## Monday, 28 December 2015

### How we write numbers

I'm thinking to start off with this image*:
And the questions are - of course - What do you notice? What do you wonder?

I don't know which way it will go then.

But I know there are lots of great ways to go from this image. I'm hoping for some observations that are historical, and perhaps some mathematical ones.

One area I'd like to explore - historically and mathematically is the changeover from Roman numerals to Hindu-Arabic numerals. It could show us how a change in the way we think brings about changes in society. It could show us the importance of cultures outside Europe. And there's some potential great exploration of different number systems and numbers in different bases.

Already F has been writing some equations in Roman numerals, and R has been talking about how our system comes from India.

I like what Graeme Anshaw has done with this:

I think we'd probably want to get our heads round how Roman numerals work. Here's a good way:
Arithmetic wasn't so easy for the Romans, so they used a counting board or abacus:
It would be interesting to make, and try to use one of these. It looks like they work like a Japanese soroban:
 source
And then, how did they do this on a counter with jetons?

Here we could investigate other ways of representing numbers from different cultures, and at different times.

Now for Hindu-Arabic numerals...
 source
How did they make that journey, from India to the Muslim world, to Europe? Lots of stories in there, and I'm pleased there's such a strong cross-cultural element.

I've just been reading Keith Devlin's Man of Numbers, about how Fibonacci introduced them into Europe. There's lots worth finding out in that story too.

And thanks to Tracy, I have a nice little picture book about him to read and contemplate with the class:

Part of why this all seems more do-able to me now, is partly that I've seen there is lots of maths that can be explored by the students, rather than simply being told about. Reading Madeleine Goutard on our base system, and other bases, I see more clearly how we could represent them:

 from Madeleine Goutard, Mathematics and Children
The bottom one is perhaps how three-toed sloths might represent numbers, in base six.
 Three-toed sloth (source: wikimedia)
How would those numbers in base six be written?

How about other bases? They could be represented by Dienes' blocks too:
 source
Only problem is, even though there's loads of sets gathering dust in maths cupboards everywhere, they're not used any more, and not sold in the catalogs. Maybe I'll make some...

So, lots of directions to explore from that first image.

And then there's that dress, with that mysterious pattern of numbers.

See How We Wrote Numbers for reflections on how this went.

*Calculating-Table by Gregor ReischMargarita Philosophica, 1503. The woodcut shows Arithmetica instructing an algorist and an abacist (represented as Boethius and Pythagoras). There was keen competition between the two from the introduction of the Algebra into Europe in the 12th century until its triumph in the 16th. Source

1. I think we tend to be too quick to dismiss Roman numerals as unwieldy and difficult to calculate with, at least as far as multiplication is concerned. They don't provide such a powerful and extendable system as offered by our place-value system, but multiplying with Roman numerals, though limited to natural numbers perhaps, is easy to make sense of - it could be said to involve repeated addition, collecting like numerals and exchange, with some scaling (especially when multiplying by 10). It also calls for partitioning (and the distributive law) in a very 'natural' way and thus could provide a useful, if limited, model of multiplication for students.
@ProfSmudge

1. Thanks Prof. I like the idea of exploring how multiplication might work.

Thinking about the abacus, I've come to the conclusion that the battle depicted between the abacists and the algorists wasn't quite the walkover the image presents. Expert users of the saroban, which seems very like the Roman system, are, I've heard, very fast.

2. Is there any evidence the Romans ever did any manipulations with their number symbols? I was always told that if they needed to do any sort of meaty calculations they'd get a Greek slave to do it using their own number system.

And I don't want to be picky, but surely our number system is base TEN rather than than base 10 ? No matter what number you're using as a base, 10 represents the value of the base. That's true whether you're working in base ten, base twelve, or - and this one's a lot of fun, I promise, base negative two.

2. Thanks Alan. Interesting thoughts.

I'm thinking, this Roman system continued up until Fibonacci's time. My impression from Keith Devlin's book about him is that a lot of the examples he gave in his book of what you could do with the new numbering involved multiplication and division.

Things like:

"If one hundredweight of linen or some other merchandise is sold near Syria or Alexandria for 4 Saracen bezants, and you will wish to know how much 37 rolls are worth, then..."

"Three men made a company in which the first man put 17 pounds, and the second 29 pounds, the third 42 pounds, and the profit was 100 pounds."

(And of course there's the famous rabbits!)

I suppose it's possible that there wasn't so much trade before the 12th century, and so this kind of calculation wasn't needed so much. Perhaps trade took off, at least for places like Pisa, and the new numbers were more attractive?

As for the base ten, thank you - that was a mistake that I was very likely to make, and yes, of course all bases are base 10. I did "new maths" at secondary school, so we learnt about different bases (along with a bit of modular arithmetic, and Boolean algebra); it was actually one of the parts of the curriculum that grabbed me most. We didn't do base negative two though - I'll have to check that out!