My response to the blog wars about multiplying negative numbers. Mostly inspired by Eric’s comment on Mike Croucher’s Walking Randomly.

Big image, links to a PDF (of vector goodness).

I wanted to put the Inkscsape SVG source inside the PNG image. But it turns out wordpress.com “optimises” the image and means my klever hack doesn’t work. Bad wordpress.com.

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This entry was posted on 2009-10-05 at 14:49:06 and is filed under maths.

2009-10-05 at 15:25:20

I prefer the “money” explanation. Having a negative amount of money means owing the money (anyone with a bank account will understand this) so to say I have -£1000 in my account makes perfect sense.

If I am paying a mortgage of £600 a month, I can work out what my balance is (absent other transactions) by multiplying by number of months from now. 2 months ahead gives me 2 * -600 pounds, i.e. -1200 pounds. 2 months in the past gives me (-2) * (-600) pounds, i.e £1200 more.

At the time I gave this any real thought (when I was doing a PGCE) I did indeed have some serious financial problems so this was foremost in my mind.

There is some suggestion that Indians (who used numbers for accounting) found -ve numbers easier to conceptualise (and therefore use) than Greek based mathematicians who were used to numbers as measures of geometric extent.

Your geometrical explanation (which is great by the way) does lapse into alegbra. I don’t think it needs to but you can see how the -ve is so much more obvious with money (which can be naturally negative) than distance (when you have to understand co-ords to do the job).

Double entry bookkeeping is of course something you use if you are a bit nervous about minuses 8-).

2009-10-05 at 15:57:45

Wikipedia on the history of negative numbers suggests that confusion about negative numbers was widespread among Europeans (even European mathematicians) as late as the 18th century: for example, in 1758 the British mathematician Francis Maseres claimed that negative numbers “darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple”.

In some sense the “minus times minus” question is an improvement on the 18th century situation, since it basically assumes the existence of negative numbers, with at least some understanding of them, even if complete understanding of how to operate on them is still missing.

Maybe at some point we will reach this level of public understanding of imaginary and complex numbers? I am cautiously optimistic.

2009-10-05 at 19:01:04

The trouble with complex numbers is motivation. Why bother? At school level you can try to do this via the completion of a field idea – complex numbers are sufficient to supply all the roots you need for any polynomial.

Of course those roots don’t look like they appear on any graph but…

Sadly, a rather slender reason for the complex plane.

The real motivation for complex numbers is that complex *analysis* is so wonderful. Sadly you can’t see that until you’ve learned some analysis – I am told that this is now solely post 16. Maybe in a generation only university students will learn it. A shame.

2009-10-05 at 16:01:18

Leo Rogers identifies three sources for the confusion:

[ed: I added LI markers to make the three points typographically clearer; sorry about the bold]