Problem 28 · 2010 Math Kangaroo
Stretch
Number Theory
fraction-to-decimal
\(\sqrt{0.\underbrace{44\ldots4}_{100\text{ times}}}\) is written as a decimal. What is the 100th digit after the decimal point?
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Answer: E — 6
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Hint 1 of 2
A long run of 4s after the point is very close to a familiar fraction.
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Hint 2 of 2
Take the square root of that fraction and read off the repeating digit.
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Approach: compare with the nearby fraction 4/9
- The full repeating decimal \(0.\overline{4}=\tfrac{4}{9}\), and \(\sqrt{\tfrac{4}{9}}=\tfrac{2}{3}=0.\overline{6}\).
- Our number (one hundred 4s) is a hair below \(\tfrac49\), so its root is a hair below \(0.6666\ldots\); the difference only shows up far past the 100th place.
- So through the 100th digit the value reads \(0.6666\ldots\), making the 100th digit 6 — choice E.
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