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The fractions 22/7, 333/106, and 355/113 as approximations to pi

Example

Among the classical fractions $\frac{22}{7}, \frac{333}{106}, \frac{355}{113},$ the best approximation to $π$ is $355 / 113$ , and it is already forced by Legendre's criterion.

Facts & Assumptions

Given: The real number $π$ and the three rational numbers above.

[L1]

The number $π$ is irrational (Ivan Niven, "A simple proof that pi is irrational", Bulletin of the AMS 53 (1947), 509).

[F1]

If $α$ is irrational and a reduced rational number $r / s$ satisfies $∣ α - r / s ∣ < 1 / (2 s^{2})$ , then it is a convergent of $α$ (Legendre's criterion for convergents).

[F2]

A convergent of an irrational number is the best approximation among all rationals with smaller next denominator (Convergents are best rational approximations of the first kind).

Verification

technique · direct

1.1

Direct decimal comparison gives. [given, algebra] $∣ π - \frac{22}{7} ∣ \approx 1.264 \times 10^{- 3}, ∣ π - \frac{333}{106} ∣ \approx 8.322 \times 10^{- 5},$ $∣ π - \frac{355}{113} ∣ \approx 2.668 \times 10^{- 7} < \frac{1}{2 \cdot 113^{2}} \approx 3.916 \times 10^{- 5} .$ Moreover $355 = 3 \cdot 113 + 16$ and $113 = 7 \cdot 16 + 1$ , so $\gcd (355, 113) = 1$ . Thus $355 / 113$ is reduced and satisfies Legendre's criterion.

givenalgebra

2.1

By [L1] and [F1], the fraction $355 / 113$ is a convergent of $π$ . Then [F2] says no. [L1, F1, F2, step 1.1] rational with denominator at most $113$ approximates $π$ more closely. Since $22 / 7$ and $333 / 106$ have denominators $7$ and $106$ , neither can beat $355 / 113$ .

F1F2step 1.1

2.2

The direct errors from step 1.1 also show that $333 / 106$ improves on. [step 1.1, algebra] $22 / 7$ , but that both are far worse than $355 / 113$ .

step 1.1algebra∎

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Sources

Bruce Ikenaga, Approximation by Rational Numbers (standard reference, not scraped)
Peter Hackman, Elementary Number Theory (standard reference, not scraped)
Ivan Niven, A simple proof that pi is irrational (standard reference, not scraped)