Question

Prove that square root of 12 is irrational.
**I don't know if I did this correctly

PF: By contrapositive, assume sqrt(12) is rational. Then there exist an a,b as integers such that a/b is written in the lowest terms, and sqrt(12)=a/b. Then by squaring both sides, 12=a^2/b^2. For a/b to be written in the lowest term a or b or both have to be odd. then a^2=12b^2 is even since an even number squared gives and even number. So, a=2n, where n is an integer, thus a^2=4n^2. So 12=4n^2/b^2. b^2 must also be even because 4n^2 can only be divided by and even number (b^2) to get the even number 12. Since both a and b are even, this contradicts at least one being odd. Therefore sqrt(12) is irrational. QED.

Could you tell me any mistakes I made, that would help alot!

Answer 1

Looks good. This is the method usually used to prove that √2 is irrational, but it works for any integer. You can find lots of proofs for that online. Look some up, and see how they use the same argument.

Answer 2

Thank you! Could you also factor sqrt(12) to be 2sqrt(3) and prove sqrt(3) to be irrational by assuming rational?

Answer 3

sure could. But the argument is the same. Might as well work with what you were given, in this case at least.