>>12392904Merely asserting the existence of the square root of negative one and working as if you've already constructed it and can manipulate it like any other number doesn't make much sense because there is no guarantee that the rules of arithmetic don't break down, i.e. that you don't get any contradictions. That's why mathematicians are actually careful with complex numbers and construct them as pairs of real numbers
(a,b) with the rules of arithmetic
(a,b)+(c,d)=(a+c, b+d),
(a,b)*(c,d)=(ac-bd, ad+bc)
1=(1,0)
0=(0,0).
Then these pairs operate just like you would expect complex numbers to operate and you DEFINE
a+bi = (a,b). Then indeed
i^2= i* i = (0,1)*(0,1)=(0*0-1*1, 0)=(-1, 0 ) = -1.
You can verify that these pairs form an associative, commutative division ring, i.e. a field.
Now with dividing by 0 this doesn't work: the rules of arithmetic break down.
1/0 * 0 =0=1.
There is no way to construct a nice number system the way you do it with complex numbers that can accommodate division by zero, unless the whole number system is just one number x such that x is both 0 and 1, x+x=x, x*x=x. This is called the zero ring.
