I can divide by zero
Leopold Kronecker is supposed to have said, “God created the integers; all the rest is the work of man.” That work has advanced by extending the natural numbers to include negative numbers, zero, fractions, square roots, pi, e, i, and more. Every time we expand the system we gain and we lose. Things become more abstract, and we have to work harder to convince ourselves what we have now is a number. Not quite in historical order, it might go like this:
The natural numbers are fine: 1,2,3,4…
The rational numbers aren’t bad. Two halves make one whole. And now Johnny can have a pie.
Zero is weird but useful, though it does introduce an exception to remember. “Zero: can’t divide by that.”
Okay, it makes sense to associate any length with a number; and so the square root of two must be a number, even though we can’t say what number exactly. We can give it its own symbol, like zero.
And as long as we’re giving out symbols, here’s pi. And here’s e. And now the numbers aren’t countable. We’ve made a system of “numbers” with which it is in principle impossible to count. That’s bad. No, wait. That’s counter-intuitive; surprising! Yeah, surprising. Surprise is good. Besides, who but us is going to even notice? We may need to redefine “counting.”
And it makes sense that all equations like this should have roots, even if they don’t really. Better to say some roots just aren’t really numbers; or not Real numbers.
These complex numbers are handy. They aren’t so strange after all – just ordered pairs of real numbers. And C is a field, just like R. But we can’t put them in order.
Hmm, what if we consider ordered pairs of complex numbers?
And we press on, eventually getting to the point where we have ordered 16-tuples of real numbers. These we can divide by zero. Of course at that point, the hypercomplex numbers we’re dividing by zero are not commutative or associative or countable or ordered. No reasonable man would say they were numbers at all.