they are terminating, purely recurring, mixed recurring or irrational:
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$A = 3,14159$
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$B = 2,\widehat{7}$
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$C = 0,1\widehat{23}$
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$D = 1,010010001\dots$
Exact or recurring? The denominator decides
Without performing the division, determine the type of decimal expansion (terminating, purely recurring or mixed recurring) corresponding to each of the following…
Without performing the division, determine the type of decimal expansion (terminating, purely recurring or mixed recurring) corresponding to each of the following fractions by analysing the prime factors of their denominators:
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$\frac{7}{20}$
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$\frac{5}{21}$
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$\frac{11}{6}$
Determine the value or values that the digit $x$ can take (from $0$ to $9$) so that the fraction $\frac{x}{15}$ yields a terminating decimal. Reason your answer based on…
Determine the value or values that the digit $x$ can take (from $0$ to $9$) so that the fraction $\frac{x}{15}$ yields a terminating decimal. Reason your answer based on the prime factorisation of the denominator.
The generating fraction: pure and mixed
Find the irreducible generating fraction of the following purely recurring decimal numbers: $3,\widehat{45}$ $0,\widehat{7}$ $12,\widehat{3}$
Find the irreducible generating fraction of the following purely recurring decimal numbers:
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$3,\widehat{45}$
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$0,\widehat{7}$
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$12,\widehat{3}$
Determine the irreducible generating fraction of the following mixed recurring decimal numbers: $0,2\widehat{71}$ $1,0\widehat{5}$ $3,12\widehat{4}$
Determine the irreducible generating fraction of the following mixed recurring decimal numbers:
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$0,2\widehat{71}$
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$1,0\widehat{5}$
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$3,12\widehat{4}$
Operating with recurring decimals without losing precision
Work out the following combined calculation, first converting the decimal numbers to their generating fractions. Express the final result as an irreducible fraction and…
Work out the following combined calculation, first converting the decimal numbers to their generating fractions. Express the final result as an irreducible fraction and as a decimal number:
$1,\widehat{3} + 0,\widehat{5} - 0,1\widehat{6}$
Two vehicles are travelling in a straight line. Vehicle A has a constant speed of $33,\widehat{3}$ m/s, while vehicle B travels at $120$ km/h. If both maintain their…
Two vehicles are travelling in a straight line. Vehicle A has a constant speed of $33,\widehat{3}$ m/s, while vehicle B travels at $120$ km/h. If both maintain their speed for exactly $1,5$ hours, calculate the exact difference in metres between the distances travelled by each. Work with fractions to avoid rounding errors.
Patterns in the period and nested intervals
In divisions with a long period, you can find a particular digit without working out the whole expansion: all you need is the length of the period and integer division.
Find the digit in the $250$th decimal place in the decimal expansion of the fraction $\frac{3}{13}$. Explain the steps followed to find the recurring sequence and…
Find the digit in the $250$th decimal place in the decimal expansion of the fraction $\frac{3}{13}$. Explain the steps followed to find the recurring sequence and calculate the position.
Observe the diagram illustrating the process of nested intervals to locate an irrational number on the number line. Given that the process continues infinitely,…
Observe the diagram illustrating the process of nested intervals to locate an irrational number on the number line. Given that the process continues infinitely, determine which famous irrational number the intervals are approaching and write an interval of width $0,001$ that contains this number.