history of mathematics - National STEM Centre

Activity 4.4
n units
This proof is an early
example of proof by
contradiction. The proof
of the irrationality of
A/2 appeared in the
Investigating and proving
unit in Book 4.
Incommensurables
4 More Greek mathematics
Here is definition 1 from Book X of Euclid's Elements.
I
Those magnitudes are said to be commensurable which are measured by
the same measure, and those incommensurable which cannot have any
common measure.
1 Explain Euclid's definition in your own words.
2 The proposition given below was at one time recorded as proposition 117 in
Book X of Euclid's Elements, but it has now been shown that it was not part of
Euclid's original text. Even so, it certainly originates in ancient Greece.
Proposition: the diagonal of any square is incommensurable with the side.
Suppose the diagonal AC, and the side AB, of a square are commensurable.
Then there exists a unit length in terms of which AC is n units and AB is m
units, and n and m are the smallest possible. This means that the numbers n and
m have no divisors in common (other than the unit). Since the square on the
diagonal AC is twice the square on the side AB, we have n 2 = 2m 1
It follows that n 2 is an even number (since it is twice some number) so
n is an even number; and so m mustbe an ode/number (because n and m
have no divisors in common). Since n is even, n = 2p (say), and its square
being equal to 2m 2 gives us (2p) 2 = 2m 2 ; so 4p 2 = 2m 2 ; so 2p 2 = m 2 .
Thus m 2 , and so m, now turns out to be an even number. But we already
know that m must be an odd number. So we have shown that m is both an
even number and an odd number, which is absurd.
Use this method to write in your own words a proof that V^ is irrational.
In spite of the fact that the main emphasis of Greek mathematics was not on the
arithmetic evaluation of quantities, some Greek mathematicians developed
sophisticated methods of evaluation. Heron of Alexandria, who lived in the 1st
century AD, wrote a book, Metrica, concerned mainly with numerical examples of
practical methods for calculating lengths, areas and volumes. Unlike Euclid's
Elements, Heron's work is more of a set of rules, not all of which were found to be
correct in the light of subsequent knowledge, for solving certain types of problems.
Activity 4.5 gives, as an example, Heron's method for calculating square roots. The
method models a square as a rectangle with sides that gradually approach the same
length.
Activity 4.5 Heron's algorithm to find square roots
The area of the square is A. To find the side
• choose any length, x, for the side of a rectangle,
A A
• calculate the other side of the rectangle as —, since x x — = A as required
x x
A
• find the average of x and — x
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