# The Sound of Proof

## Introduction

In The Sound of Proof (2017–), mathematician Marcus du Sautoy and composer Jamie Perera have taken five classic mathematical proofs from Euclid’s Elements and explored how to turn them into sound so you can hear the journey of the proof. Euclid’s Elements represents the beginning of mathematicians’ obsession with proof. It shows how from very basic axioms about geometry and number we can deduce extraordinary new revelations about circles, triangles, prime numbers and fractions. A proof is a journey which takes the axioms and twists and turns them into something new. This journey has a very musical quality to it. Jamie and Marcus have tried to capture this by finding a way to sonify Euclid’s proofs. From drawing a circle in sound to squaring a melody, they have explored creating a dictionary that translates mathematical ideas into sound. Welcome to the Sound of Euclid’s Elements.

The challenge of #PRiSM-X2 Math Music Match is to listen to the sounds and to read the proofs and then to see if you can match the sound to the proof. Can you hear the maths in the music?

## #PRiSM-X2 Math Music Match

#### Instructions

1. Listen to Proofs A to E and read Proofs 1 to 5 below.
2. Drag the text Proofs (1 to 5) into the correct order to match the audio Proofs (A to E).
3. Find out the answers and see how you did.
• Proof A
• Proof B
• Proof C
• Proof D
• Proof E
• Proof 1
• Proof 2
• Proof 3
• Proof 4
• Proof 5

## Proof 2

### Proof that the square root of 2 is irrational

If you draw a square with sides of length one unit then Pythagoras’s theorem implies that the diagonal across the square has length the square root of 2. But what is this length? The Pythagorean’s thought that there might be a fraction whose square is 2. But it turned out that there is no fraction whose square is 2. To prove this they used a classic trick in the mathematician’s arsenal: proof by contradiction. Suppose there is a fraction and then show why this leads to the contradictory conclusion that a number is both odd and even at the same time. This is impossible so the original assumption that the square root of 2 is a fraction must be wrong.

#### Formal proof

Let L be a number whose square is 2 and suppose that L is actually equal to a fraction

L = A/B.

We will prove that this leads to a contradiction therefore proving that L is irrational (which means it is not a fraction).

We can assume that one of A or B is odd. If they are both even we can keep dividing both top and bottom by 2 until one of the numbers becomes odd.

Since L2 = 2 it means that

A2/B2 = 2.

Multiply both sides by B2

A2 = 2 x B2.

So is A odd or even? Well A2 is even so A must be even because odd times odd is still odd. So A = 2 x N for some number N. Since A is even that means that B must be the odd number. But hold on…

2 x B2 = A2 = (2 x N)2 = 2 x 2 x N2

so we can divide both sides of the equation by 2 to get

B2 = 2 x N2.

Remember that we’d worked out that B was the odd number. So B2 is also odd. But the right hand side of this equation is an even number! So if L is expressible as a fraction it would imply that odd=even. That’s clearly absurd so our original assumption that L can be written as a fraction must have been false.