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.)

I have previously blogged (scroll down a bit) about a game I sometimes play on Sydney trains. Each carriage has a four-digit identification number, and the challenge is to get those digits to equal 10 by any combination of standard mathematical operations. I usually don’t go to the extent of wondering whether it is a prime number or, if not, what its prime factorisation is.

Yesterday, though, inspired by the book, I tried it. The number was 8105, which, ending in 5, is automatically not prime. The first task, making it equal 10, was easy: 8 + (10/5). The second task, factorisation, was harder. Calculating the long way round, I found 5 x 1621. (I later remembered that the easy way to divide by 5 is to double then divide by 10.)

So is 1621 prime or composite? There is no easy way to tell and my mental arithmetic is too rusty, so I used the calculator app on my mobile phone to calculate 1621 / 3, 1621 / 7, 1621 / 11 and so on up to 1621 / 49, at which point I gave up and searched the interweb for ‘1621 prime’ and yes, indeed, I had stumbled on a four-digit composite number (8105) which has a four-digit prime factor (1621). There are many four-digit prime numbers, but they are comparatively rarer than one-, two- or three-digit ones: of the 9 one-digit numbers, 4 are prime (44%); of the 90 two-digit numbers, 21 are prime (23%); of the 900 three-digit numbers, 143 are prime (15%) and of the 9000 four-digit numbers, 1071 are prime (11%). (Calculated from the list here.)

I changed trains at Central, and the second train had the number 5608, which is automatically not prime. I got as far as 2 x 2804 before I had to get off. Calculating in my head now, we then get 1402 x 2 x 2 and 701 x 2 x 2 x 2. Is 701 prime? In my head, I can get as far as dividing by 3, 7, 11 and 13, all of which are not factors. Short-cut: the webpage I linked to above, which shows that 701 is indeed prime, so 5608 = 701 x 2 x 2 x 2.

I am not going to make a habit of this.

(A brief search shows that there are prime factorisation calculators on the interweb, but I couldn’t find a page of numbers and their prime factors.)

(PS Alex Bellos’s webpage says that these two books were published in the USA as Here’s looking as Euclid and The grapes of math, respectively. The second pun doesn’t work for me. I pronounce mathS with a PAT vowel, and wrath with a POT vowel. gives the pronunciation as ‘rath, rahth or, esp. British, rawth‘, so I am ‘none of the above. I don’t think I can seriously say rahth  or rawth. I could just say rath if I was quoting the movie Se7en or the poem Jabberwocky.)

(PS two days later – one of the numbers today was 6568, which factorises as 2 x 2 x 2 x 821, so perhaps three-digit prime factors are as unusual as  I first thought.)


4 thoughts on “Adventures in Numberland

  1. Pingback: geology, geography and geometry | Never Pure and Rarely Simple

  2. When my kids were growing up we played two driving games based on the license plates of the cars around us. CA standard plates are a single number, followed by three letters, followed by three numbers, i.e. 3RTW783. The game involving numbers was to make the last three numbers equal the first number exactly the same way you play your carriage game. As the kids grew, of course, the maths they used to get their answers became more complex.
    The game involving the letters is actually the one they loved the most. They had to make as many words as possible by adding additional letters to the existing ones. They still tend to play this game any time they are in a car.


    • That has the added interest and challenge of having a different target number each time. But with only three digits to play with, I suspect that it would be harder to make an equation.
      We always used the letters, making a sentence or a least a set of words which made sense. My cousin’s wife’s car had KZR, so she called it ‘Kate’s Zippy Racer’.

      Liked by 1 person

      • I love Kate’s Zippy Racer for a car name!I have always named my cars, but the names never had anything to do with the license plate letters. A college boyfriend did use that method. The car he drove when we dated was a Volkswagen bug. The plate’s letters were GAK and so, of course, he called it the “GAK”). My first car, a 1967 GTO, was The Goat (you know how I came up with that!), my old 1960 Porsche was named Tubs, the van I drove when the kids were growing up was Monstro (named by my eldest after the whale in Pinocchio), my Mustang was named, inartfully, Stang. My dad also always named our cars.


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