What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?

Question

What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?

In: Mathematics

Answer 1

You can’t have such a number and still have arithmetic with the rules we’re familiar with. More specifically, with real or complex numbers multiplication is associative, which means that for any 3 numbers a,b and c it holds

(a•b)•c=a•(b•c)

This eule is extremely fundamental. Now suppose we defined some number x to be equal to be 1/0. Then we should have that 0•x=1. But then, since zero times any real number is zero, we see that

(2•0)•x=0•x=1 while 2•(0•x)=2•1, so associativity fails.

So a number system that includes 1/0 will be very different from the way we’re used to numbers behaving. On the other hand, complex numbers behave very similar to real numbers, in the sense that they satisfy (essentially) all the same laws