Prove or disprove that the sum of two irrational numbers is irrational

Question

Prove or disprove that the sum of two irrational numbers is irrational. How do i answer this? Thanks.

I don't know how to define an irrational number, root a doesn't work and i can't think of another way — Heelloppp, Aug 31 '15 at 10:54
Well understanding the definitions is really the beginning of how to work with these questions, and questions such as this are asked to help you to think through what the definitions imply. So you need to make sure you do know the definition. — Mark Bennet, Aug 31 '15 at 11:01
Perhaps you meant to ask: Given $a, b \in \mathbb R \setminus \mathbb Q, a \ne -b$, does $a + b \in \mathbb R \setminus \mathbb Q $ ? — Dor, Aug 31 '15 at 11:10
This is a duplicate of many, many questions, but I can't find them. Oh well, it's also missing context, so I will close it as that. — Caleb Stanford, Aug 31 '15 at 11:25

Yes · Accepted Answer · 2015-08-31T11:12:21.477

8

Since $-\sqrt{2} + \sqrt{2} = 0$, so the sum of two irrationals may be rational.

edited Aug 31 '15 at 11:12

answered Aug 31 '15 at 10:54

Yes

20,910

How is that the sum though, isn't that the difference? Isn't it the same as saying root 2 - root 2? – Heelloppp Aug 31 '15 at 10:55
a sum takes the form $a_{1}+a_{2} + \cdots$... – Yes Aug 31 '15 at 10:56
So does that prove or disprove the question? It says the sum of two irrational numbers is irrational, but can't it be both rational and irrational? – Heelloppp Aug 31 '15 at 10:58
1

What do you think "irrational" means, @Heelloppp ? – Thomas Andrews Aug 31 '15 at 10:58
It disproves the proposition. – Yes Aug 31 '15 at 10:58
It's like asking if a woman can both be pregnant and not pregnant… – J. M. ain't a mathematician Aug 31 '15 at 10:58
@Heelloppp: A real number that is not rational is called irrational. Very simple, please be noted. – Yes Aug 31 '15 at 10:59
@Heelloppp $\sqrt 2+(-\sqrt 2)=0$ and you will admit that $-\sqrt 2$ is irrational, I think. – Mark Bennet Aug 31 '15 at 11:36

score 3 · Answer 2 · answered Aug 31 '15 at 11:29

($2+\sqrt3$)+($2- \sqrt3$) $=$ $4$ which is rational. So , the statement " sum of two irrationals is irrational" is disproved since counter example is found.

When a statement like that is given to prove or disprove, " sum of two irrationals is irrational" , it is proved if it is found to be always true and disproved if at least one counter example can be given.

In fact, sum of two irrationals can be either rational or irrational. Not necessarily irrational all the time.

Prove or disprove that the sum of two irrational numbers is irrational

2 Answers2