**1.**

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

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

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

"The amount of two rational number is irrational."

**2.** I understand this declare is true (if ns am correct) however how come prove it"s true?

"The sum of two irrational number is irrational"

I supplied the instance $sqrt2+ sqrt3 = 3.14$

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

If 2 numbers room rational we deserve to express their sum as$$fracab + fraccd$$which is equal to $$fracad + bcbd.$$Hence, rational.

The amount of two irrational numbers may be irrational. Consider $2+sqrt2$ and also $3+sqrt2$. Both room irrational, and also so is their amount $5+2sqrt2$.

For one, the comes directly from the closure of addition on $hunterriverpei.combbQ$, but I don"t think that"s the prize they would expect.

Let $a = dfracp_1q_1$ and $b = dfracp_2q_2$ be rationals in $hunterriverpei.combbQ$ and also $q_1, q_2 eq 0$:$$a + b = dfracp_1q_1 + dfracp_2q_2 = dfracp_1q_2 + p_2q_1q_1q_2 in hunterriverpei.combbQ$$

For the 2nd one, how around $dfracsqrt22 + dfracsqrt22 = sqrt2$. A single example is enough to prove the claim.

For bonus points, have the right to you prove that $dfracsqrt22$ is irrational?(Hint: Contradiction. Intend it"s rational, and also use the closure of addition on $hunterriverpei.combbQ$ the was proven.)

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

=$fracpz+qzqz$,which is again in the form $frac ab$ so that is bound to be rational and also $qz$ is no equal to $0$.

Sum that irrational might be irrational is true yet it is always rational if the sum is composed of the irrational number and its an unfavorable and then the amount will yield $0$.Sum of 2 irrational numbers that you expressed together a decimal is no true and only an approximation.

The amount of 2 irrational number is not necessarily irrational. Because that example, $sqrt2$ and $-sqrt2$ space two irrational numbers, yet their sum is zero ($0$), which subsequently is rational.

Thanks for contributing solution to hunterriverpei.comematics Stack Exchange!

Please be sure to*answer the question*. Carry out details and also share your research!

But *avoid* …

Use hunterriverpei.comJax to style equations. hunterriverpei.comJax reference.

See more: How Far Is Sacramento From Los Angeles, Ca, How Far Is Los Angeles From Sacramento

To discover more, watch our advice on writing great answers.

article Your answer Discard

By clicking “Post your Answer”, friend agree come our regards to service, privacy policy and also cookie policy

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

Please help me point out the error in mine "proof" the the sum of 2 irrational numbers must be irrational

exactly how to recognize when can I usage proof through contradiction to prove operations v irrational/rational numbers?

site design / logo © 2021 ridge Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.11.12.40742

your privacy

By clicking “Accept all cookies”, friend agree ridge Exchange have the right to store cookies on your machine and disclose info in accordance with our Cookie Policy.