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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?

J. M. ain't a mathematician
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Justin L.
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    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. – mau Jul 26 '10 at 07:32
  • @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. – Justin L. Jul 26 '10 at 07:38
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    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. – Isaac Jul 26 '10 at 07:40
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    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. – mau Jul 26 '10 at 07:43
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    This is also a proposition of Euclid. Isaac's and mau's constructions are similar. – Larry Wang Jul 26 '10 at 08:00
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    "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. – Silverfish Jun 10 '16 at 15:02
  • @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". – Abbas Jaffary Mar 11 '17 at 05:23

5 Answers5

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If you have a segment $AB$, place the unit length segment on the line where $AB$ lies, starting with $A$ and in the direction opposite to $B$; let $C$ be the other point of the segment. Now draw a semicircle with diameter $BC$ and the perpendicular to $A$; this line crosses the semicircle in a point $D$. Now $AD$ is the square root of $AB$.

$\triangle BCD$ is a right triangle, like $\triangle ACD$ and $\triangle ABD$; all of these are similar, so you find out that $AC/AD = AD/AB$. But $AC=1$, so $AD = \sqrt{AB}$.

See the drawing below:

constructing square root of a line segment

mau
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Without the unit-length segment--that is, without something to compare the first segment to--its length is entirely arbitrary, so can't be valued, so there's no value of which to take the square root.

Let the given segment (with length x) be AB and let point C be on ray AB such that BC = 1. Construct the midpoint M of segment AC, construct the circle with center M passing through A, construct the line perpendicular to AB through B, and let D be one of the intersections of that line with the circle centered at M (call the other intersection E). BD = sqrt(x).

AC and DE are chords of the circle intersecting at B, so by the power of a point theorem, AB * BC = DB * BE, so x * 1 = x = DB * BE. Since DE is perpendicular to AC and AC is a diameter of the circle, AC bisects DE and DB = BE, so x = DB^2 or DB = sqrt(x).

edit: this is a special case of the more general geometric-mean construction. Given two lengths AB and BC (arranged as above), the above construction produces the length BD = sqrt(AB * BC), which is the geometric mean of AB and BC.

Isaac
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    FYI: The theorem relied on here is Euclid's proposition 35 (Book III). – Larry Wang Jul 26 '10 at 08:04
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    @Kaestur: Ahh, yes, that's what I've called the "power of a point" theorem (or, at least, that's the relevant form where the point is inside the circle). – Isaac Jul 26 '10 at 08:07
  • Given a concrete segment with some length, the unit length sometimes can be constructed like 3、sqrt(2), etc. – user1206899 Aug 21 '22 at 06:54
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take a line AB of 1 unit. draw a line segment BC perpendicular to AB and join CA. take the radius of CA and with centre of compass on A draw an arc cutting the extension of line AB. that point is square root 2.(this is in grade 9 syllabus for us). what MAU answered is also another way.

lucky
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This answer focuses on OP's last question about the unit-length line.

I would answer the rest of the question exactly as mau. It can be seen that in mau's drawing, the unit length segment is a necessary element, and without that segment this construction is not possible. Now I add a small point here:

If we don't have the unit length segment, we may still be able to complete the construction if we know the length whose square root we are seeking. The rationale is that the value of the length contains information. It implies the relative size of a segment with respect to the unit length. So it is useful!

In particular, in the case where the said length is an integer, we are able to use that information to construct a unit length segment (how?). Then we can apply mau's method.

Saeed
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Distances are reckoned in terms of some unit, so in that sense, a unit length segment is always necessary. That said, you don't necessary have to append on a segment with the number who's square root you want to find, at least in special cases.

For example, suppose you want to find the square root of 5. Construct a right triangle with side lengths 1 and 2. This can be done with straight edge and compass. Then the hypotenuse has length $\sqrt{5} $(times the unit).

The procedure can get more complicated. Since 7 is a prime which is not expressible as the sum of two perfect squares, you need the right triangle to have one side of length 2 and the other side of length $\sqrt{3}$ where $\sqrt{3}$ had been constructed before

TurlocTheRed
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