Powell’s Pi Paradox: the genius 14th century Indian solution
Mathologer・2 minutes read
The Powell’s Pi Paradox involves a formula for Pi constructed from odd numbers, showcasing ultra slow convergence and unexpected digit pair matches. Madhava's use of correction terms led to new Pi formulas with quickly converging series, providing more accurate approximations of Pi.
Insights
- Martin Powell discovered a paradoxical property of Pi in 1983, where consecutive pairs of digits in the series match unexpectedly, showcasing ultra slow convergence.
- Ancient Indian mathematicians like Madhava made significant contributions to Pi's approximation, developing power series expansions for trigonometric functions that aided in speeding up the convergence of Pi series, ultimately leading to more accurate approximations of Pi.
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What is Powell’s Pi Paradox?
A mathematical gem involving Pi formula from odd numbers.
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Summary
00:00
"Powell's Pi Paradox: Unveiling Pi's Mysteries"
- Powell’s Pi Paradox is a mathematical gem involving a formula for Pi constructed from odd numbers.
- By multiplying both sides of the Pi formula by 4, one can compute Pi by adding more terms of the series.
- Summing the first million terms of the series only yields 6 correct digits of Pi, showcasing ultra slow convergence.
- The paradox arises when consecutive pairs of digits in the series match unexpectedly, contrary to random probability.
- This paradoxical property of Pi was discovered by Martin Powell in 1983, despite over 16 million known digits of Pi.
- Ancient Indian mathematicians, like Madhava, discovered the Pi formula before Leibniz and Gregory, using advanced calculus.
- Madhava’s discoveries included power series expansions for trigonometric functions, aiding in speeding up the convergence of Pi series.
- By adjusting the final terms of the series smartly, one can obtain better approximations of Pi, as demonstrated with the 355/113 approximation.
- Spotting patterns in the differences between partial sums and approximations like 355/113 can lead to even more accurate approximations of Pi.
- Further corrections involving multiples of 4 in the correction terms of the series can provide increasingly precise approximations of Pi.
14:04
Refining Pi with Correction Terms
- Three different approximations for Pi discussed: sum of first six terms, using 1/6th as correction term, and a new refined correction term.
- Correction term for adding N terms revealed as 1 over (N+ 1/(4N)).
- Indian mathematicians discovered a new refinement of the correction term for Pi.
- Approximation of Pi improved from 3-digit agreement to 15-digits using the super-duper correction term.
- Explanation of Powell’s Pi paradox using correction terms provided.
- Madhava's third correction term fails to explain all coincidences between the millionth sum and Pi.
- Fourth correction term, missed by Madhava, resolves remaining coincidences for the millionth sum.
- Madhava's use of correction terms led to new Pi formulas with quickly converging series.
