Hey can you guys help me out with this problem:

It is given that is rational, where a,b,c and d are positive integers and c is not a square. Show that as a consequence, . Use this result to show that is not rational.

Thanks!

Printable View

- March 6th 2011, 08:12 PMeskimogeniusSurds problem
Hey can you guys help me out with this problem:

It is given that is rational, where a,b,c and d are positive integers and c is not a square. Show that as a consequence, . Use this result to show that is not rational.

Thanks! - March 7th 2011, 05:18 AMSoroban
Hello, eskimogenius!

I have a truly "clunky" approach to the first part . . .

Quote:

. .

.

We have: .

Long division:

. .

Since the division produces a rational answer, the remainder must be zero.

. .

Therefore: .

- March 7th 2011, 01:13 PMeskimogenius
- March 7th 2011, 11:28 PMeskimogenius
Never mind about that second part, my math teacher has solved it. Thank you very much Soroban.

For those who want to know the solution:

The first fraction that was given when compared to the second fraction: it can be seen that b = 1. Now for it to be true, the above proven equation has to hold.

When you sub b=1 into the proven equation above, it can be seen that

However, as a, c and d are positive integers, that fraction has to be less than one, and hence this is impossible because c is a positive integer.

Therefore, the second fraction is never rational.

PS. By fraction I mean everything that is need to be proved irrational and everything that is need to be proved irrational.