Years ago I registered a Wordpress blog, because Wordpress had support for LaTex. I've just started trying to use it. But I've created a page https://cawtech.wordpress.com/partial-differentiation-puzzle-from-an-old-book/ when I intended to create a post. I'll fix that sometime. At least the mathematics shows up.

Meanwhile, I've noticed that there are lots more HTML entities than I remember. So I can write a ∈ {a,b,c} and {a} ⊂ {a,b,c} while ℘({a,b}) = {∅,{a},{b},{a,b}}. Since there is a square root sign I can write my favourite proof of the irrationality of √2

Suppose that √n∈ℚ. Write √n = a/b where a and b are coprime. Then b√n=a. Also, by Bezout's Theorem there are r,s∈ℤ such that ra+sb=1. Multiply by √n

ra√n + sb√n = √n

We already have b√n=a so sb√n is an integer. Does anything similar happen for ra√n ? Yes, just multiply b√n=a by √n obtaining bn=a√n, from which we deduce that ra√n must also be an integer. Thus √n is the sum of two integers and itself an integer.

That is awkward; a rational square root always turns out to be a whole number! Since 1<2<4 we know that 1<√2<2. But there is no integer between one and two so √2 is irrational.
The election to The City of Edinburgh Council is fast approaching. I have until 4th May to decide who to vote for. So far I only have a leaflet from the Green Party candidate, Claire Miller.

> Work to introduce rent controls so that the city centre becomes a truly affordable place to live

Google "Rent control or bomb damage" to find out why I will not be voting for Claire.

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alan_crowe

September 2023

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