Math & Data Science Q&A - Get answers from our AI that are verified by human experts
First time here? Checkout the FAQs!
x

*Math Image Search only works best with zoomed in and well cropped math screenshots. Check DEMO

0 like 0 dislike
71 views
Prove that \(\sqrt{6}\) is irrational.
in Mathematics by Diamond (71,587 points) | 71 views

1 Answer

0 like 0 dislike
Best answer
Suppose that \(\sqrt{6}\) was rational. We show that this leads to a contradiction. We may write \(\sqrt{6}=x / y\) where \(x\) and \(y\) are integers with \(y \neq 0\) and \(\operatorname{gcd}(x, y)=1\). Squaring this equation and cross-multiplying we get that \(6 y^{2}=x^{2}\) or \(2 \cdot 3 \cdot y^{2}=x^{2}\). Therefore, 2 divides \(x^{2}=x \cdot x\). Since 2 is prime we must have that 2 divides \(x\). Similarly, 3 divides \(x^{2}=x \cdot x\).

And since 3 is prime we must have that 3 divides \(x\). Since \(2 \mid x\) and \(3 \mid x\) and \(\operatorname{gcd}(2,3)=1\), by the first part of this problem, we have that \(6=2 \cdot 3\) must divide \(x\). So \(x=6 u\) where \(u\) is a non-zero integer. Subbing this into \(6 y^{2}=x^{2}\) gives us that \(6 y^{2}=6^{2} u^{2}\). Thus \(y^{2}=6 u^{2}\). Following the same reasoning as above, this forces that 6 must divide \(y\). Therefore, 6 is a common divisor of \(x\) and \(y\) which contradicts the fact that \(\operatorname{gcd}(x, y)=1\).
by Diamond (71,587 points)

Related questions

0 like 0 dislike
0 answers
0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Jul 29, 2022 in Mathematics by Gauss Diamond (71,587 points) | 103 views
0 like 0 dislike
1 answer
0 like 0 dislike
0 answers
asked Apr 5, 2021 in Mathematics by MathsGee Platinum (163,814 points) | 200 views
0 like 0 dislike
0 answers
1 like 0 dislike
1 answer
0 like 0 dislike
1 answer
asked Jun 14, 2021 in Mathematics by Student SIlver Status (11,376 points) | 199 views
1 like 0 dislike
1 answer
1 like 0 dislike
1 answer

Join MathsGee, where you get expert-verified math and data science education support from our community fast.


On the MathsGee, you can:


1. Ask and answer questions


2. Get expert-verified answers


3. Vote on questions and answers


4. Tip your favorite community members


5. Join expert live video sessions (Paid/Free)


6. Earn points by participating


7. Start a Fundraiser



Posting on MathsGee


1. Remember the human


2. Act like you would in real life


3. Find original source of content


4. Check for duplicates before publishing


5. Read the community guidelines




MathsGee Rules


1. Answers to questions will be posted immediately after moderation


2. Questions will be queued for posting immediately after moderation


3. Depending on the number of messages we receive, you could wait up to 24 hours for your message to appear. But be patient as posts will appear after passing our moderation.




MathsGee Android Q&A

MathsGee Android Q&A