Sunday, February 17, 2008
That √2 is irrational
Rudin's text on mathematical analysis provides a proof that √2 is irrational by showing that the set of rational numbers A consisting of all rational numbers whose square is greater than 2 has no least element and the set B of all rational numbers whose square is lesser than 2 has no greatest element. He considers A first and shows that for any p in A, we can find
a number q that is lesser than p and yet in A. He conjures up from apparently nowhere the rational number
q = p – (p^2 - 2)/(p + 2) ……….(1)
The number q is immediately seen to satisfy the needs of the proof for the set A.
How did he arrive at (1)? Some algebra shows that it does not matter what denominator is chosen in (1) as long as it is a rational greater than p + √2. So this could have been 2p for example. This it turns out is sufficient for us to work up a number such as q in (1) ourselves.
Since p > √2, we expect 2/p < √2. So √2 and hence the rational q we are searching for lies between p and 2/p, and of course q should be greater than √2. But first let us find out by what ratio the number √2 divides the segment (2/p,p). Let us say the ratio is µ. Then
√2 = μ 2/p + (1 - μ) p = μ (2/p-p) + p……….(2)
From which, we get = p / (√2 + p). The above equation suggests that as μ varies between 0 and p / (√2 + p), the number μ 2/p + (1 - μ) p spans the segment (√2,p), which is exactly the interval in which we are looking for rational numbers q. We now can find q, by choosing a appropriate value for μ. Now since p >
√n, we can choose μ = p / (p + p) < p/(√2 + p), so we have q = p/2 + 2/2p = p/2 + 1/p.
How close is this number to the in (1)?. As I said earlier in the post this would be the same number if we replaced the denominator in (1) by (p+p) instead of p + 2.
1) why "..Some algebra shows that it does not matter what denominator is chosen in (1) as long as it is a rational greater than p + √2."?
2) why ".. g lies between p and 2/p"? I see √2 < g < p NOT 2/p < g < p as you wrote.
3) I don't understand from "..what ratio the number √2 divides.." since the end, including expression #2. What means µ as ratio?)
Please, can you explain it easier?
thank you very much
benny
Regarding your first question: Remember that we are interested in showing that there is no smallest number in A so, if one chooses a rational smaller than p + √2 say 's' in the denominator then
q = p - (p^2 - 2)/s < p - (p^2 - 2)/ ( p + √2) = p - (p - √2) = √2
Since we wanted q to be greater than √2, such a choice of 's' will definitely not do in the denominator of (1).
Regarding your second question:
you are right q > √2. But read what I written:
Since p > √2, we expect 2/p < √2. So √2 and hence the rational q we are searching for lies between p and 2/p, and of course q should be greater than √2.
As you noted q must be between √2 and p. The reason I choose 2/p as the lower bound, is that it is easier to pick rationals between 2/p and p rather than √2 and p.(agree?) But we must make sure that the rational we pick between 2/p and p must be greater than √2.
So we first measure using the number μ, how far √2 is from 2/p.
Now as μ varies between 0 and 1 the number μ 2/p + (1 - μ) p attains all values between 2/p and p and in particular when μ= p / (√2 + p) we have equality μ 2/p + (1 - μ) p = √2.
And you choose a value of μ greater than p / (√2 + p) we would have μ 2/p + (1 - μ) p > √2.
That is what I mean by saying that the number μ 2/p + (1 - μ) p spans the segment (√2,p) as μ varies between 0 and p / (√2 + p) and it is in the interval (√2,p) that we are searching for our rational q.
Finally note that μ 2/p + (1 - μ) p is rational if μ is rational.
So if we choose any rational value for μ between (0, p / (√2 + p) ) we would be getting a rational. Since p > √2, I have given one example for μ namely p/( 2p )=1/2 which is definitely less than (p/(√2 + p). When μ = 1/2, the number q = μ 2/p + (1 - μ) p = 1/p + p/2 is a rational greater than
√2 and less than p. So you choose various other rational values for μ such as p/(p + 1000000000) as long as they it is less than p/(p + √2) we would be getting different values for q less than p and greater than √2.
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