Question

Write an example where difference of two irrational is a rational number

Answer 1

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The short answer to your question is that is not necessarily true. For instance, 2–√,2–√−1,1−2–√ are all irrational but

2–√+(1−2–√)=1∈Q

and

2–√−(2–√−1)=1∈Q

However, it is worth noting that if x and y are irrational, then either x+y or x−y is irrational i.e. x+y and x−y cannot be both rationals. The proof for this is given below.

Proof

If both x+y and x−y are rational, then

we have that

x+y=p1q1

and

x−y=p2q2,

where p1,p2 ∈Z

and q1,q2 ∈ Z∖{0}.

Hence,

x=p1q1+p2q22=p1q2+p2q12q1q2

and

y=p1q1−p2q22=p1q2−p2q12q1q2.

Now

p1q2+p2q1,p1q2−p2q1∈Z,

whereas 2q1q2∈Z∖{0}.

This contradicts the fact that x and y are irrationals. Hence, if x and y are irrational then either x+y is irrational or x−y is irrational.