Chapter 1 Fundamentals 1.1 Real Numbers Express Each Repeating Decimal as a Fraction

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Precalculus: Converting a Repeating Decimal into a Fraction

This lesson demonstrates how to convert a repeating decimal into an exact fraction. The example comes from a precalculus text and illustrates how concepts from algebra, sequences, series, and geometric series can be connected to produce an exact answer. :contentReference[oaicite:0]{index=0}

The Problem

Express the repeating decimal as a fraction:

$0.\overline{57}$

The answer provided by the textbook is:

$\frac{19}{33}$

The goal is to verify this result using a geometric series approach. :contentReference[oaicite:1]{index=1}

Understanding the Repeating Pattern

The repeating decimal can be expanded as:

$0.\overline{57}=0.57+0.0057+0.000057+\cdots$

Writing each term as a fraction gives:

$0.\overline{57}=\frac{57}{100}+\frac{57}{10000}+\frac{57}{1000000}+\cdots$

Notice that every term contains a factor of 57. :contentReference[oaicite:2]{index=2}

Factor Out the Common Term

Factoring out 57 produces:

$0.\overline{57}=57\left(\frac1{100}+\frac1{10000}+\frac1{1000000}+\cdots\right)$

This can be rewritten as:

$0.\overline{57}=57\sum_{n=0}^{\infty}\left(\frac1{100}\right)^{n+1}$

The repeating decimal has now been transformed into a geometric series. :contentReference[oaicite:3]{index=3}

Many repeating decimals can be converted into fractions by recognizing the hidden geometric series.

The Geometric Series Formula

For a geometric series:

$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}$

provided:

$latex |r|<1&bg=ffffff&fg=000000&s=4$

In this problem:

$a=\frac1{100}, \qquad r=\frac1{100}$

Since:

$latex \left|\frac1{100}\right|<1&bg=ffffff&fg=000000&s=4$

the geometric series converges. :contentReference[oaicite:4]{index=4}

Apply the Formula

Substituting into the geometric series formula:

$0.\overline{57}=57\left(\frac{\frac1{100}}{1-\frac1{100}}\right)$

Simplify the denominator:

$1-\frac1{100}=\frac{99}{100}$

Therefore:

$0.\overline{57}=57\left(\frac{\frac1{100}}{\frac{99}{100}}\right)$

Cancel the factors of 100:

$0.\overline{57}=57\left(\frac1{99}\right)$

Thus:

$0.\overline{57}=\frac{57}{99}$

Reduce the Fraction

Both numerator and denominator are divisible by 3:

$\frac{57}{99}=\frac{19}{33}$

Therefore:

$\boxed{0.\overline{57}=\frac{19}{33}}$

Why This Example Matters

This example demonstrates how ideas from several different mathematical subjects connect together:

Decimals
Fractions
Exponents
Sequences
Series
Geometric Series
Algebraic Manipulation

As students progress through algebra, precalculus, calculus, and differential equations, recognizing these connections becomes increasingly important. :contentReference[oaicite:5]{index=5}

Final Thoughts

Repeating decimals are more than arithmetic curiosities. They provide an excellent introduction to infinite series and show how seemingly different areas of mathematics work together. Recognizing patterns, rewriting expressions, and applying known formulas are core skills that appear throughout higher mathematics. :contentReference[oaicite:6]{index=6}

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