>>12360884>You cannot take the square root of a negative number because there is no number squared that produces a negative result.you're absolutely right. You can't take the square root of a negative number, because the domain of the square root is the non-negative reals. And there indeed is no real number that when squared gives you a negative number.
>Despite this, negative square roots are still defined as i^2 = -1.That's why i is not a real number (real as in an element of the real numbers, not the schizo "hurr durr not real" way). If we use fancy language, complex numbers are an algebraic extension of the reals, that is the reals are a subset of the complex numbers that satisfy the same algebraic properties, but the converse is not true.
>Why can't I define division by zero as 0*j = 1?You can do this only for the trivial ring {1}, i.e. if you define 0=1, and that is the only element of the ring (a set with addition and multiplication). For any other ring, for example the integers, this cannot hold, because that would contradict the axioms of addition and multiplication. Here is a proof:
There's an axiom that tells us that 0 is an identity element of addition. All that means is
0 + a = a + 0 = a
for any number a. Similarly 1 is an identity element of multiplication:
1*a = a*1 = a
We also have the distributive axiom
a*(b + c) = a*b + a*c
and the inverse axiom of addition that says that for any number a there exists a number (-a) such that
a + (-a) = 0
Therefore we can derive the following chain of equalities
a*1 = a
a*(0+1) = a
a*0 + a*1 = a
a*0 + a = a
a*0 + a + (-a) = a + (-a)
a*0 = 0
QED
of course then you can say "but why can't we define something with different axioms?". You sure can. Now whether or not what you define is first of all consistent with itself and most importantly useful at all is actually what matters. Hope this helps.
