Irrational Lengths and The Root Spiral

When calculators were not yet invented, it was hard for mathematicians to approximate irrational numbers. The irrational numbers was the offshoot of the discovery that the side length of the diagonal of a square with side length 1 is irrational. But how do mathematicians of the ancient time approximate a segment with length, say, \sqrt{5}? Can they draw a segment whose length is exactly \sqrt{5}?

root spiral

With the knowledge of the Pythagorean theorem, it is possible to create a right triangle with side 1 unit making the diagonal \sqrt{2}. The diagonal can then be used as the side of another right triangle whose shorter side length is 1. This process can go on producing the figure above.

In the figure above, the diagonals produce lengths that are square root of natural numbers, so it is possible to create such segments using only the tools  (compass and straightedge) known to ancient mathematicians.

The figure above is called the square root spiral, root spiral, or spiral of Theodorus.  It was said to be first constructed by Theodorus of Cyrene, a mathematician who lived in the 5th century BC. It is said that Theodorus also proved that all the square root of the natural numbers from 3 to 17 are irrational, except those which are perfect squares.

Image via Wikipedia

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