46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).


51.0.456̅ = 0.456456456…

Verified step by step guidance

1. Identify the repeating decimal part. Here, the repeating block is "456", which repeats every 3 digits after the decimal point.
2. Express the decimal as a sum of its non-repeating part plus the repeating part as a geometric series. Since the decimal is 0.456456456..., it can be written as 0.456 + 0.000456 + 0.000000456 + ...
3. Rewrite the repeating part as a geometric series: \( 0.456 + 0.456 \times 10^{-3} + 0.456 \times 10^{-6} + \cdots \). This is a geometric series with first term \( a = 0.456 \) and common ratio \( r = 10^{-3} \).
4. Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to find the sum of the repeating part: \( S = \frac{0.456}{1 - 10^{-3}} \).
5. Convert the decimal 0.456 to a fraction (\( \frac{456}{1000} \)) and simplify the expression \( \frac{0.456}{1 - 10^{-3}} \) to get the fraction form of the repeating decimal.

Key Concepts

Repeating Decimals

A repeating decimal is a decimal number in which a sequence of digits repeats infinitely. For example, 0.456̅ means the digits '456' repeat endlessly. Understanding this pattern is essential to express the decimal as a series or fraction.

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. Repeating decimals can be represented as geometric series by expressing the repeating block as terms with decreasing powers of 1/10.

Conversion of Geometric Series to Fraction

The sum of an infinite geometric series with first term 'a' and common ratio 'r' (|r|<1) is a/(1-r). This formula allows converting the geometric series representation of a repeating decimal into a fraction, expressing it as a ratio of two integers.