4+1=V unless we’re Italian…. Also some stuff about rabbits

Only about 5 five people in the world realise I’m in second level of a BA in pure mathematics (well, now the almost 6000 a week who visit this blog … minus the 5985 who are Chinese spam bots….)

Anyway, ever heard of Leonardo da Pisa? No, not the painter. That’s Vincci. This chap is also known as Fibonacci. He liked rabbits.

Take the following example. We have two rabbits (one male, one female). That’s one pair of rabbits. They have two babies. We now have two pairs of rabbits. The second generation is still growing, but meanwhile the first generation have some more babies. We now have three pairs of rabbits.

The next month, the first generations plus the second generation reproduce again. We have… five pair of rabbits. What happens the next month when these rabbits who are of reproductive age have a further litter? See the sequence? Calculate it out, and we start to see a complicated sequence appearing, consisting of the current number plus the previous number:

1,1,2,3,5,8,13,21,34,55,89,144,233,377…… etc etc etc

What Fibonacci didn’t realise is that this sequence appears all over the natural world. I’m not going to go into this…. Mind you, next time you look at a flower, check out the sequence of petals. 🙂 It helps the Spanish work out how many sunflower seeds they can sell per pack.

Let us take a 21st century blogger. He is writing a weekly business report which, for some bizarre, reason, he wants to title in Roman Numerals. He is unsure about how to title edition number 41… XXXXI seems logical, but it should be XLI. (10-50+1=41). Which is why Roman Numerals should have died a happy death millennia ago, they are bloody useless, in as much as they force you to do mental arithmetic with no mental tools. And what’s worse – the militaristic idiots had no negative numbers, so 10-50=40. Which appears to be the same counting system the banks use when they ask for a taxpayer bailout.

Hang on, how could the Romans decide that 10-50+1=40 is XLI if they had no concept of zero? How can 10 less 50 equal 40? If I have 10 pizzas and promise you 50, I don’t have 40 pizzas in stock!

In fact, most modern scholars agree that XLI is probably an 18th century attempt to make the Romans seem wiser than they really were…… the brutes probably did write XXXXI for 41, and our 21st century blogger is historically correct, but politically incorrect.

We use what is known as the Arabic numeral system, based on a decimal counting system. Also known as the Hindu-Arabic numeral system.

So we have 10 numbers for the concepts between no ice-creams and 9 ice-creams. 9+1? We move to a double numbering system, in which the first position is the decimal and the second the unitary. That’s 1+0 ice-creams. 40 newsletters? Add one, it’s 41. If I have 5, and want to buy something worth 8, so I borrow 3 from you, I end up with an asset of 8 but a debt of -3, which together total my original 5. Nice and simple. Unless we’re discussing the differences between Spanish and English billions. (9 or 12 zero’s?)

Anyway, Fibonacci realised this in around 1200. He had some cash as a kid, and travelled throughout the Arabian world, which was slightly ahead of Europe in those days. When he got back to whatever Italian town (clue is in the name) he lived in, he became a clerk. Of course, in those day, Europe had Roman numerals.

In short, Fibonacci was the first to twig on that if I have 10 tunics, but some fat bloke comes into my shop and orders 15, I need to borrow a further 5 until I’m paid for the full amount… and so accounting was born. And since in Roman numerals I can’t have negative numbers, how do I account for the fact that for a short period I owe 5 tunics? You can’t, in Roman numerals – they had no concept of zero, or of negative numbers. Instead, they had to write out a complicated longhand logic stating “Mr P promises to pay me 10 ducats if I lend him 5 ducats for the period of one year and so this allows me to borrow 4 mules from Mr D until the 7th of Feb allowing me to sell my 5 donkeys plus the 4 mules from Mr D for the amount of 12 ducats, which means next year I shall have …. “

In short, under the Roman system, if I write down that I have 10 and owe 50, I end up with owning 40 which I must pay to a third party. Whereas we say you have -40. The semantic difference is enormous. For example, the whole concept of debt in a legal situation becomes possible.

So Fibonacci published a book called Liber Abaci (book of the abacus), and became the first European (and so the first person?!!) to recognise the concept of accounting debt. Which sort of created capitalism.

So what he did was introduce our modern numbers from 0 to 9. And then built upon these to create the idea of addition, subtraction, multiplication and division. And then the rest of the book was a collection of number puzzles.

Which makes life easier. For example, how do we multiply in Roman numerals?

Multiplication is rather obvious once you realize that the Roman symbols are additive; that is, CXI is really C + X + I. To multiply two multinomial expressions in algebra like (a+b)(x+y+z) we multiply each term in the first by every term in the second and add the results. This is the approach for multiplying Roman numbers.

First we need a multiplication table. Due to the nature of the values of the Roman symbols (every value involves only 5’s and 10’s) this is easy to form and learn:

times     I    V    X    L    C    D
I    I    V    X    L    C    D
V    V    XXV    L    CCL    D    MMD
X    X    L    C    D    M    V
L    L    CCL    D    MMD    V
C    C    D    M    V
D    D    MMD    V

The value V, of course, may be written as MMMMM.

The algorithm for multiplication is very similar to the one for addition with just five steps:

Substitute for any subtractives in both values; that is; “uncompact” the Roman values.

For each symbol in one value form the product with every symbol in the second and catenate them all together.

Sort the symbols in order from left-to-right with the “largest” symbols on the left.

Starting with the right end, combine groups of the same symbols that can make a “larger” one and substitute the single larger one.

Compact the result by substituting subtractives where possible.

As an example, perform XXI•XVII.

1.    Substitute for any subtractives to obtain:    XXI•XVII
2.    Form products and catenate to obtain:    CLXX CLXX XVII
3.    Sort to obtain:    CCLLXXXXXVII
4.    Combine groups to obtain:    CCLLLVII
CCCLVII
5.    Substitute any subtractives to obtain:    CCCLVII

In step two, we get X “times” XVII to obtain CLXX. The second X in the first value gives the same, and finally the I “times” gives XVII. It is these three values that are concatenated together to form the result. (Note the catenation performs the addition of the intermediate partial products.)

Simple, isn’t it (!).

Let’s do this in Arabic numerals: 21•17 is 21 multiplied by 21 17 times. Which is basically counting on your fingers. Unless you memorise your times tables.

This is not… what the Romans did. We decided to do this, because we knew how to do it, and felt it was necessary to force this knowledge upon the Romans.

Back to Fibonacci.

Now, I ignore his ramblings on the golden ratio. Also his translations of Arabic number theory into Latin. And, let us be honest, Fibonacci wasn’t a mathematical genius. He was a tourist who recognised the advanced theories he saw and pinched the ideas to translate them into Latin. A clever man but not a genius.

But it’s thanks to him that we can casually saw “I lent him 4 euros” or “one day she’ll return the 4 euros she owes me” or “my bank gives me compound interest of 2.3% APR on the base capital annual amount” or even “here’s edition number 41”.

And all that because some old Roman thought his grandfathers way of counting rabbits wasn’t as good as some moro over the sea counting on his fingers.

But what is even more amazing. Fibonacci introduced these ideas into Europe at a time when we didn’t habitually use spaces between words. Sowewrotelikethisbecauseanyonewhocanwriteonlywantspeoplelikehimselftoread.

Interesting, isn’t it?