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Chapter 2: Problem 16

Express \(\frac{1}{7}\) and \(\frac{2}{19}\) as periodic decimals.

Short Answer

Expert verified
The periodic decimal forms of \(\frac{1}{7}\) and \(\frac{2}{19}\) are 0.\(142857\) and 0.\(105263157894\), respectively.

Step by step solution

01

Converting \(\frac{1}{7}\) to a decimal

To do this, the numerator 1 is divided by the denominator 7. This gives 0.142857142857..., which means that the decimal representation of \(\frac{1}{7}\) is 0.142857142857 and so on, and it repeats every 6 digits.
02

Converting \(\frac{2}{19}\) to a decimal

Just like in step 1, divide 2 by 19. This gives 0.10526315789473684..., which means that the decimal representation of \(\frac{2}{19}\) is 0.105263157894 and so on, and it repeats every 18 digits.
03

Expressing the answers in correct form

The decimal representation of \(\frac{1}{7}\) can be expressed as 0.\(142857\), the underscore indicating the repeating part. The decimal representation of \(\frac{2}{19}\) can be expressed as 0.\(105263157894\).

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Most popular questions from this chapter

Show that sup \(\\{1-1 / n: n \in \mathbb{N}\\}=1\)

(a) Give the first four digits in the binary representation of \(\frac{1}{3}\). (b) Give the complete binary representation of \(\frac{1}{3}\).

If \(0

Modify the argument in Theorem \(2.4 .7\) to show that if \(a>0\), then there exists a positive real number \(z\) such that \(z^{2}=a\)

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