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!
3 answers
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.
Thank you! Could you also factor sqrt(12) to be 2sqrt(3) and prove sqrt(3) to be irrational by assuming rational?
sure could. But the argument is the same. Might as well work with what you were given, in this case at least.
1 answer