Project Euler

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# [[Problem 57]]

## Square Root Convergents

It is possible to show that the square root of two can be expressed as an infinite continued fraction.

<p style="text-align:center">$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$</p>

By expanding this for the first four iterations, we get:

$1 + \frac 1 2 = \frac  32 = 1.5$

$1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4$

$1 + \frac 1 {2 + \frac 1 {2+\frac 1 2}} = \frac {17}{12} = 1.41666 \dots$

$1 + \frac 1 {2 + \frac 1 {2+\frac 1 {2+\frac 1 2}}} = \frac {41}{29} = 1.41379 \dots$

The next three expansions are $\frac {99}{70}$, $\frac {239}{169}$, and $\frac {577}{408}$, but the eighth expansion, $\frac {1393}{985}$, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.

In the first one-thousand expansions, how many fractions contain a numerator with more digits than the denominator?

## 平方根の近似分数

2の平方根は無限に続く連分数で表すことができる.

<div style="text-align:center">√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...</div>

最初の4回の繰り返しを展開すると以下が得られる.

1 + 1/2 = 3/2 = 1.5

1 + 1/(2 + 1/2) = 7/5 = 1.4

1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...

1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...

次の3つの項は99/70, 239/169, 577/408である. 第8項は1393/985である. これは分子の桁数が分母の桁数を超える最初の例である.

最初の1000項を考えたとき, 分子の桁数が分母の桁数を超える項はいくつあるか?

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