Question

Please show that the sum of two irrationals can be rational

Answer 1

\[(\sqrt{2}) + (-\sqrt{2}) = 0\]

Answer 2

Two irrational numbers very RARELY add up to a rational number.

Any integer, finite decimal, or repeating decimal is a rational number. A
rational number can be represented as a fraction. An irrational number
cannot. It IS true that two RATIONAL numbers add up to a rational number:
two fractions always add up to a fraction. Here are two examples:
(1/2)+(1/3)=(5/6), 1+3=4. The latter applies because you can represent it
as (1/1)+(3/1)=(4/1).

However, sqrt(2) is not rational because there is no fraction, no ratio of
integers, that will equal sqrt(2). It calculates to be a decimal that never
repeats and never ends. The same can be said for sqrt(3). Also, there is
no way to write sqrt(2)+sqrt(3) as a fraction. In fact, the representation
is already in its simplest form. To get two irrational numbers to add up to
a rational number, you need to add irrational numbers such as [1+sqrt(2)]
and [1-sqrt(2)]. In this case, the irrational portions just happen to
cancel out, leaving:
[1+sqrt(2)]+[1-sqrt(2)]=2. 2 is a rational number (i.e. 2/1).