Your question, #-sqrt(12)#, can be simplified as follows,
#-sqrt(12)#
#-sqrt(4 * 3)#
#-(sqrt(4) * sqrt(3))#
#-(2 * sqrt(3))#
#-2sqrt(3)#
A rational number is any number that can be expressed as a fraction of two whole numbers. In our case, the two is rational since it can be expressed as,
#2/1# or #4/2# or #800/400# and so on...
However, we have a problem with #sqrt(3)#. It's approximated value is around 1.732... (and it doesn't repeat).
That is just the problem. There are no two real integers that can exactly express #sqrt(3)#. The fact that we had to approximate it's value already raises some flags - since by definition, a rational number when expressed as a decimal always terminates or repeats.
Overall, we have a rational number (-2) times an irrational one (#sqrt(3)#). There are some proofs online (the one I saw proves by contradiction) that will reassure you that the product of a rational number and an irrational number is irrational.
