How much more?

In Australia, a standard bottle of wine contains 750 millilitres. Today I bought a bottle containing 1 litre. The label proclaims “OVER 33% THAN A 750 ML BOTTLE”.

This is true. 250 ml is 33 point 33 recurring percent more than a 750 ml bottle, which is indeed over 33 percent (or more than, if that’s what your style guide says). To be fair, the winery may not legally be able to claim “33 percent more” when there’s actually “33 point 33 recurring more”. But is anyone going to complain that they got 2.5 ml more than promised, which is what the difference between 33 point 33 recurring percent and 33 percent actually comes down to?

I’ve just got to remember not to drink 33 point 33 recurring percent more of it.

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Trains of thought, revisited

I have previously written (here, scroll down to the last paragraph) about a little mental game which I (and apparently some other people) play with the carriage numbers on Sydney’s trains – the point being to make the four digits of the number total 10 using any standard mathematical process.

A few days ago I travelled in carriage number 6472. I quickly figured out (6 x 4) – (7 x 2) and (6 – 4) x (7 – 2), which seemed neatly and satisfyingly symmetrical. Is there a general pattern here? No and yes. In the second case, 4250, 5361, 6472, 7583 and 8694 all equal 10, but in the first case, 4250 equals 8, 5361 equals 9, 6472 equals 10, 7583 equals 11 and 8694 equals 12 (which is another pattern of its own). So these two equations are equivalent only when the first number is 6. I’m sure there’s a way of proving this mathematically, but my skills are too rusty.

one, two three, first, second, third

There is statistical law called Benford’s law or the first-digit law, which states that in many naturally occurring collections of numbers, the first digit is significantly more likely to be 1, 2 or 3, and significantly less likely to be 7, 8 or 9. 1 is the first digit about 30% of the time, and 9 about 5%.

This also generally applies the written words one, two, three etc. Google Ngrams shows that one to six appear in exactly that order, then ten, eight, seven and nine. Ten gets a boost because of its use as the base for the decimal system, while eight is a power of two, and we prefer counting in even numbers.

Adventures in Numberland

Two days ago I bought Alex through the looking glass by Alex Bellos, an exploration of ‘how life reflects numbers and numbers reflect life’. His previous book was Alex’s adventures in Numberland (which the bookshop didn’t have, otherwise I would have bought it as well; I’m going to inquire at bookshops near home or work, and if they don’t have it, order it online [edit: I bought it at a bookshop in the city on Monday]), so he’s obviously got a thing for Lewis Carroll. (Unlike Alice, these books are entirely non-fiction.)

The first chapter is about numbers, and he starts with an account of a retired taxi driver with Asperger’s, whose hobby is to divide every number he sees into prime numbers. (The fundamental theorem of arithmetic states that every positive integer has a unique prime factorisation). The examples the man provides are 2761 = 11 x 251, 2762 = 2 x 1381, 2763 = 3 x 3 x 307 and 2764 = 2 x 2 x 691. (I don’t know how often one of the prime factors is so big; Wikipedia’s example is 1200 = 2 × 2 × 2 × 2 × 3 × 5 × 5. The bigger and the ‘odder’ the original number, the bigger any one factor might be, but the comparatively rarer prime numbers get.)

Trains of thought

I’m back to spending large amounts of time sitting on Sydney trains. Three trains of thought (haha!) arose recently.

Throughout the carriages are a number of posters advising of good behaviour on trains, usually in rhyming couplets. One which is not a direct rhyme is:

Cover your cough
or sneeze please

This is undoubtedly meant to be ‘Cover your (cough or sneeze) please’, but the line break means that it could be interpreted as ‘(Cover your cough) or (sneeze) please’ – that is, instructing us to sneeze.

2,520

A few days ago I read the factlet that 2,520 is the smallest number which is divisible by every number from 1 to 10. After a bit of playing with my calculator, I found that this is equal to 9 x 8 x 7 x 5. Why not 10, 6, 4, 3 and 2? Well, if a number is divisible by 40 (8 x 5), then it is also divisible by 10; if is is divisible by 9, then it is also divisible by 3; and if it is divisible by 8, then it is also divisible by 4 and 2. 6 is less immediately obvious, but 2,520 is divisible by 72 (9 x 8), so it is also divisible by 6 (and 12). Another way of reaching the same conclusion is to break 9, 8, 7 and 5 into their smallest divisors (3 x 3) x (2 x 4) x 7 x 5, which equals 3 x (3 x 2) x 4 x 5 x 7, which is clearly divisible by 6. That, in fact, which I didn’t notice at first but the Wikipedia page about the number has just informed me, makes it equal to 3 x 4 x 5 x 6 x 7.

2520 is the postcode for Wollongong, but I don’t think there’s anything mathematically significant about that.