For many centuries prior to the actual proof, mathematicians had thought that pi was an irrational number. The first attempt at a proof was by Johaan Heinrich Lambert in 1761. Through a complex method he proved that if x is rational, tan(x) must be irrational.
library.thinkquest.org/C0110195/what/irrational.html library.thinkquest.org/C0110195/what/irrational.html
The ratio of the circumference of a circle to its diameter is the number called pi, the Greek letter . This number has fascinated people down through the ages. ... Back now to what Lambert is most remembered for. In 1768, he proved that if is a nonzero rational number, then tan must be irrational. So, since ,
www.hardycalculus.com/calcindex/IE_lambert.htm www.hardycalculus.com/calcindex/IE_lambert.htm
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Johann Lambert proved that pi is irrational in 1761.
http://wiki.answers.com/Q/When_Lambert_proves_pi_is_irr...
Prove sqrt 2 is irrational? Prove that pi is irrational? Prove that Pi is an irrational number? When did Lambert prove pie irrational? How do you prove that pi is irrational? When did lambert prove Pi was irrational?
wiki.answers.com/Q/How_do_you_prove_that_Pi_is_an_irrat... wiki.answers.com/Q/How_do_you_prove_that_Pi_is_an_irrational_number
The first proof of the irrationality of PI was found by Lambert in 1770 and published by Legendre in his "Elements de Geometrie". increases faster than a^(2n+1), so there is some value of n beyond which this ratio will be less than 1. This proves the quantity can't be an integer, so we have a contradiction.
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Lambert is best known, however, for his work on Pi . Euler had already established in 1737 that e and e ^2 are irrational. Lambert was the first to provide a rigorous proof that Pi is irrational.
library.wolfram.com/examples/quintic/people/Lambert.htm... library.wolfram.com/examples/quintic/people/Lambert.html
Euler derives a very rapidly converging arctangent series. 1761; Lambert proves that pi is irrational. He publishes a more general result in 1768. He also shows that the functions e^x and tanx cannot assume rational values if x is a non-zero rational number.
rpimath.topcities.com/irrationals/pi.html rpimath.topcities.com/irrationals/pi.html
This page also uses pi for the irrational number 3.14159..., phi for the golden mean (which is the irrational number 1 Lambert proves that pi is irrational. He publishes a more general result in 1768. He also shows that the functions e^x and tanx cannot assume rational values if x is a non-zero rational number. 1814;
rpimath.topcities.com/irrationals/e.html rpimath.topcities.com/irrationals/e.html
a short and simple proof that pi is irrational ... That is irrational, i.e. not the quotient of two integers, is much easier to prove, only a good math education of a grammar school is needed. The first proof of the irrationality of , given by J.H. Lambert in 1768, is more complicated than the one below but also more general.
www.lrz-muenchen.de/~hr/numb/pi-irr.html www.lrz-muenchen.de/~hr/numb/pi-irr.html