About: Proof that 22/7 exceeds π   Sponge Permalink

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This proof uses integral calculus. If 22/7 > π, then 22/7 - π is a positive number. We claim that this integral comes out to 22/7 - π. * That the integral is positive follows from the fact that the integrand is a quotient whose numerator and denominator are both nonnegative, being sums or products of even powers of real numbers. So the integral from 0 to 1 is positive. It remains to be shown that this integral evaluates to 22/7 - π. To wit: {| Thus, 22/7 - π > 0 and it follows that 22/7 > π

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  • Proof that 22/7 exceeds π
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  • This proof uses integral calculus. If 22/7 > π, then 22/7 - π is a positive number. We claim that this integral comes out to 22/7 - π. * That the integral is positive follows from the fact that the integrand is a quotient whose numerator and denominator are both nonnegative, being sums or products of even powers of real numbers. So the integral from 0 to 1 is positive. It remains to be shown that this integral evaluates to 22/7 - π. To wit: {| Thus, 22/7 - π > 0 and it follows that 22/7 > π
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abstract
  • This proof uses integral calculus. If 22/7 > π, then 22/7 - π is a positive number. We claim that this integral comes out to 22/7 - π. * That the integral is positive follows from the fact that the integrand is a quotient whose numerator and denominator are both nonnegative, being sums or products of even powers of real numbers. So the integral from 0 to 1 is positive. It remains to be shown that this integral evaluates to 22/7 - π. To wit: {| |- | | |- | | |(expanded terms in numerator) |- | | |(performed w:polynomial long division, an important aspect of formulating w:algebraic geometry) |- | | |(definite integration) |- | | |(substitute one for x, then zero for x, and subtract them—arctan(1) = π/4) |- | | |(addition) |} Thus, 22/7 - π > 0 and it follows that 22/7 > π
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