Is the sum and difference of two irrationals always irrational?

If x and y are irrational, is x+y irrational? Is xy irrational?

Answer

The short answer to your question is that is not necessarily true. For instance, 2,21,12 are all irrational but 2+(12)=1Q and 2(21)=1Q


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

Proof

If both x+y and xy are rational, then we have that x+y=p1q1 and xy=p2q2, where p1,p2Z and q1,q2Z{0}.

Hence, x=p1q1+p2q22=p1q2+p2q12q1q2 and y=p1q1p2q22=p1q2p2q12q1q2.

Now p1q2+p2q1,p1q2p2q1Z, whereas 2q1q2Z{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 xy is irrational.


Below are some statements worth knowing.


1 Sum of two rationals is always a rational.

Proof: Let the two rationals be p1q1 and p2q2, where p1,p2Z and q1,q2Z0. Then p1q1+p2q2=p1q2+p2q1q1q2
where p1q2+p2q1Z and q1q2Z{0}. Hence, the sum is again a rational.


2 Sum of a rational and an irrational is always irrational.

Proof: Let the rational number be of the form pq, where pZ and qZ{0} while the irrational number be r. If r+pq is a rational, then we have that r+pq=ab for some aZ and bZ{0}. This means that r=abpq=aqbpbq where aqbpZ and bqZ{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.


3 Sum of two irrationals can be rational or irrational.

Example for sum of two irrationals being irrational

2 is irrational. 2+2=22 which is again irrational.

Example for sum of two irrationals being rational

2 and 12 are irrational. (Note that 12 is irrational from the second statement.) But, 2+(12)=1 which is rational.

Attribution
Source : Link , Question Author : Miguel Mora Luna , Answer Author : user642796

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