Repeating decimals are a natural place to see infinity in ordinary numbers. A decimal like 0.777... never ends, but it still has an exact value on the number line. Calculus gives a clear way to understand this value by treating the decimal as an infinite series.
This matters because the same idea is used in limits, power series, and many approximation methods.
Key Facts
- 0.777... = 0.7 + 0.07 + 0.007 + ...
- An infinite geometric series has the form a + ar + ar^2 + ar^3 + ...
- If |r| < 1, then a + ar + ar^2 + ... = a / (1 - r)
- For 0.777..., a = 0.7 and r = 0.1, so 0.777... = 0.7 / 0.9 = 7/9
- For a repeating block with n digits, the common ratio is r = 10^(-n)
- 0.abcabcabc... = abc / 999 when abc is a three digit repeating block
Vocabulary
- Repeating decimal
- A repeating decimal is a decimal whose digits eventually repeat in a fixed pattern forever.
- Infinite series
- An infinite series is a sum with infinitely many terms.
- Geometric series
- A geometric series is a series in which each term is found by multiplying the previous term by the same common ratio.
- Common ratio
- The common ratio is the constant multiplier between consecutive terms in a geometric sequence or series.
- Limit
- A limit describes the value that a sequence, function, or partial sum approaches as the input or number of terms grows.
Common Mistakes to Avoid
- Treating 0.777... as approximately 7/9 only, which is wrong because the infinite geometric series sums exactly to 7/9.
- Using a = 7 instead of a = 0.7 for 0.777..., which shifts the decimal place and makes the sum ten times too large.
- Forgetting the condition |r| < 1, which is wrong because the formula a / (1 - r) only gives a finite sum when the infinite geometric series converges.
- Using 99 instead of 999 for a three digit repeating block, which is wrong because the denominator must match the number of repeating digits.
Practice Questions
- 1 Write 0.444... as an infinite geometric series and find its fraction value.
- 2 Convert 0.363636... to a fraction by identifying the repeating block and using a geometric series.
- 3 Explain why 0.999... equals 1 using the geometric series formula rather than rounding.