1 July 2011

Consider a bronze-age shepherd or herder; tending to his animals, sometimes a few animals get lost, a few get eaten by predators, and a few are stolen by others. A few are also legitimately subtracted from his flock, by being bartered away with other tribespeople, and by being slaughtered for food. At the end of the week, the flock looks thin, compared to what he started out with, and he needs to take stock of his stock. Keeping the count of animals in his head is getting more and more difficult as the number of animals in his care grows.

So, he makes marks on his shepherd's crook, one for each animal. At the end of the day, he can count them again, and see if they match the total Like this:

And again, as he runs out of space on the stick, he adopts a short-hand, crossing out every fifth count like this:

Now, he's quite happy! He doesn't lose count anymore, and he can ensure that he returns all (or at least, most) of the animals under his care.

The Roman numeral system is known to all of us from BBC copyright notices
and clock faces. It apparently evolved from tally marks used by Etruscan
shepherds while counting their flocks, this was the system in place during most
of the medieval period in Europe. Every fifth tally was a V-shaped score, and
every tenth one was an X-shaped one. As the numbers they counted grew, the
Romans started adding other symbols like L (50), C (100) and M (1000). As a
system for serially counting several objects, it serves the purpose, and is
easier to teach a wandering Etruscan shepherd. Some other civilizations, like
the Egyptians, used similar numbers, and these are collectively called
*sign-value systems*.

In general, the basic feature of these systems is that each sign has a value, and to represent multiples of those values, the signs are repeated as many times as required. So, for example, to write twenty-eight in Roman numerals, we would use two tens (XX), one five (V) and three ones (III), giving XXVII. This is very similar to making up a sum of currency using various denominations, and reaching the total by using different coins. Except, instead of using coins, we're using numeral symbols.

But just try performing calculations with this system; even basic addition and
subtraction! In fact, the Romans themselves seem to not have used their system
for mathematics; they used a special device called a *counting board*
(something like an abacus) to perform any calculations, and then converted the
numbers back to their sign-value system for representation.

To make matters worse, the Roman system even has a *subtractive* notation,
where to save space, numbers like four and nine are written by
*subtracting* from the closest whole sign - IV instead of IIII and IX
instead of VIIII.

And now, the competition…

The other numeral system of note is the *place-value* system In these
systems, there are ten symbols, each representing the numbers from 0 to 9, and
their *place* within a number represents the power of ten which they
represent. To write the same twenty-eight as above, all one has to do is write 2
and 8 together, to form 28. Calculations in this system are also far easier than
having to reach for the counting board or abacus for every simple operation.

Here's Aryabhata describing the place value system:

एकं दश च शतञ्च सहस्रमयुतनियुते तथा प्रयुतं ।

कोट्यर्बुदञ्च वृन्दं स्थानात्स्थानं दशगुणं सयात् ॥२२॥

( आर्यबटूीयं - गणितपाटा )

The numbers from eka (one), dasa (ten), sata (hundred), sahasra (thousand), ayuta (ten thousand), niyuta (hundred thousand - one lakh), prayuta (million), koti (ten million - one crore), arbuda (hundred million) and vrnda (thousand million - one billion) are from place to place each ten times the preceding.

(Aryabhatiyam Chapter 2 “Ganitapada / Mathematics”, Verse 2)

Contrary to popular opinion however, Aryabhata probably didn't invent this system. Brahmi numerals themselves have existed from the first century CE, and at least one work which contains these numbers - the text in the Bakhshali manuscript, is probably dated to before him. In any case, such a system could not have just appeared in even a genius like Aryabhata's mind out of nowhere.

Nevertheless, the Aryabhatiyam is the first work to explicitely describe the
system so clearly. Apart from fragmentary evidence in other places, this is the
first concrete explanation of the decimal place-value system that we use to
this day. ^{1}This wasn't the only number system that Aryabhata
described. The other one is horrendously complex, using alphabetic symbols to
represent numbers in a very contrived fashion, to make them fit a verse form

Over the centuries, the Persians and Arabs, through trade and cultural contacts (and not a small number of wars), ended up adopting the Indian system by the 9th century or so. The great Persian polymath, Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, from who's name we get the word “Algorithm” was one of those who studied Aryabhata's works, as well as the works of other Indian mathematicians like Brahmagupta and Bhaskara. When he wrote his book, ”The Compendious Book on Calculation by Completion and Balancing” (al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala) (talk about a ponderous title), he used what the Arabs and Persians called “Hindu” numerals, and in a follow-up book, called “On the Calculation with Hindu Numerals”, written in 825, he expanded on the place-value system. This book was taken to Europe, and translated as “Algoritmi de numero Indorum”, or “al-Khwārizmī on the Indian numerals”. At this point, Indian numerals entered Europe, which was still in the dark ages. As I said earlier, abandoning the Roman system for almost everything is what finally led to the European Renaissance.

But this wasn't the only time that the world has seen a place-value system. Pretty much wherever you find a scientifically or technologically oriented civilization, sooner or later you find a place-value system. For example, the Babylonians used a kind of base-60 system, the remnants of which are our use of 360 for the number of divisions of a circle, or 60 for the number of divisions in an hour or minute. The Greeks, some of the earliest scientists whom we can call scientists, gave values to each of their alphabets, and used that to form a place-value system that's still in use in modern Greece. And the Mayans had a base-20 system, which is what their notorious long count calendar (the one that ends on 21-12-2012) is written in. Instead of 1s, 10s and 100s, they had 1s, 20s and 400s

One wonders if they used their toes to count to 20!

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