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

**Answer**

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.

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,p2∈Z and q1,q2∈Z∖0. Then p1q1+p2q2=p1q2+p2q1q1q2

where p1q2+p2q1∈Z and q1q2∈Z∖{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 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.

**3** Sum of two irrationals can be rational or irrational.

**Example for sum of two irrationals being irrational**

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

**Example for sum of two irrationals being rational**

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

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