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Chapter 7: Problem 30

Explain why 0.2 and the repeating decimal \(0.199999 \ldots\) both represent the real number \(\frac{1}{5}\).

Short Answer

Expert verified
By representing the repeating decimal 0.1(9) as a variable x, we multiply x by 10 and then subtract the original x to obtain \(9x = 1.8\). Solving for x gives \(x = \frac{1}{5}\). Since 0.2 is also equal to \(\frac{1}{5}\), both 0.2 and 0.1(9) represent the same real number \(\frac{1}{5}\).

Step by step solution

01

Variable Representation of 0.1(9)

Let x represent the decimal number 0.1(9): \( x = 0.19999 \ldots \)
02

Multiply x by 10

Multiply the given x by 10: \( 10x = 1.9999 \ldots \)
03

Subtract the original x from 10x

Subtract x from 10x: \( (10x - x) = (1.9999 \ldots - 0.19999 \ldots) \) \( 9x = 1.8 \)
04

Solve for x

Divide both sides of the equation by 9: \( \frac{9x}{9} = \frac{1.8}{9} \) Which gives: \( x = \frac{1}{5} \)
05

Compare Decimal and Fraction Representations

Now we have: \( 0.1(9) = \frac{1}{5} \) And, as it is known: \( 0.2 = \frac{1}{5} \)
06

Conclusion

Since both 0.2 and 0.1(9) have the same fractional representation, which is 1/5, they both represent the same real number, 1/5. Therefore, we have shown that 0.2 and the repeating decimal 0.199999... represent the real number \(\frac{1}{5}\).

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