Is it by definition that i^2=-1 or is it worked out from more fundalmental laws? i^2 could also have been 1. It depends how one evaluates the square. 1. Multiply the two numbers inside the two squareroots first than square root the product (i.e. sroot(a)sroot(a)=sroot(a*a)=a) or 2. Cancel the squareroot and leave the number (that was before inside the square root) alone (i.e. sroot(a)sroot(a)=(sroot(a))^2=a) From the definition of i^2, the second option was chosen. For real numbers it did not matter which option one chose but complex numbers posed a problem. I think Euler first used option 1 for this operation. I got this information from 'Mathematics: The Loss of Certainty' by Morris Klein. If it is an axiom than why option 2 instead of 1?