Talk:Geometry for Elementary School/A proof of irrationality

Proof for irrationality far too advanced
This is supposed to be an elementary school text, but this proof employs algebraic manipulation very casually. This should be removed or rewritten. Papna (talk) 02:22, 15 February 2009 (UTC)
 * I was thinking the same thing. Another question: How does this even relate to geometry? It appears to be mostly redundant to High School Mathematics Extensions/Mathematical Proofs, so maybe this page could be deleted. Belteshazzar (talk) 20:17, 25 May 2010 (UTC)


 * It is also at Famous_Theorems_of_Mathematics/√2_is_irrational, but I don't see this a redundancy. Different books should be encouraged to cover material as appropriate even if it was covered elsewhere.  The number of modules here that introduce the concept of a set is also quite high.  Believe it or not I did follow Papna's suggestion and rewrote the article somewhat (compare the current article to how it looked when Papna made the above comment .   I tried to make this proof more accessible by recasting it terms of even and odd numbers instead of using words like "coprime" and trying to make clear exactly which  number divides both the numerator and denominator.  As far as appropriateness of the subject goes, I can only say that I have taught the same subject to sixth graders with some success, but whether or not sixth grade is stilled considered elementary school varies by district.  But I am sure the other active editors agree with you.  If the consensus is to delete it I will not take it too personally.


 * As far as geometry goes there is some this was historically a rather important result. In fact it was so startling geometrically (to the ancient greeks) that the drowned the person who discovered this.  Remember, in Greek times they only used a ruler and compasses.  The greeks believed that every pair of line segments were commensurable.  Meaning you could choose a very little line segment and measure any two line segments by a whole number of these little line segments.


 * Or if you prefer you could choose a new unit of length so that those two particular line segments would both have lengths that are whole numbers. You might need to choose a new system of measurement for every pair of segments, for any fixed two line segments you could find some unit of measure giving them both whole number lengths.  This idea still seems to appeal to many peoples geometric intuition, and this theorem shows it is false.  Specifically the right triangle whose side lengths were (1,1,√2) has the property that there is no unit unit of measure that makes its hypotenuse and one of its legs have whole number length.  For the greeks this would have been a very real counterexample to their intuition. The greeks would have been uninterested in the fact that there was an irrational number, but the fact that there was a triangle that they could construct whose sides were not commensurable was a real problem.  Historically this lead the greeks to make their first study of ratios.  From which we have come about the modern notion of fractions.  (Believe it or not fractions have a long involved history. But I will try to restrain myself from launching into it here.)


 * The other reason this page was here was that this book at one point in time was attempting to give an introduction not so much to geometry, but to Euclidean geometry. Even more it was trying to stick to things that appear in Euclids elements and this proposition is in the later books. Thenub314 (talk) 15:21, 26 May 2010 (UTC)


 * My main concern about this module is not its level of relating to geometry, but its difficulty. Anything beyond the Pythagorean Theorem, I believe, is too difficult for young learners. The only irrational number one should learn at that age would be pi, and even proving that pi is irrational would be too difficult. Introducing e to elementary school students would be, to use a Chinese phrase, pulling on the sprouts to help them grow. Kayau ( talk &#124; email &#124; contribs ) 13:31, 27 May 2010 (UTC)