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Chapter 2: Problem 67

Find the lengths of the recurring blocks for: (a) \(\frac{1}{6}, \frac{5}{6}\) (b) \(\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}\) (c) \(\frac{1}{11}, \frac{2}{11}, \frac{3}{11}, \frac{4}{11}, \frac{5}{11}, \frac{6}{11}, \frac{7}{11}, \frac{8}{11}, \frac{9}{11}, \frac{10}{11}\) (d) \(\frac{1}{13}, \frac{2}{13}, \frac{3}{13}, \frac{4}{13}, \frac{5}{13}, \frac{6}{13}, \frac{7}{13}, \frac{8}{13}, \frac{9}{13}, \frac{10}{13}, \frac{11}{13}, \frac{12}{13}\)

Short Answer

Expert verified
(a) 1, (b) 6, (c) 2, (d) 6.

Step by step solution

01

Understand the Problem

We need to find the recurring block lengths for fractions of the form \(\frac{k}{n}\), where \(k\) is an integer and \(n\) is the denominator. The recurring block refers to the digits after the decimal that repeat indefinitely.
02

Analyze \(\frac{1}{6}\) and \(\frac{5}{6}\)

Determine the decimal representations:- \(\frac{1}{6} = 0.1\overline{6}\) (repeats every 1 digit).- \(\frac{5}{6} = 0.8\overline{3}\) (repeats every 1 digit).Hence, both fractions have a repeating block of length 1.
03

Analyze \(\frac{1}{7}, \ldots, \frac{6}{7}\)

Calculate the decimal representations:- \(\frac{1}{7} = 0.\overline{142857}\)- \(\frac{2}{7} = 0.\overline{285714}\)- \(\frac{3}{7} = 0.\overline{428571}\)- \(\frac{4}{7} = 0.\overline{571428}\)- \(\frac{5}{7} = 0.\overline{714285}\)- \(\frac{6}{7} = 0.\overline{857142}\)All have a recurring block length of 6.
04

Analyze \(\frac{1}{11}, \ldots, \frac{10}{11}\)

Find the decimal representations:- \(\frac{1}{11} = 0.\overline{09}\)- \(\frac{2}{11} = 0.\overline{18}\)- \(\frac{3}{11} = 0.\overline{27}\)- \(\frac{4}{11} = 0.\overline{36}\)- \(\frac{5}{11} = 0.\overline{45}\)- \(\frac{6}{11} = 0.\overline{54}\)- \(\frac{7}{11} = 0.\overline{63}\)- \(\frac{8}{11} = 0.\overline{72}\)- \(\frac{9}{11} = 0.\overline{81}\)- \(\frac{10}{11} = 0.\overline{90}\)Each fraction has a repeating block of length 2.
05

Analyze \(\frac{1}{13}, \ldots, \frac{12}{13}\)

Determine their decimal representations:- \(\frac{1}{13} = 0.\overline{076923}\)- \(\frac{2}{13} = 0.\overline{153846}\)- \(\frac{3}{13} = 0.\overline{230769}\)- \(\frac{4}{13} = 0.\overline{307692}\)- \(\frac{5}{13} = 0.\overline{384615}\)- \(\frac{6}{13} = 0.\overline{461538}\)- \(\frac{7}{13} = 0.\overline{538461}\)- \(\frac{8}{13} = 0.\overline{615384}\)- \(\frac{9}{13} = 0.\overline{692307}\)- \(\frac{10}{13} = 0.\overline{769230}\)- \(\frac{11}{13} = 0.\overline{846153}\)- \(\frac{12}{13} = 0.\overline{923076}\)All have a repeating block length of 6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Decimals
Fractions are a way to represent parts of a whole. When we convert a fraction to a decimal, we often encounter either a terminating or a repeating decimal. Fraction decimals describe this conversion. Terminating decimals finish after a certain number of digits, while repeating decimals continue indefinitely. For example, \'\(\frac{1}{4} = 0.25\'\) is a terminating decimal, but \'\(\frac{1}{3} = 0.\overline{3}\'\) is a repeating decimal. Each can have different lengths of repeating sequences, known as recurring blocks. When converting fractions to decimals, dividing the numerator by the denominator will show if the decimal representation is terminating or repeating. Keeping an eye on the sequence of digits can help in determining the type of decimal you have.
Recurring Blocks
Recurring blocks in decimals are the sequences of digits that repeat indefinitely in non-terminating decimals. Understanding recurring blocks is crucial when working with fraction decimals. For instance, when you divide 1 by 7, you get \'\(0.\overline{142857}\'\), where \'\(142857\'\) is the recurring block. Recognizing and pointing out these blocks helps to identify patterns in fractions converted to decimals. These blocks are not random and directly correlates with the denominator of the fraction. The length of the recurring block is influenced by factors like whether the denominator has factors other than 2 or 5, which permits repetition. This insight helps bridge understanding to more complex mathematical concepts.
Decimal Representation
Decimals are another way to represent fractions. With decimal representation, every fraction can be expressed in the form of digits separated by a decimal point. It unveils patterns or repetitions within numbers. When a fraction is expressed as a decimal that repeats, such as \'\(\frac{2}{11} = 0.\overline{18}\'\), the series of digits repeating is important. The decimal point represents the division, where the position of the digits indicates parts smaller than one. This representation simplifies arithmetic and enhances comprehension in various mathematical situations. It's a system that allows numbers to be expressed in a potentially more understandable way for quick calculations and assessment of size or order of fractions.
Prime Denominators
A prime number is a number greater than 1 with no divisors other than 1 and itself. Prime denominators in fractions often lead to repeating decimals in their decimal form. When the denominator of a fraction is a prime number, as seen with 7, 11, or 13, the converted decimal form generally has recurring elements. The unique property of primes is their non-divisibility by other numbers, contributing to the complexity of the decimal form they create. Given this, recognizing when denominators are prime assists in predicting if a fraction will have a repeating decimal or not, essentially linking prime numbers directly to behaviors in decimal expressions.

