A proof regarding Rational numbers

[thudda]

I have found some trouble in trying to prove this question.please help mw with that.

Q1) If (a+b)/2 is a rational number can we say that a and b are also rational numbers.? Justify your answer.


I have tried the sum in the following way.

Assume (a+b)/2=p/q (As it is rational)
Lets assume a and b are also rational. Then a=m/n , b=x/y where m,n,x,y ε Z and n,y not equal to 0.

∴ p/q = (my+nx)/2ny = (a+b)/2

∴ (a+b)/2 = m/2n + x/2y
= 1/2(m/n+x/y)

for a and b to be rational they has to be equal to m/n and x/y..That is not the case always so we can't say if (a+b)/2 is rational a and b are also rational.

I doubt that this proof is wrong.Please correct that if there's any wrong.

 

[lurflurf]

We do not want to start with assume a and b are also rational as that is what we are trying to show.
The equations do note prove anything.
The question does not make it clear what a and b are, I assume they are from some common number system like real, complex, or algebraic numbers.

The most straight forward thing to do is think of an example of a and b so that (a+b)/2 is rational and a and b are not.

 

[thudda]

I thought about that too but could'nt figure out 2 examples for a and be..But just figured out we can use a=√2 and b=-√2 so that a+b=0 and (a+b)/2=0 which is rational. Any way thanks a lot.

 

[tiny-tim]

hi thudda!

But just figured out we can use a=√2 and b=-√2 so that a+b=0 and (a+b)/2=0 which is rational.

yes that's fine …

the answer to the question is "no", and you've justified it by showing a counter-example!

 

[Natron]

it would seem that if a is irrational and b= (a rational number) - a , then this would be a perfect example of your equation coming out rational with a+b as the numerator. for example a = pi and b = (6-pi). add them together and you get 6 even though both numbers are irrational.