Approximation/Examples/22 over 7

Theorem

$\dfrac {22} 7$ is a convenient approximation to $\pi$:

$\dfrac {22} 7 = 3 \cdotp \dot 14285 \dot 7$

Proof

\(\ds \dfrac {22} 7\)

\(=\)

\(\ds 3 \cdotp \dot 14285 \dot 7\)

\(\ds \pi\)

\(\approx\)

\(\ds 3 \cdotp 14159265\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {22} 7 - \pi\)

\(\approx\)

\(\ds 0 \cdotp 0012645\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\frac {22} 7 - \pi} \pi\)

\(\approx\)

\(\ds 0 \cdotp 0004025\)

\(\ds \)

\(\approx\)

\(\ds 0 \cdotp 04025 \%\)

$\blacksquare$

Sources

• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): approximation
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): approximation
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): approximation