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If is rational, then there are natural numbers a and b, such that = . Let’s insist that this is the most reduced form of the fraction, so there is no way of rewriting as when m and n are natural numbers less than a and b.

If = , then by squaring both sides of the equation, 2 = which we can rewrite as a2= 2b2.

Whatever the value of b2, 2b2 must be even since multiplying any natural number by two makes an even number. If 2b2 is even, then a2 is even. Now, since the square of an odd number is always an odd number, and the square of an even number is always even, this means that a must be even.

If a is even, then there is a number c less than a such that a = 2c, and therefore that a2 = (2c)2 = 4c2.

By replacing a2 with 4c2 in the equation above, we get 4c2 = 2b2. Which reduces to b2 = 2c2. By the same reasoning, this means that b2 is even, so b is even. If b is even, there is a number d less than b such that b = 2dp>

Therefore can be rewritten , or since the 2s cancel. We have our contradiction! From above we have stipulated that is the most reduced form of the fraction, which means that there are no values for c and d less than a and b such that = . Since we have arrived at a contradiction by assuming that can be written as it must be the case that cannot be written in such a way, so is irrational.

Appendix Three

In Franklin’s 16 × 16 magic square all the lines and columns add up to 2056. It is not a true magic square since the diagonals do not add up to 2056, but it is so rich in properties that Clifford A. Pickover writes that ‘it is no exaggeration to say that one could spend a lifetime contemplating its wonderful structure’. For example, every 2 × 2 subsquare (and there are 225 of those) adds up to 514, which means that every 4 × 4 subsquare adds up to 2056. Many other symmetries and patterns are also contained in the square.

Appendix Four

The principle behind Gijswijt’s sequence is to look for repeating blocks of numbers in the previous terms of the sequence. The ‘block’ has to be at the end of the sequence of previous terms, and the number of times it is repeated provides the next term.

Mathematically, the sequence is described as follows. Start with a 1, and then each subsequent term is the value k, when the previous terms are multiplied in order and written xy k for the largest possible value of k.

The sequence is 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1…

I think it is easiest to understand what is at work here by considering the first time a 3 appears, which is in position 9. The previous terms multiplied in order are 1 × 1 × 2 × 1 × 1 × 2 × 2 × 2. What Gijswijt requires us to do is to transform this sum into a term xy k for the largest value of k. In this case, we get (1 × 1 × 2 × 1 × 1) × 23. So, the following term is a 3. We are looking for the largest repeating block of numbers at the end of the sequence of previous terms, although in this case the block is a single number, 2, repeated three times.

But often the block will have several digits. Consider position 16. The previous terms multiplied tther are 1 × 1 × 2 × 1 × 1 × 2 × 2 × 2 × 3 × 1 × 1 × 2 × 1 × 1 × 2. This can be written (1 × 1 × 2 × 1 × 1 × 2 × 2 × 3) × (1 × 1 × 2)2. So, the 16th term is a 2.

Going back to the beginning now, the second term is a 1 since the previous term 1 is not multiplied by anything. The third term is a 2 since the previous terms multiplied in order are 1 × 1 = 12, and the fourth term is 1 since the previous terms result in (1 × 1 × 2) × 1, where the final 1 is not multiplied by itself.

Appendix Five

We want to show that the harmonic series diverges, in other words that

will add up to infinity. This is done by showing that the harmonic series is larger than the following series, which does add up to infinity:

Let’s compare terms of the harmonic series in groups of two, four, eight and so on, starting from the third term. They are listed below. Because is bigger than must be bigger than which is . Likewise, since and are all bigger than ,