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Re: Number line
Posted:
Mar 1, 2009 10:02 PM
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A modest but knowledgeable reader of this list wrote me off-list:
>Pythagoreans didn't ever discover irrationals, and could not have discovered >irrationals given their concept of number.
>
>What was discovered (between 415 & 405 BC, and not likely to have been by a >Pythagorean) was the existence of non-co-measurable magnitudes, which are not >irrationals, but a pair-wise property of two magnitudes on not having a common >unit. Thus root-2 and 2 x root-2 are co-measurable... which is why non-co->measurables are not irrationals.
My correspondent was thus saying that the Greeks did not have the concept of the number line because they conceived of "magnitudes" rather than numbers in the way we think of them.
I won't argue. However, I would like to point out one Greek who may have had a partial idea of a number line, namely Zeno. In his "Dichotomy" he argues that one cannot cover a distance of length 1 because first one must reach 1/2 the distance, then 3/4, then 7/8, and so on. What he has done is to show what we would call a one-to-one correspondence between the members of the infinite sequence 1/2, 3/4,...2^n-1/2^n and a subset of the points of the interval [0,1]. Zeno may not have realized it but he was pairing numbers with the points of a line.
James A. Landau
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Northrop-Grumman Information Technology
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