- #1
Constructivist
- 5
- 2
Hi,I am learning pure math say real analysis from Rudin. Now, I am getting confused on philosophy of learning. Well, if I read Rudin, somehow with struggle, I am able to fill the gaps in the proof which he gives. Though I can completely understand the proof, my question is how was the proof thought of in the first place. To give an example , as shown in the screenshot,Rudin takes q as p - ((p^2 - 2)/(p+2)) to prove that for every rational number p (where p^2 < 2), there exists a rational number q >p such that q^2 <2. Well, I can fill in the gap in the proofs mostly by myself or ask someone for help say in PF. What I am asking over here i, s how was this particular q = p - ((p^2 - 2)/(p+2)) thought of in the first place? So, I was pondering on this. let's say we take q = (sqrt(2) + p)/2 which is a reasonable guess but we cannot take this number as sqrt(2) is irrational and q will turn an irrational. So, next guess is q = (1.4+p)/2. In that case, q is rational but what if p itself is 1.4? Clearly, there lie number between 1.4 and sqrt(2) and we do not know if they are rational or not? So, q has to be a function of p alone and let q = f(p). Also, the properties of the function are:
1. q = f(p) which is greater than p
2. (f(p))^2 < 2.
3. if p is rational number tending to square root of 2, then q also tends to a rational number tending to square root of 2.
So, intuition is to have q = p + (r- p)/m . Where r is rational number closer to square root of 2 and m is a rational number (Here , (r- p)/m is the small value if m is big ). But, this cannot be the case. As we have discussed before, the proof fails if p tends to fixed r. The fact the square root of 2 is irrational is a pain. So, how can we avoid it? Discuss the variables in-terms of squares. Example is let p be rational and p^2 < 2. The trick is to take a q such that 2>q^2>p^2 and have this q as rational. Since p^2 -2 <0, we need to have a q^2-2 <0.
So, let q^2-2 = (p^2 -2)/m where m should be positive. The one concern we have for m is: Is it a rational number say 5 or 10 or 1000 or is it a function of p. If m were a fixed number, then q^2 = ((p^2 -2)+2m)/m and q = sqrt((p^2 -2)+2m)/m). There are some problems with this m. Can (p^2 -2)+2m)/m be rational ? ----a first question. Even, if it is rational, the problem is ' Is q = sqrt((p^2 -2)+2m)/m) rational ?' (Example: 2 is rational but sqrt(2) is not). So, why cannot m be perfect square like 4, it is the same issue (q becomes sqrt(p^2 +6)/2 . Then is sqrt(p^2 +6)/2 is rational? Let m be Ap+B where A nd B are rational. In that case, m is rational. But wait, we never know if sqrt(Ap+B) is rational. So, the better solution is to go with m = (Ap+B)^2. In that case, q becomes sqrt(p^2 -2 + 2 (Ap+B))/(Ap+B). So, if (p^2 -2 + 2 (Ap+B)) is a perfect square, then q is also rational. So, our task is to find A relation between A and B and make this term a perfect square. The equation I got is 2A^2 -B^ + 1 = 0 . A is 1/sqrt(2) and B is sqrt(2). And hence we get q = p - ((p^2 - 2)/(p+2)).My question is do I need to go this depth or understanding the proofs just enough? Do we need to think on how they investigated to get the proof as we read along Rudin or can I just fill in the holes in proofs. I feel such an approach will give some experience in deriving newer formulae in future. Your suggestions are welcome.
1. q = f(p) which is greater than p
2. (f(p))^2 < 2.
3. if p is rational number tending to square root of 2, then q also tends to a rational number tending to square root of 2.
So, intuition is to have q = p + (r- p)/m . Where r is rational number closer to square root of 2 and m is a rational number (Here , (r- p)/m is the small value if m is big ). But, this cannot be the case. As we have discussed before, the proof fails if p tends to fixed r. The fact the square root of 2 is irrational is a pain. So, how can we avoid it? Discuss the variables in-terms of squares. Example is let p be rational and p^2 < 2. The trick is to take a q such that 2>q^2>p^2 and have this q as rational. Since p^2 -2 <0, we need to have a q^2-2 <0.
So, let q^2-2 = (p^2 -2)/m where m should be positive. The one concern we have for m is: Is it a rational number say 5 or 10 or 1000 or is it a function of p. If m were a fixed number, then q^2 = ((p^2 -2)+2m)/m and q = sqrt((p^2 -2)+2m)/m). There are some problems with this m. Can (p^2 -2)+2m)/m be rational ? ----a first question. Even, if it is rational, the problem is ' Is q = sqrt((p^2 -2)+2m)/m) rational ?' (Example: 2 is rational but sqrt(2) is not). So, why cannot m be perfect square like 4, it is the same issue (q becomes sqrt(p^2 +6)/2 . Then is sqrt(p^2 +6)/2 is rational? Let m be Ap+B where A nd B are rational. In that case, m is rational. But wait, we never know if sqrt(Ap+B) is rational. So, the better solution is to go with m = (Ap+B)^2. In that case, q becomes sqrt(p^2 -2 + 2 (Ap+B))/(Ap+B). So, if (p^2 -2 + 2 (Ap+B)) is a perfect square, then q is also rational. So, our task is to find A relation between A and B and make this term a perfect square. The equation I got is 2A^2 -B^ + 1 = 0 . A is 1/sqrt(2) and B is sqrt(2). And hence we get q = p - ((p^2 - 2)/(p+2)).My question is do I need to go this depth or understanding the proofs just enough? Do we need to think on how they investigated to get the proof as we read along Rudin or can I just fill in the holes in proofs. I feel such an approach will give some experience in deriving newer formulae in future. Your suggestions are welcome.