Question

Prove that (5-3 1/2 )(2 1/2 +5 1/2 ) is a irrational number

Answer 1

 DukeZhang: Just because two numbers are irrational doesn't mean that their sum cannot be rational. 

Consider 1 - sqrt(2), and sqrt(2) 

Both are irrational, yet their sum is 1 and is perfectly rational... 

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You can prove it by contraction 

First of all, if you assume that sqrt(5) + sqrt(2) is rational, then it can be written: 
a / b = sqrt(5) + sqrt(2) 

Then if we square both sides we get: 
a^2 / b^2 = 5 + 2sqrt(5)sqrt(2) + 2 

Now if we move the two and the 5 to the LHS we get; 
a^2 / b^2 - 7 = 2sqrt(5)sqrt(2) 

Combining the two square roots gives: 
a^2 / b^2 - 7 = 2sqrt(10) 

We can see that LHS is rational. So if the expression is rational, then so is sqrt(10), but we know sqrt(10) is irrational, hence the expression must also be irrational. 

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To really completely the proof you should prove that sqrt 10 is irrational. 

Do it by contradiction: 

assume sqrt(10) is rational and that the fraction is cancelled to its lowest terms (that is a and b share no factor), then: 
sqrt(10) = a / b 

Square both sides; 
10 = a^2 / b^2 

Now multiply through by b^2; 
10b^2 = a^2 

Fro this we can see the LHS is a multiply of 10, so must be RHS be a multiple of 10 as well. 

The only way we can get a multiple of 10 after squaring a number, is if the number itself is also a multiple of 10. hence as have a = 10k, put this in: 
10b^2 = (10k)^2 
10b^2 = 100k^2 

Cancel a 10 from both sides; 
b^2 = 10k^2 

And the same thing applies, we can see the RHS is a multiple of 10, and so is the LHS, hencen so is b. So we have both a and b divisible by 10. 

But we assumed that a and b have no common factor. 

By this we see that sqrt(10) is not rational, so is irrational.