Expert: Ahmed Salami - 6/13/2004

Question
Determine whether the following statement is true or false.

The sum of any two irrational numbers is also an irrational number.

Answer
Hi Stef,
Sorry for the delay.
Let me start by saying a few things.
(1)x and -x are either both rational or irrational.
(2)The sum of a rational and an irrational number is always irrational.
proof:
On the contrary, let us assume the sum is rational and let the numbers be x and (a/b), we have
x + (a/b)= c/d
x = c/d - a/b
this is a contradiction because x would then be rational.
(3)The sum of two rational numbers is always rational, for
a/b + c/d = (ad + bc)/bd

Now to our question,
Let the two numbers be x and y and let us assume that the sum is rational. Therefore,
x + y = a/b
x = a/b - y = a/b + (-y)  
x is therefore the sum of a rational and an irrational number which is irrational as proven before. This is consistent with our assumption.
On the other hand, assume x,y,z are irrational such that
x + y = z
then,
x = z - y = z + (-y)
x is now also the sum of two irrational numbers which is irrational as before. It is therefore also consistent.
The statement is therefore not always true.

considering numerical examples, the numbers (1 - sqrt2), sqrt2 and are both irrational.
But,
(1 - sqrt2)+ sqrt2 = 1 (which is rational)
sqrt2 + sqrt2 = 2sqrt2 (which is irrational)

I hope this makes sense to you. If you are unsure of anything, get back to me.
Regards.