Question

Please Help me!!!! Square root of 3 is rational or irrational? Why?

Answer 1

Do you know what a rational and irrational number is?

Answer 2

Suppose \(\sqrt{3}=\dfrac{m}{n} \) where \(\dfrac{m}{n}\) is reduced. Then \(3=\dfrac{m^2}{n^2}\implies3n^2=m^2\) so \(m^2\) is a multiple of \(3\) and so then is \(m\). So \(3n^2=(3k)^2\) for some \(k\in \mathbb{Z}\). So \(3n^2=9k^2\implies n^2=3k^2\) so then \(n^2\) is a multiple of \(3\) and thus \(n\) is also a multiple of \(3\). So \(\dfrac{m}{n}\) can be reduced. Contradiction.

Answer 3

May I ask what grade you are in?

Answer 4

me?

Answer 5

No, I meant @brianjrthemonkeycr :)

Answer 6

Thanks

Answer 7

np. This proof will work for any prime. You might want to show that if n^2 is a multiple of prime \(p\) then so is \(n\). But it is pretty easy.

Answer 8

ok, I was confuse thanks

Answer 9

\(n^2=p\implies n*n=p*k\) for some integer \(k\). Since \(n\) divides the left side, it must divide the right, but \(n\) does not divide \(p\) so that \(n\) must divide \(k\) so \(n=pk_0\) where \(k_0= \dfrac{k}{n}\in \mathbb{Z}\).

Answer 10

assuming \(n\ne 1\) of course.

Answer 11

Does this all make sense?

Answer 12

I was going to provide a more basic explanation that would be more suitable if you're in middle school.

A rational number is a number that can be written as a clean cut fraction or ratio. For example, 0.25 can be written as 1/4.
0.3333333.... is also rational, since it can be written as 1/3

Irrational, therefore, are numbers are any numbers that are not rational.
The square root of 3 is 1.73205080757....with ongoing digits
This number cannot be expressed as a clean cut fraction or ratio, and therefore, is irrational.

Answer 13

that should say 'if \(n^2\) is a multiple of prime \(p\) then \(nn=pk\) for some k...'

Answer 14

Supposedly that proof got someone thrown off a boat about 2500 years ago. Math used to be much cooler :)

Answer 15

Thanks to all of you

Answer 16

No problem, good luck!