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Re: Who discovered irrational numbers?
Posted:
Mar 27, 2009 2:08 AM
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A brief history of the square and the square root of 2
(answering a question posed by Hossam in another thread)
Mathematics began with calendars. Laying out calendar
patterns using shells and small pebbles in regular and
syncopic lines involved squares (Lebombo Africa 30,000
years ago, Lascaux Europe 18'000 years ago). Later on
the square became of interest as a geometrical figure,
and for a long time a simple definition was in use
(my claim):
A square has four sides that are equally long, each one
measuring five paces, or a multiple of five paces, and
a pair of diagonals that are again of the same length,
each one measuring seven paces or a multiple of seven
paces. And if the side measures a multiple of seven
paces, the diagonal a multiple of ten paces, or two
times five paces.
An Egyptian mathematician made a big step forward by
adding the numbers: five and seven are 12, seven and
ten are 17, and two times twelve is 24. Proceeding
in both directions he obtained a number column whose
first lines are 1 1 2, 2 3 4, 5 7 10, 12 17 24,
29 41 58, 70 99 140, 169 239 338, 408 577 1970 ...,
providing many handy values and thus allowing easy
calculations. How long is the diagonal of a square
if the side measures 10 royal cubits? Ten cubits are
70 palms, the diagonal measures 99 palms or 14 cubits
1 palm. I found ample evidence for the use of this
number column for the calculation of the square and
the octagon in the Rhind Mathematical Papyrus (read
on the advanced level). The Babylonians divided 1393
by 985, obtained 1;24.51,10,3,2..., let go the small
figures and kept 1;24,51,10 as excellent value for
the square root of 2 as mentioned on the Babylonian
clay tablet YBC 7289 from around the same time as
the Rhind Mathematical Papyrus.
Then came the Greeks and made one more big step forward
by converting the above number column into the continued
fraction (1;2,2,2,2...), having studied geometry in the
Nile Valley, paying homage to the Egyptians, for example
Aristotle: peri Aigypton hai mathaematikai proton technai
synestaesan.
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