You are watching: The sum of 2 rational numbers is rational

I know this statement is false (if I am correct) but how to prove it"s false?

"The sum of two rational numbers is irrational."

2. I know this statement is true (if I am correct) but how to prove it"s true?

"The sum of two irrational numbers is irrational"

I used the example $\sqrt{2}+ \sqrt{3} = 3.14$

But i may need to use proof by contradiction or contaposition.



If two numbers are rational we can express their sum as$$\frac{a}{b} + \frac{c}{d}$$which is equal to $$\frac{ad + bc}{bd}.$$Hence, rational.

The sum of two irrational numbers may be irrational. Consider $2+\sqrt2$ and $3+\sqrt2$. Both are irrational, and so is their sum $5+2\sqrt2$.


For one, it comes directly from the closure of addition on $\altoalsimce.orgbb{Q}$, but I don"t think that"s the answer they would expect.

Let $a = \dfrac{p_1}{q_1}$ and $b = \dfrac{p_2}{q_2}$ be rationals in $\altoalsimce.orgbb{Q}$ and $q_1, q_2 \neq 0$:$$a + b = \dfrac{p_1}{q_1} + \dfrac{p_2}{q_2} = \dfrac{p_1q_2 + p_2q_1}{q_1q_2} \in \altoalsimce.orgbb{Q}$$

For the second one, how about $\dfrac{\sqrt{2}}{2} + \dfrac{\sqrt{2}}{2} = \sqrt{2}$. A single example is sufficient to prove the claim.

For bonus points, can you prove that $\dfrac{\sqrt{2}}{2}$ is irrational?(Hint: Contradiction. Suppose it"s rational, and use the closure of addition on $\altoalsimce.orgbb{Q}$ that was proven.)


$\frac pq$+$\frac xz$ $(q,z \neq 0)$(by formula of rational numbers).

=$\frac{pz+qz}{qz}$,which is again in the form $\frac ab$ so it is bound to be rational and also $qz$ is not equal to $0$.

Sum of irrational may be irrational is true but it is always rational if the sum consists of the irrational number and its negative and then the sum will yield $0$.Sum of two irrational numbers that you expressed as a decimal is not true and only an approximation.


The sum of two irrational numbers is not necessarily irrational. For example, $\sqrt{2}$ and $-\sqrt{2}$ are two irrational numbers, but their sum is zero ($0$), which in turn is rational.

Thanks for contributing an answer to altoalsimce.orgematics Stack Exchange!

Please be sure to answer the question. Provide details and share your research!

But avoid

Asking for help, clarification, or responding to other answers.Making statements based on opinion; back them up with references or personal experience.

Use altoalsimce.orgJax to format equations. altoalsimce.orgJax reference.

See more: What Should I Do If My Dog Ate A Pencil Will He Be Ok, What Should You Do If Your Dog Ate A Pencil

To learn more, see our tips on writing great answers.

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged irrational-numbers rational-numbers rationality-testing or ask your own question.

Please help me spot the error in my "proof" that the sum of two irrational numbers must be irrational
How to know when can I use proof by contradiction to prove operations with irrational/rational numbers?
site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.11.5.40661

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.