# complex numbers – \$i^2neq1\$, but what is the mistake I have made?

I was doing a little bit of fiddling around with complex numbers when I came across this:

$$x=1$$

$$x=sqrt{1}$$

$$x=sqrt{-1 times -1}$$

$$x=sqrt{-1} times sqrt{-1}$$

$$x=itimes i$$

$$x=i^2$$

$$1=i^2=-1$$

from that, we get that $$1=-1$$ which, of course, doesn’t make sense. Do the square root laws not work for factorizing $$1$$ with negative $$1$$s? Do complex numbers not have the same properties as real numbers? What is the mistake I’ve made?