This property of the mediant was used by Gómez Morin  to find the roots of all orders. Here we describe his method for finding square-roots.
The idea is to generate a sequence of nested intervals, each containing the square root that is sought. Then by the Nested Interval Property of real numbers, the real number representing the square root is the limit of the sequence.
Suppose we wish to find the square root of 2.
Take two positive fractions f1 and f2 whose product is 2. Say f1 < f2. Form their mediant M.
Then f1 < M < f2.
CLAIM: f1 < 2/M < f2.
To prove the claim, suppose that the opposite is true.
That is either 2/M £ f1 or 2/M ³ f2.
If 2/M £ f1, then 2 £ Mf1. But 2 = f1f2. So f1f2 £ Mf1. That is, f2 £ M, a contradiction.
Similarly we find that 2/M ³f2 is also impossible.
Since both M and 2/M are in the interval (f1,f2). We now continue by taking the mediant of M and 2/M and produce a new interval that is contained in (M,2/M).
Thus we obtain a sequence of nested open intervals. Each of these intervals contains √2 - see proof elsewhere that √Q is always in the interval (M,Q/M) when Q is not an integer that is the square of another integer.
|f1 = 1/1 || f2 = 2/1 |
|M = 3/2 ||2/M = 4/3 |
|7/5 ||10/7 |
|17/12 ||24/17 |
|41/29 ||58/41 |
|... ||... |
 Gómez Morín, Domingo. “La Quinta Operación Aritmética, Media Aritmónica”, 2nd. Edition, 2006. ISBN:980-12-1671-9
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