Pi is IRRATIONAL: simplest proof on toughest test

TL;DR

A high school level proof shows that pi is irrational using calculus and a clever recurrence relation.

Transcript

Welcome to another Mathologer video. A couple of weeks ago I showed you an animation of the first ever proof that pi is irrational by Johann Lambert who published it in 1761. Really a huge milestone in the history of mathematics. It came more than 2,000 years after the ancient Greeks first ran into those annoying irrational numbers. After watching ... Read More

Key Insights

• 🤨 The proof presented in the video offers a streamlined and concise demonstration of pi's irrationality.
• 😒 The use of integration by parts and a recurrence relation allows high school students to understand and grasp the proof.
• 👨‍💼 The significance of the sine function in the proof relates to its representation of the circle and its properties.

Questions & Answers

Q: How does the proof of pi's irrationality in the video compare to Johann Lambert's proof?

The proof shown in the video is a simplified and streamlined version of Lambert's proof, making it more accessible to high school students. It relies on integration by parts and a recurrence relation, while Lambert's proof is longer and easier to motivate.

Q: Why does the proof assume that pi is a fraction?

Assuming that pi is a fraction allows the proof to establish a recurrence relation for a sequence of numbers. By showing that all terms of the sequence are integers, a contradiction is reached, proving that pi is irrational.

Q: What significance does the sine function have in the proof?

In the proof, the sine function represents the circle and its properties are at the core of the recurrence relation. The proof leverages the fact that the area under the sine curve is equal to 2.

Q: Can the proof be expanded to show the irrationality of other numbers?

The proof is specific to pi, but it demonstrates the general idea of using a recurrence relation and a contradiction to prove irrationality. It is possible to apply similar techniques to other numbers, but the specific proof would differ.

Summary & Key Takeaways

• The video presents a streamlined proof of the irrationality of pi, based on a problem from an Australian math exam.

• The proof involves using integration by parts to establish a recurrence relation for a sequence of numbers.

• By assuming that pi is rational, the proof leads to a contradiction, demonstrating that pi is in fact irrational.