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High definition

By Lucy Rycroft-Smith. Posted October 18, 2019

We asked #bigmathoff competitor Lucy Rycroft-Smith to tell us a little about her latest project – CM Define It, an app aiming to collect and define mathematical vocabulary, which launches today.

When you teach mathematical vocabulary, how do you define its meaning?

Are you exact, choosing your words specifically? Do you give a written definition? Do you give multiple explanations? Do you use diagrams? Metaphors? Connect to previous vocabulary?

As part of our work at Cambridge, creating a Framework for mathematics learning, we are creating a network of semantic links across nodes in our mathematical layer – and we initially thought we could just import a mathematical glossary from somewhere else to populate this. But we found so many inconsistencies, technical errors, and definition loops in many existing glossaries that we decided to make an app to ask the mathematical community what they thought, with the aim of developing a crowdsourced, multi-layered collage which takes into account different layers of mathematical experience.

Buzz in when you think you know the answer

By Andrew Taylor. Posted March 14, 2019

Aperiodical guest author Andrew Taylor writes about an intriguing piece of number theory which turns out to also be something else.

How many ways are there of writing some natural number $n$ as the sum of two squares?

$$ n = p^2 + q^2 $$

I don’t want an answer for some particular $n$. I don’t even want a general formula. I just want to know… on average.

Almost all above average

By Peter Rowlett. Posted February 23, 2012

This morning James Grime tweeted about a BBC News article, “‘Third of UK postcodes’ have slow broadband speeds“. This quotes Julia Stent, director of telecoms at uSwitch saying:

Britain might be riding the wave of a super-fast broadband revolution, but for 49% who get less than the national average broadband speed, the wave isn’t causing so much a splash as a ripple.

Now, the thrust of the article, that broadband speeds are undesirably slow in some parts of the country, might be valid, but the appeal to the “average” is a very weak argument (Update [23:47]: Although, please see the comment below). The result that 49% are below average should not come as a big surprise!

James, rightly, questions which average is most appropriate, but I am more interested in a tweet by Ian Preston:

We can make almost everyone above average if we are happy for one person to be really badly off.

This, of course, is quite right.

Unless I’m reading it wrong (and I may well be), the Bank of England’s Lending to Individuals December 2011 has outstanding net lending to individuals as £1451.4 billion. The UK Office for National Statistics gives the Public Sector Net Debt excluding financial interventions as £988.7 billion (January 2012) and Total population (UK) as 62.3 million (mid-2010).

If we gave one person all that debt and everyone else zero, then a simple average would be £2.44 trillion divided by 62.3 million people, which is £39,167.

Since almost everyone is worth zero, we would almost all be 39 thousand pounds above average. Sound good? This would sort out Government debt and make almost all of us “above average”. And if we’re “above average” then, erm, everything is fine, right?

(Flaws in the argument left as an exercise for the reader!)