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

to you in grade school.

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

Every wife had 7 sacks

Every sack had 7 cats

Every cat had 7 kits

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.

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