Is the square root of - 11.2 a rational number?

No, it is an imaginary number.

Can't take the square root of a negative number without creating a method termed an imaginary number, which is the square root of - 1 which they have used " i " to signify.

Then square root of - 11.2 would be written Sqrt (11.2) i. The Sqrt (11.2) = 3.3466

so the answer is 3.3466i