| Posted on Thursday, 08 May, 2003 - 12:49 pm: |
I was stuck on the following questions and was wondering if you could help. 1) Let a, b be rational and x irrational. Show that if (x+a)/(x+b) is rational then a=b. 2) Let x, y be rationals such that (x2 + x + Ö2)/(y2 + y + Ö2) is also rational. Prove that either x=y or x+y = -1. 3) Prove that if n is any positive integer then Ön + Ö2 is irrational. Thanks.
| Posted on Thursday, 08 May, 2003 - 02:57 pm: |
For
1>
assume a not equal to b, then take (x+a)/(x+b) = c/d (since (x+a)/(x+b) is rational) then simply try and get a contradiction.
for 2> let (x2 +x+Ö2)/(y2 +y+Ö2)=c/d performing dividendo we get,
now simply finish this one off using some logical arguments.
for 3> if n is any positive integer then Ön is either rational or irrational.
Now what do we know about Ö2??
and then what can we say about Ön + Ö2??
love arun
P.S-> if anyone finds any mistake above please do point out and correct them if possible.All comments are welcome.
assume a not equal to b, then take (x+a)/(x+b) = c/d (since (x+a)/(x+b) is rational) then simply try and get a contradiction.
for 2> let (x2 +x+Ö2)/(y2 +y+Ö2)=c/d performing dividendo we get,
|
love arun
P.S-> if anyone finds any mistake above please do point out and correct them if possible.All comments are welcome.
| Posted on Thursday, 08 May, 2003 - 03:14 pm: |
For (2) you can also use (1) to show that x2 + x = y2 + y.
(3) If expression is rational then show that Ön - Ö2 is also rational. Now show that Ö2 must also be rational, a contradiction. [Do you know how to prove that Ö2 is irrational ?]
| Posted on Thursday, 08 May, 2003 - 04:00 pm: |
I know now how to solve 1)
I was wondering if you could show me how to do 2)
For 3) I am thinking that since (2)^0.5 is irrational and (n)^0.5 can be either rational or irrational. Looking at the case when (n)^0.5 is rational, we can prove by contradiction that a rational-irrational=irrational. When (n)^0.5 is irrational we know that in general,irrational-irrational=(rational or irrational). But the only way of obtaining a rational as an answer is if the two irrationals that we are subtracting cancel each other out or a real number occurs. This can't occur for this case because the irrational is of the form (n^0.5) therefore for this question an irrational-irrational=irrational.
Thanks
| Posted on Thursday, 08 May, 2003 - 04:24 pm: |
For (2) did you understand what Demetres said? We have to show (x2 + x + Ö2)/(y2 + y + Ö2) is irrational unless x=y or x+y = -1. From (1) we know that if r is irrational then (r+a)/(r+b) is also irrational unless a=b. In our example in (2) the irrational is Ö(2) and a = x2 + x, b = y2 + y.
Thus, we get to the point, x2 + x = y2 + y
We can solve this by rearranging:
x2 - y2 + x - y = 0
(x + y)(x - y) + (x - y) = 0
(x + y + 1)(x - y) = 0, giving the required results...
In Arun's method if we assume the numerator is not zero then it is rational but the denominator is irrational (make sure you can see why) and hence the fraction is irrational (again, make sure you can see why) contradicting the fact that (c - d)/d is rational. Thus, the numerator must be 0 and the result follows as before...
For (3), to be honest I don't quite get your reasoning of why it can't be a case of irrational - irrational = rational...
Try following Demetres hint for what seems to be the simplest solution...
Marcos
P.S. If I'm wrong anywhere please correct me anyone
Also,
if you need a better explanation don't hesitate to
ask
...
| Posted on Thursday, 08 May, 2003 - 05:39 pm: |
Hi Marcos
Question 2 is crystal clear but for question 3, I know how to prove that (2)^0.5 is irrational by contradiction but I was wondering if you could show me how to solve the whole of question 3.
Thanks
| Posted on Thursday, 08 May, 2003 - 07:41 pm: |
Consider Ön + Ö2)(Ön - Ö2) = n-2 . Now, if we assume Ön + Ö2 is rational, it follows (by the rationality of the RHS) that
I hope that clears things up now,
Marcos