Next | Continued Fraction Arithmetic | 46 |

This is probably the most important property of continued fractions

Suppose you truncate the continued fraction for

*n*You get a

*very good*rational approximation for*n*In fact, you get the

**best possible**rational approximation for*n*(In the sense that no fraction with a smaller denominator is closer)

Consider = [4; 8, 8, 8, 8, ...] = 4.12310562561...:

[4] = 4/1 =4[4; 8] = 33/8 =4.125 [4; 8, 8] = 268/65 =4.1230769... [4; 8, 8, 8] = 2177/528 =4.12310606 [4; 8, 8, 8, 8] = 17684/4289 =4.123105619...

268/65 has a denominator smaller than 4.12 = 412/100

But it is more than 100 times as accurate

Even 33/8 is more accurate than 412/100

Conclusion: You needn't calculate the whole expansion of a continued fraction

If you stop early, you get a

*good*approximation

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