## Tuesday, December 27, 2011

### 57: Really Old Math

How long has mathematics been around?  Well, by some measures,
almost as long as the human race, since nearly as long as people have
existed they have probably been able to count simple numbers of
objects around them.  But a more interesting question is how long ago
we have documented history of nontrivial human mathematical activity?
Looking at the timeline of mathematics at Wikipedia, we can see
references to ambiguous artifacts like marked rocks and bones from as
long as 70000 BC, and a numerical system was known to have been
invented by the Sumerians in the 3000s B.C.  But the first interesting
mathematical documents are probably the ones known as the Moscow and
Rhind Papyruses, dated from the 1800s B.C.
Both of these ancient Egyptian scrolls are very practical
documents, stating problems and providing solutions.  The Moscow
Papyrus is most noted for describing calculations of the volume of a
frustrum, or cut-off pyramid.  Strangely, the Egyptians seemed more
interested in calculating this than the volume of a full pyramid.  The
Rhind papyrus is much more elaborate, and contains such achievements
as calculating pi to within 1%, and solving first-order linear
equations.  But because modern attitudes, philosophies, and notations
had not been invented yet, the Egyptians had to take some very
convoluted approaches.
For example, today we consider simple fractions like one-fifth,
two-sevenths, or three-eighteenths, to be a basic foundation of our
arithmetic system.  But to the Egyptians, the only "legal" fractions
were unit fractions, with 1 in the numerator.  Thus the value we
easily express as 2/61sts had to be written by the Egyptians as 1/40th +
1/244th + 1/488th + 1/610th.  This looks to me like a real pain in the
butt, but I guess the easy way wasn't as obvious when nobody taught it
Multiplication and division were also not very well-understood in
the Rhind Papyrus-- while these operations were required for practical
purposes, they were implemented by repeated addition and subtraction,
combined with the concept of doubling or halving values.  So, to
multiply a number by 6, they would multiply it by 2, then by 2 again,
to get the number multiplied by 4.  Then they would add it to the
original number multipled by 2.  So, the overall result of this
cumbersome process would be the number correctly multiplied by 6,
though today we consider the calculation much easier.
For more complicated problems, the papyrus often presents an
answer and then verifies it, without showing any actual way the answer
could have been calculated.  This may be simply because the practical
Egyptians solved most problems by trial-and-error:  they would try an
answer, and if it didn't work, tweak it in the right direction until
it did.
But probably the most surprising problem is one that involves
adding together powers of 7, though of course couched in very
convoluted language.  The solution describes 7 houses, 49 cats, 343
mice, 2401 stalks of wheat, and 16801 bushels of grain.  This
coincides nicely with the trick question in the classic 18th century
rhyme:
As I was going to St Ives
I met a man with 7 wives
Kits, cats, sacks, and wives,
How many were there going to St Ives?
But this rhyme was penned before the papyrus was discovered!  Was
this just an odd coincidence?  Or could variants of this problem,
starting in ancient Egypt, been passed around through the subconscious
of Western culture for two millenia, and directly led to the poem?
Another interesting aspect of the Rhind Papyrus is that it
contains mistakes.  Was the author just not that good at math,
repeating calculations by rote without fully understanding them?  Or
was the problem due to a non-math-literate scribe whose copy is our
only version of the document?  Today there is no way for us to tell.
But even if he wasn't that great at math or made a few mistakes, we
owe that ancient scribe, Ahmes, a huge debt for providing us this
amazing insight into the mathematical world of ancient Egypt.
And this has been your math mutation for today.

• Rhind Papyrus at Wikipedia
• Moscow Papyrus at Wikipedia
• Timeline of Mathematics at Wikipedia
• Review of 'The World of Mathematics', my main source on the Rhind Papyrus.
• "As I Was Going To St Ives" at Wikipedia