Question

5. Prove that if x an irrational number and y is rational number, then x+y is an irrational
number.

Answer 1

Let x+y be rational, then
\[x+y=\frac{ p }{q },(say),wher p and q are integers.\]
\[x=\frac{ p }{ q }-y=\frac{ p-qy }{y }=\frac{ s }{ t },(say)where s and t are integers.\]
or x is a rational number.
which is a contradiction.
because x is a rational number. (given)
Hence our assumption is wrong.
or x+y is an irrational number.

Answer 2

Sum of a rational and an irrational is always irrational.

Proof: Let the rational number be of the form pq, where p∈ℤ and q∈ℤ∖{0} while the irrational number be r. If r+pq is a rational, then we have that r+pq=ab for some a∈ℤ and b∈ℤ∖{0}. This means that r=ab−pq=aq−bpbq where aq−bp∈ℤ and bq∈ℤ∖{0}. This contradicts the fact that r is irrational. Hence, our assumption that r+pq is a rational is false. Hence, r+pq is a irrational.

Answer 3

Thank you

Answer 4

give me a medal :)