From 0 to Infinity in 26 Centuries (3 page)
Read From 0 to Infinity in 26 Centuries Online
Authors: Chris Waring
Zero interest
Although this seems like a pretty decent number system, not unlike our own, there were a couple of problematic areas. The first problem was that until
c.
500
BC
the Mesopotamians had no symbol for zero, which means they had no way of showing an empty column. For example, if I write down 205, the zero tells you that I mean two-hundred and
five, and nothing else. The Mesopotamian number system was flawed because empty columns in the middle or at the end of a number were missing. For example:
This looks like 60 + 10 = 70. But there could be an empty column in the middle, in which case the number would be 3600 + 0 + 10 = 3610. Or there could be an empty column at the
end, in which case we would have 216000 + 60 + 0 = 216060 – quite a large difference. Apparently, Mesopotamians tended to rely on the context in which the numbers were used in order to read
them in the most reasonable way.
Early Arithmetic
Many people consider the abacus to have been the ancient world’s version of the electronic calculator. In fact, the counting frame with beads
– which most people think of when they hear the word abacus – is a relatively modern piece of technology, first made popular in China after
AD
1000.
The word ‘abacus’ is thought to come from the Hebrew word for ‘dust’, and the first abacuses were simply that – a board or level surface
strewn with dust that could be used as a scratch pad for calculations. Eventually the dust was replaced by a board with tokens that could be placed in columns to allow for the addition of large
numbers without having to be able to count higher than ten.
Later, the Romans used pebbles or, in Latin,
calculi
(from which we get the words calculus and calculation). In England, we called the tokens
‘counters’, which is why shops had a counter-top to put their counting board on.
Multiplication madness
The second problem arose when the Mesopotamians tried to multiply numbers together. Whichever way you multiply using our decimal system, you need to have memorized your times
tables up to 9 x 9 (because 9 is the highest digit we have). However, according to the Mesopotamian system, you needed
to know your times tables up to 59 x 59! We think they
used a few key times tables, written on small tablets, to help, but even so their times-table tests at school must have been a nightmare.
Archaeologists have found many hundreds of clay tablets littered with Mesopotamian mathematics. It seems the Mesopotamians were able to use fractions, to work out the areas of rectangles and
triangles, and to solve quite complicated equations. My favourite fact is that many tablets found appear to have been maths homework! But maybe you need to be a maths teacher to appreciate that
...
A
NCIENT
E
GYPTIAN
M
ATHEMATICS
The Ancient Egyptians were a talented lot. In addition to building the pyramids, many of which are still standing over 4,000 years later, they also turned their hands to
committing the written word to papyrus, a paper-like material made from interwoven reeds. Papyrus was a much more forgiving material to write on than the clay tablets the Mesopotamians were using
further north. As such, unlike the Mesopotamians, the Ancient Egyptians were not limited to using a single symbol. However, because papyrus rots, especially if it gets wet, it does mean that a vast
majority of the writing of Ancient Egypt has been destroyed over time.
Systems in place
It also seems that the Egyptians were not limited to one writing system.
Egypt is famous for its hieroglyphics – pictograms they carved on to their monuments, and which remained a complete mystery until French soldiers unearthed the Rosetta Stone in 1799.
Hieroglyphics were the Ancient Egyptian equivalent of calligraphy – decorative writing for use only on wedding invitations and inscriptions. The Egyptians had another writing system called
hieratic, which they used for everyday stuff – a much easier and faster way to write script that scribes would then use for their calculations.
Hieroglyphic numbers had symbols for 1, 10, 100, 1000, etc., which the Ancient Egyptians would combine to make the required number:
So if an Egyptian wanted to refer to Rameses’ 1,234 chariots on his latest obelisk, he would have used the following symbols:
Because they added up the symbols in order to generate a total, the Egyptians could write the symbols in any order and direction they pleased – a
handy tool when they wanted to be decorative.
