Irrationality of Square Roots

Simon has started a little video series about the Irrationality of Square Roots.

In Part 0, Simon talks about what square root of 2 is and in Part 1, he presents an algebraic proof that root 2 is irrational. He learned this from Numberphile.

In Part 2,┬áSimon presents a geometric proof that root 2 is irrational. Based on Mathologer’s videos.

Parts 3 and 4 following soon!

 

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A new tour of Simon’s sketch book

The fertility formula, to predict the population the following year:

A fake number (called “Wau”) to imagine infinity (via Numberphile):

Drawing a square root of 5 (via James Grime):

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Pebbling a Chessboard (via Numberphile):

Kolakoski Sequence:

Proof for probabilities in a Wythoff’s game

Probability that everyone will be eliminated simultaneously in Simon’s “Hat Game” (a card game he invented):

Finite List of Primes:

Creating consecutive numbers by using various operators to connect four fours:

River Crossings Puzzle

This is a Japanese version of the famous River Crossings Puzzle that Simon learned from the Scam School channel (yes, our little programming and math nerd actually watches Scam School, a channel dedicated to social engineering at the bar and in the street!)

The answer, a sequence of 17 moves:

Simon showing the classic River Crossings puzzle to friends

Math graphs for solving the simple and the more advanced River Crossings puzzles using minimum vertex covers and Alcuin Numbers (learned via Numberphile):

Ramanujan’s Taxicab Numbers

Ramanujan’s Taxicab Numbers are the smallest numbers that can be written as a sum of two numbers to the power of n in two different ways. We only know such numbers if n is 1, 2, 3 and 4. We don’t know the smallest number that can be written as a sum of two fifth powers in two different ways, or any other powers larger than four. The name “Taxicab Numbers” comes from the story about Ramanujan’s visit to Cambridge, when he got picked up by a cab with 1729 on the number plate.

“Every possible integer is one of Ramanujan’s personal friends”, Simon tells me after he sketches the Taxicab numbers at a beach cafe. “Which is a bit uncomfortable because you have infinitely many friends then”.

1 + 2 + 3 + 4 +… = – 1/12? Or does it break math?

Simon keeps coming back to this irresistible math problem about whether the infinite sum 1 + 2 + 3 + 4 +… converges to negative 1/12. He was quite disappointed to hear Mathologer explain that it’s highly contraversial and that many mathematicians don’t believe in the negative 1/12 solution, at least not as long as they use the “equal” sign. Then, about a week later, on May 7 he suddenly told me he is now completely convinced that “Numberphile’s proof (that 1 + 2 +3 + 4 +… does converge to -1/12) breaks math!”