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Most popular questions from this chapter

(a)(i) Generate the first twelve terms of the Fibonacci sequence: $$F_{0}, F_{1}, \ldots, F_{11}$$ (ii) Use this to generate the first eleven terms of the sequence of "differences" between successive Fibonacci numbers. Then generate the first ten terms of the sequence of "differences between successive differences". (iii) Find an expression for the \(m^{\text {th }}\) term of the \(k^{\text {th }}\) sequence of differences. (b)(i) Generate the first twelve terms of the sequence of powers of 2 : $$2^{0}, 2^{1}, 2^{2}, \ldots, 2^{11}$$ (ii) Use this to generate the first eleven terms of the sequence of "differences" between successive powers of \(2 .\) Then generate the first ten terms of the sequence of "differences between successive differences". (iii) Find an expression for the \(m^{\text {th }}\) term of the \(k^{\text {th }}\) sequence of differences. (b)(i) Generate the first twelve terms of the sequence of powers of 2 : $$2^{0}, 2^{1}, 2^{2}, \ldots, 2^{11}$$ (ii) Use this to generate the first eleven terms of the sequence of "differences" between successive powers of 2 . Then generate the first ten terms of the sequence of "differences between successive differences". (iii) Find an expression for the \(m^{\text {th }}\) term of the \(k^{\text {th }}\) sequence of differences.

The decimals listed here all continue forever, recurring in the expected way. Calculate: (a) \(0.55555 \cdots+0.66666 \cdots=\) (b) \(0.99999 \cdots+0.11111 \cdots=\) (c) \(1.11111 \cdots-0.22222 \cdots=\) (d) \(0.33333 \cdots \times 0.66666 \cdots=\) (e) \(1.22222 \cdots \times 0.818181 \cdots=\)

Let \(N\) be a positive integer written in base \(2 .\) Describe and justify a simple test, based on the digits of \(N_{\text {base } 2}\) : (i) for \(N\) to be divisible by 2 (ii) for \(N\) to be divisible by 3 (iii) for \(N\) to be divisible by 4 (iv) for \(N\) to be divisible by 5 .

Players \(A\) and \(B\) specify a real number between 0 and \(1 .\) The first player \(A\) tries to make sure that the resulting number is rational; the second player \(B\) tries to make sure that the resulting number is irrational. In each of the following scenarios, decide whether either player has a strategy that guarantees success. (a) Can either player guarantee a "win" if the two players take turns to specify successive digits: first \(A\) chooses the entry in the first decimal place, then \(B\) chooses the entry in the second decimal place, then \(A\) chooses the entry in the third decimal place, and so on? (b) Can either player guarantee a win if \(A\) chooses the digits to go in the odd-numbered places, and (entirely separately) \(B\) chooses the digits to go in the even-numbered places? (c) What if \(A\) chooses the digits that go in almost all the places, but allows \(B\) to choose the digits that are to go in a sparse infinite collection of decimal places (e.g. the prime-numbered positions; or the positions numbered by the powers of \(2 ;\) or \(\ldots) ?\) (d) What if \(A\) controls the choice of all but a finite number of decimal digits?

(a)(i) Start the process of generating prime numbers by choosing your favourite small prime number and call it \(p_{1}\). (ii) Then define \(n_{1}=p_{1}+1\). (b)(i) Since \(n_{1}>1, n_{1}\) must be divisible by some prime. Explain why \(p_{1}\) is not a factor of \(n_{1}\). (What is the remainder when we divide \(n_{1}\) by \(p_{1}\) ?) (ii) Let \(p_{2}\) be the smallest prime factor of \(n_{1}\). (iii) Define \(n_{2}=p_{1} \times p_{2}+1\) (c)(i) Since \(n_{2}>1, n_{2}\) must be divisible by some prime. Explain why \(p_{1}\) and \(p_{2}\) are not factors of \(n_{2}\). (What is the remainder when we divide \(n_{1}\) by \(p_{1},\) or by \(\left.p_{2} ?\right)\) (ii) Let \(p_{3}\) be the smallest prime factor of \(n_{2}\). (iii) Define \(n_{3}=p_{1} \times p_{2} \times p_{3}+1\) (d) Suppose we have constructed \(k\) distinct prime numbers \(p_{1}, p_{2}, p_{3}, \ldots, p_{k}\). Explain how we can always construct a prime number \(p_{k+1}\) different from \(p_{1}, p_{2}, \ldots, p_{k}\) (e) Apply the above process with \(p_{1}=2\) to find \(p_{2}, p_{3}, p_{4}, p_{5}\).
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