#1: Let:
Assume is rational.
Multiply both sides by the conjugate:
Add and :
Can you see the contradiction?
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#2: What's the question?
I have two questions in my packet that I'm not completely show how to do. Any help is appreciated.
Q1) prove that the sum of square root of 2 and square root of 3 is irrational. [hint: prove square root of six is irrational and then use this as a lemma]
Q2) Suppose a,b,c,d is a set of integers such that a/c and b/c. Show that ab/cd.
Hello -
This is neat. As a matter of interest, it works for any expression of the form , where and and are individually irrational. Just put , multiply both sides by , and continue as before.
Also, if you do want to prove that is irrational, do it in the same way as proving is irrational, as follows:
Suppose is rational, and write it as , where , and and are co-prime. (In other words the fraction is in its lowest terms.)
Then:
has a factor of (and as well, but you don't actually need that)
has a factor , since is prime
So write , say, where
So
has a factor
has a factor
and have a common factor .
Contradiction.
is irrational.
Grandad