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Given

  1. A straight line of arbitrary length
  2. The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude yields its own magnitude.

Is there a way to geometrically construct (using only a compass and straightedge) the a line with the length of the square root of the arbitrary-lengthed line? What is the mathematical basis?

Also, why can't this be done without the unit line length?

  • 5
    I cannot understand your last remark. If the lenght of a segment is 4 units, its square root is 2 units, that is its half; but if it is 16 units, its square root is 4 units, that is a quarter. Thus to make sense of the notion of a square root you must specify which is the unit you use, which is tantamount to have a unit length segment.2010-07-26
  • 0
    @mau I guess it was a bad way of asking why the square root along one dimension is fundamentally different, geometrically, then, say, bisection, which can be done without a unit-length segment.2010-07-26
  • 3
    Maybe because a square root does not have a geometric meaning in one dimension (and maybe not an inherent geometric meaning at all), whereas bisection has to do with ratios, which have geometric interpretation. Square roots are related to the geometric mean, which does have geometric meaning.2010-07-26
  • 4
    So probably the best answer is that square root is not e dimensional invariant; you are mixing lengths (the square root itself) and areas (the original number, that you must see as an area). Greek geometry was very strict in it; only with the rise of algebra such distinctions were lost.2010-07-26
  • 2
    This is also a proposition of Euclid. Isaac's and mau's constructions are similar.2010-07-26
  • 3
    "Also, why can't this be done without the unit line length?" Consider an initial line segment 25 cm long. Should its "square root" be longer or shorter? Sounds simple - you might say it is obvious that $\sqrt{25}=5$ so we should have a rather shorter segment 5 cm long... but we might just as well say that our initial segment was 0.25 metres long, and $\sqrt{0.25}=0.5$ suggesting we need a new segment 50 cm long, double the initial length! Unless we declare a "unit" to be metres or centimetres or inches or whatever, then our initial length is arbitrary and there's no way to square root it.2016-06-10
  • 0
    @Silverfish I was toying w a similar problem: given a segment of length k, can we "reconstruct" a unit length? Depends on k. In my case, say k is the golden ratio. Then if I square it and subtract k from its square, I obtain my unit length. But I can't square arbitrary k w/o defining an absolute unit length? Likewise, I can't divide an arbitrary segment of length k by itself without explicitly knowing the unit length? I can bisect 2, trisect 3, but arbitrary k is not possible, such as a segment "defined" as length "pi" relative to a "unit length".2017-03-11

3 Answers 3