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Infinity a bit closer to home

In The Observer on the 14 February 2010 there was an item about how many teachers had failed a simple maths test. Whoever had administered the test (answer below) was disgusted to find that ’only six teachers (4%) knew that the answer to 2 divided by 0 was infinity’. This made me very angry, and I fully intended to write to the editor. I didn’t, but someone else did. He was a mathematician, and the good news was his dismissal of this ’answer’. The bad news was that his explanation was was a bit technical.

So here is my argument...

When you divide one number by another, what you are asking is this: how many lots of the divisor (the number you are dividing by) can you fit into (or get out of - it really doesn’t matter) the dividend (the number you are dividing the dividend into)?

Basic question: can you really have a lot containing zero items?

I tried entering 99 divided by zero in my Windows 7 calculator. The answer? ’Cannot divide by zero’. Microsoft Excel came up with ’#DIV/0!’ - a bit cryptic, but it amounts to the same thing. As a triple check, I did the same with’s Calc spreadsheet, and got the same cryptic but unequivocal response.

I also remembered, from my early days of programming in BBC Basic, ’Division by zero error’ popping up when I got my maths wrong.

Go on - tell me that it must be a number because the calculator and the spreadsheets do allow you to multiply other numbers by it. Let’s apply the same logic as I used for division (’ many lots of the divisor can you fit really doesn’t matter) the dividend?’). When you multiply any number - say 9 - by zero you are asking ’How many items are there in no lots of 9 items?’. I’m not at all sure that’s real arithmetic.

QED. My conclusion? Whoever set this test for the teachers is a mathematical illiterate. Or an innumerate. Or just a dickhead.

’The test was set by Richard Dunne, a former teacher and the author of Maths Makes Sense,’ said the Observer article.

Maths Makes Sense? I was sense - common sense - that told me that the question was absurd. Perhaps Channel 4 should have asked Marcus du Sautoy, who is a real mathematician and has succeeded The Dreadful Doctor Dawkins as Professor of the Public Understanding of Science...

Whether zero is actually a number, rather than a convenient symbolic representation of bugger-all, is a philosophical question rather than a mathematical one. Personally, I’m sure it isn’t, in the sense that you get a number when you count things - and you can’t count nothing.

Then there’s the question of whether infinity is a number. I’m no mathematician (as I may already have inadvertenty revealed on this page!), but I don’t think it is - because you can’t get infinity by counting, either. Even if you had all the time in the world (literally) you couldn’t, because however big a number you get to you can always add another to it and produce yet another finite number. The sequence ’12345...’ itself is infinite, in the sense that you can go on counting forever. Infinity is a concept, a notion - but no way is it a number.

That’s why it doesn’t appear on calculator and computer keyboards: you can’t do arithmetic with infinity, so it isn’t a number. QED again, Mr Dunne.

Update from the weird world of the quantum

10 March 1020 While reading Quantum: a guide for the perplexed, I came across the assumption that any number divided by zero does indeed euqal infinity. This contradicts what I have said here, demonstrating to my own satisfaction (if to nobody else’s) that neither zero nor infinity are actually numbers. Now I’d be the first to admit that, when it comes to a disagreement on anything mathematical between the 67-year-old holder of a bare pass in maths at O-level, gained 51 years ago, and a professor of physics, there is no contest. However, I still hold to my view that zero and infinity aren’t numbers in arithmetic.

Personal site for Paul Marsden: frustrated writer; experimental cook and all-round foodie; amateur wine-importer; former copywriter and press-officer; former teacher, teacher-trainer, educational software developer and documenter; still a professional web-developer but mostly retired.

This site was transferred in June 2005 to the Sites4Doctors Site Management System, and has been developed and maintained there ever since.