The hieratic number system was a little more complicated because it used different symbols for each unit, each ten, each hundred and each thousand. The symbols for 40 and 50, for example, bore
no relation to each other. It seems that this system relied on the fact that the scribes would be familiar with the symbols and would be able to perform calculations either in their heads, or by
converting to the hieroglyphic system for tricky sums.
The business of numbers
Three important sources of mathematical information were left behind from Ancient Egypt: the Rhind Papyrus, the Moscow Mathematical Papyrus and the Berlin Papyrus. All three
documents contained mathematical problems in arithmetic and geometry, alongside, interestingly, the first written information about pregnancy tests.
From these three sources we have learned that the Egyptians used fractions. However, they only used fractions that had a numerator of 1 – that is, the number on the top of the fraction
could only be 1. They would talk about more complicated fractions by adding these
unit fractions
together. So, for example, they would think of ¾ as ½ + ¼. Although
slightly cumbersome, this method stood the test of time – unit fractions were still used by mathematicians in medieval times.
The pyramid builders obviously had a pretty good grasp of
geometry; the papyri contain detail about how they set about making these ancient structures. The pyramids were
made with stacks of stone blocks in layers, and the steepness of a pyramid depended on the size of the overlap between two layers – the larger the overlap, the steeper the pyramid. The
Egyptians devised a series of methods to work out what size of overlap was needed for different gradients. It has also been suggested the Egyptians had some idea of Pythagoras’ theorem (see
here
), which would have enabled them to work out the third length of a right-angled triangle if they knew the length of the other two sides.
There are, of course, many more far-fetched theories regarding the Ancient Egyptians, including the super-high technology they appropriated from the legendary island of Atlantis (or from aliens,
or time travellers...). I cannot say whether such things were true, but I do know the Egyptians were pretty clever fellows.
A Tall Order
The fact about the Ancient Egyptians that I always find most extraordinary is that the Great Pyramid, which was completed
c
. 2560
BC
, was the tallest building in the world until the central towers of Lincoln Cathedral were raised in
AD
1311 – that’s the best part of 4,000
years!
T
HE
M
AYANS
By the first millennium
AD
Mayan civilization had reached a level of cultural and mathematical development similar to the Mesopotamians and the
Egyptians. They declined somewhat as time went by, but when the Spanish conquistadores arrived in the early 1500s the Mayans had managed to recapture their previous levels of sophistication.
Born in isolation
The Mayans left behind a raft of evidence that demonstrated how they conducted their mathematics, but unfortunately virtually all of it was destroyed when the Spanish invaders
arrived and sought to convert the region’s heathens to Catholicism.
The Dresden Codex
is one of three surviving examples of Mayan writing. Although it was badly damaged during the
Second World War, the book still contains a great deal of insight into the Mayan development of mathematics. Many surviving monuments in modern-day Mexico and Guatemala contain numerical
information, such as dates, inscribed upon them.
Unlike the cross-pollination that occurred between the Mesopotamian and Egyptian cultures, the Mayans developed in complete isolation. They also failed to fulfil the last two criteria of
‘civilization’: they did not possess the wheel, perhaps because there were no beasts of burden in the parts of Central America where they lived; they also did not seem to be able to
smelt metal. However, despite technically still existing in the Stone Age, the Mayans were able to build great cities, some of which contained populations of over 50,000 people.
Number crunching
So, what of their mathematics?
The Dresden Codex
is concerned only with astrology and astronomy, so everything we know about the Mayans’ mathematics is shone
through this lens.
The Mayans used a base-20 system, within which lay a base-5 system (much like the Mesopotamians’ base-60 and base-10 system).
Like modern mathematics, Mayan mathematics had a grasp of
place value
: the value attached to the position of each digit. Unlike modern mathematics, Mayan mathematics
placed numbers in vertical stacks, with the highest place value positioned at the top. Because the Mayans counted in groups of twenties, each level in the stack was twenty times the value of the
level below. From the bottom it went something like this: 1s, 20s, 400s, 8000s, etc. So our number 8577 is one 8000, one 400, eight 20s and seventeen 1s, which in Mayan looked like this:
