Proof that π is irrational | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-01-17

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational


00:00:43 1 Lambert's proof
00:01:54 2 Hermite's proof
00:02:05 3 Cartwright's proof
00:20:49 4 Niven's proof
00:24:25 5 Bourbaki's proof
00:25:49 6 Laczkovich's proof
00:28:46 7 See also
00:30:02 8 References
00:30:25 f (k)(0). Therefore, f (k)(π) is also an integer and so F(π) is an integer (in fact, it is easy to see that F(π)
00:31:51 cos and cos'
00:33:18 sin π
00:33:30 – cos π
00:42:10 Bourbaki's proof
00:47:43 Laczkovich's proof
00:53:58 ay and fk + 1(x)
00:54:12 fk + 1(x), a
00:54:24 1 if fk(x)
00:54:42 b/a and define y
01:01:43 cos(π/2)
01:04:26 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9494823080068393
Voice name: en-US-Wavenet-F


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In the 1760's, Johann Heinrich Lambert proved that the number π (pi) is irrational: that is, it cannot be expressed as a fraction a/b, where a is an integer and b is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich.
In 1882, Ferdinand von Lindemann proved that π is not just irrational, but transcendental as well.