what does 3b^2=a^2 mean???
(I am trying to prove that 3 is an irrational number and after some algebraic some manipulation got left with that.)

Question

what does 3b^2=a^2 mean??? 
(I am trying to prove that 3 is an irrational number and after some algebraic some manipulation got left with that.)

akshay_budhkar · Answer

what are a and b here?? natural numbers,integers,real numbers?

anonymous · Answer

natural numbers

anonymous · Answer

would help if I showed you what I done before hand??

akshay_budhkar · Answer

yea it would be great :D

anonymous · Answer

I am trying to prove √3 is a irrational number by contradiction right, so
√3 = a/b ( where a and b are simplified to lowest terms)
Next I raised the equation to the ^2 to get 3 = a^2/b^2
Next I multiplied both sides of the equation by b^2 to get 3b^2=a^2.

akshay_budhkar · Answer

Assume that sqrt3 is rational and equal to a/b in lowest terms with a and b integers. Squaring both sides gives 3 = a^2/b^2 so 3b^2 = a^2. If a^2 is a multiple of 3 and a is an integer then a itself must be a multiple of 3 (see question a few prior to this one). Since a is a multiple of 3 replace it by 3c. Then 3b^2 = (3c)^2 = 9c^2 which reduces to b^2 = 3c^2. Again b^2 a multiple of 3 means b itself is a multiple of 3. We have now shown that a, b are both multiples of 3 contradicting the statement that a/b was in lowest terms, i.e. no common factors. This proves that the original assumption that sqrt3 is rational must be false.

akshay_budhkar · Answer

:D

akshay_budhkar · Answer

this is how i had replied once to some1 who needed help in proving that.. hope it helps :)

akshay_budhkar · Answer

you get it right?

anonymous · Answer

I dont get why we have to substitute 3c in for a^2

akshay_budhkar · Answer

as it is a multiple of 3.. say it was 9 so we substituted it as 3*3

anonymous · Answer

so what relationship do 3c and a^2 have?

akshay_budhkar · Answer

we have a=3c

anonymous · Answer

and why is it 3c and not just 3?

akshay_budhkar · Answer

see we have a^2 is a multiple of 3 right?? as it is 3b^2..so as a^2 is a multiple of 3 we have by implication even a is a multiple of 3... it is a multiple so we dont write just 3 but as 3c as a multiple can be any number other than 3 as well

anonymous · Answer

ok great

anonymous · Answer

so a rational number can be expressed as a fraction in lowest terms and a irrational number cant