1.

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

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

"The amount of 2 rational number is irrational."

2. I recognize this declare is true (if i am correct) yet how to prove it"s true?

"The sum of 2 irrational numbers is irrational"

I offered the example $sqrt2+ sqrt3 = 3.14$

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


*

*

If two numbers are rational we can express their amount as$$fracab + fraccd$$which is equal to $$fracad + bcbd.$$Hence, rational.

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


*

For one, that comes straight from the closure of addition on $starrkingschool.netbbQ$, but I don"t think that"s the price they would certainly expect.

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

For the second one, how about $dfracsqrt22 + dfracsqrt22 = sqrt2$. A solitary example is adequate to prove the claim.

For bonus points, have the right to you prove that $dfracsqrt22$ is irrational?(Hint: Contradiction. Expect it"s rational, and use the closure of enhancement on $starrkingschool.netbbQ$ that was proven.)


*

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

=$fracpz+qzqz$,which is again in the form $frac ab$ so the is bound to be reasonable and additionally $qz$ is not equal to $0$.

Sum that irrational may be irrational is true but it is constantly rational if the sum consists of the irrational number and also its negative and then the amount will yield $0$.Sum of two irrational numbers that you expressed as a decimal is no true and only one approximation.


*

The amount of 2 irrational numbers is no necessarily irrational. For example, $sqrt2$ and also $-sqrt2$ room two irrational numbers, but their amount is zero ($0$), which in turn is rational.


Thanks for contributing response to starrkingschool.netematics Stack Exchange!

Please be certain to answer the question. Carry out details and share her research!

But avoid

Asking because that help, clarification, or responding to other answers.Making statements based upon opinion; ago them increase with referrals or an individual experience.

Use starrkingschool.netJax to style equations. starrkingschool.netJax reference.

See more: Where Is Bank 1 On A 2000 Subaru Which Side Is Bank 1 Bank 2

To discover more, check out our advice on writing great answers.


article Your answer Discard

By click “Post your Answer”, girlfriend agree come our regards to service, privacy policy and also cookie plan


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


Please help me spot the error in my "proof" the the amount of two irrational numbers should be irrational
exactly how to know when have the right to I usage proof through contradiction to prove operations through irrational/rational numbers?
site architecture / logo © 2021 stack Exchange Inc; user contributions licensed under cc by-sa. Rev2021.11.10.40696


her privacy

By clicking “Accept every cookies”, friend agree stack Exchange deserve to store cookies on your device and disclose info in accordance with our Cookie Policy.