The basic principles of learning mathematics are : (a) learning should be related to each child individually (b) the need for mathematics should develop from an intimate acquaintance with the environment (c) the child should be active and interested (d) concrete material and wide variety of illustrations are needed to aid the learning process (e) understanding should be encouraged at each stage of acquiring a particular skill (f) content should be broadly based with adequate appreciation of the links between the various branches of mathematics (g) correct mathematical usage should be encouraged at all stages.
– Ronwill
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It follows that, the ratio of the lengths of the original line segments could be expressed as the ratio of integers. But what would be that case, if the ratio cannot be represented in terms of integers.
This leads to the concept of incommensurable line segments and corresponds to the modern proof that √2 is irrational. The incommensurability of the diagonal of a square with its side can be found in Euclid’s Elements.
We had used the following theorem to prove the above statement.
Theorem. Let a and b be integers, not both zero. Then a and b are prime to each other if and only if there exist integers u and v such that au + bv = 1.
Method of contradiction is an important tool to prove various results. The gist of this method is to start with the negation statement and then proceed further.
The well ordering property of N (set of all natural numbers) states that every non-empty subset of N contains a least element. This means that if S be a non-empty subset of N, there is some natural number a in S such that a ≤ x for all x in N. The idea behind the following proof is to use the well ordering property.
The method of infinite descent is an important concept, used to prove that something does not happen. It’s theoretical basis rests on the fact that, there is no such thing as an infinite decreasing collection of positive integers. In other words, we cannot find an infinite collection of positive integers such that n1 > n2 > n3 > · · ·. Let us use this method to prove the irrationality of √2.
The American mathematician Tennenbaum discovered an ingenious proof of the irrationality of √2. His proof is as given below.
Consider a right-angled triangle with sides p, q and r, where we may assume r > p, q. Using Pythagoras theorem, we can write r2 = p2 + q2. Geometrically this means, the area of the two smaller squares is equal to the area of the big square.
We now consider an isosceles triangle, with side lengths as positive integers a, b. Then the theorem implies that a2 + a2 = b2, i.e., 2a2 = b2. Then, √2 = b / a, the ratio of two positive integers.
Without any loss of generality, we may assume that a and b are two smallest positive integers with this property. This means the above squares (green and blue) are the smallest such square with this property.
Tennenbaum’s idea was to fit the two smaller squares, each of area a2, into the larger square of area b2. This is shown in the figure below.
Of course, the two smaller squares must overlap (if they did not, we would have a2 + a2 < b2).
Now PR = b and QR = a. Then, PQ = PR – QR = b – a and VW = a – (b – a) = 2a – b.
In this way, we can show that two green and the black colour quadrilaterals are actually squares. Since the black square region is covered twice, it must have the same area as the region omitted, the sum of the areas of the two green squares. This means
(b – a)2 + (b – a)2 = (2a – b)2 or 2(b – a)2 = (2a – b)2.
This contradicts the assumption that a and b are the minimal values (or the assumption that our original green and blue squares was the smallest such square).
We conclude that no such numbers a and b exist. Therefore, √2 is an irrational number.
As a continuation to this, there will be a second part of this article, will come under the heading Few Irrational Numbers of Importance in Mathematics.
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