Subject: Re: [HM] Binary Numbers
From: Lambrou Michael (lambrou@itia.math.uch.gr)
Date: Mon Jan 31 2000 - 07:50:18 EST
> Christian Marinus Taisbak wrote:
>
> If I am not mistaken, the ancient egyptian method of multiplication
> presupposes the knowledge that any natural number can be represented
> by a (unique) sum of different powers of 2.
>
Yes, the Rhind Mathematical Papyrus requires expressing numbers to be
multiplied as sums of powers of 2. But I need to make two small comments
on Marinus' note.
First there is no need to require that numbers are so expressed in a
UNIQUE way. The method is just as good even if numbers could be expressed
in many ways as sums of powers of 2.
Secondly, there is no evidence to suppose that Ahmes, the author of
the celebrated papyrus, knew that EVERY natural number can be expressed
as a sum of powers of 2. To claim that he knew, is to claim that he knew
induction (let alone that he probably did not even know what proof is, as
we understand it to-day, or as in, say, the Elements). All that Ahmes
needed was an ad hoc method to write any number he met, as the mentioned
sum decomposition: I asked my neighbour's youngest son (good in maths) to
express various numbers that appear in RMP as the said sum. He could do
all of them in no time. When I asked him to convince me that ALL numbers
could be so decomposed, I received the answer I was expecting: a blank
face.
All the best,
Michael
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