Dividing by zero - irrational numbers, stage perspective

In the world of numbers, we have all been told a story - a tale that says you can't divide by zero. It's a script that calculators, computers, and teachers recite like a sacred mantra. But what if,
just for a moment, we dared to challenge the well-rehearsed lines and welcome the world where numbers defy the very rules that bind them - a place where dividing by zero is not an error but an exploration leading us to discover some hidden math wonders? This peculiar notion, often dismissed as a mathematical enigma, has a friend in the world of theater - a stage where creativity knows no bounds.

Imagine the stage is set, the lights go down, and the spotlight falls on the paradox of division by zero. The audience, a captive mix of skeptics and dreamers, leans in. “Can't we simply, just as mathematicians did with the square root of a negative one, throw in an inverse of zero and see what happens?”

It's true that in the real numbers, neither division by 0 nor taking the square root of a negative one is defined, and yet mathematicians have found a way to do - minus square root of 1 = i. If you throw in a number whose square is (-1) and then follow the normal rules of algebra, you will end up with the familiar complex numbers. They may have some weird properties compared to the reals, but they are a perfectly good system of numbers to work with.

Can’t we just do the same thing with division by 0 ? Can we just throw in an inverse of zero
and see what happens ?

Given that you have been told your entire life that you cannot divide by zero, you might expect that, for reasons you do not fully understand, it is not possible to simply insert the inverse of
a number in the same manner as a solution to minus square root of 1 equals i ; nevertheless, this is not the issue. The process of inserting the inverse of a number or a set of numbers and observing the results is known as localization. And it's no harder than throwing a square root of a negative one. Let's pause for a moment to understand this theatrical localization—it is like a plot twist in the mathematical script. We enter a world where integers are the main characters, but something is missing—a void left by the absence of halves.

Fear not; in order to multiply numbers as usual, we need to add in anything that's an integer divided by a power of two. For example, we introduce 1/4, the inverse of 1/2, and suddenly the stage expands into the dyadic or binary rationals—an unexpected turn that adds a touch of whimsy to our mathematical play. And we end up with another weird but perfectly valid number system.

If we also threw in 1/3, 1/4, 1/5, and 1/6 inverses of every positive whole number, then we end up with all the rational numbers—ratios of integers where the dominator is not zero.

So, what about the 0 inverse?—someone from the audience asks.

Well, let’s start with the real numbers, throw in an inverse of zero, and let’s see what happens. The script warns us of dire consequences, but what if we defy the mathematical gravity and see
where this move leads? The familiar algebraic rules become our guide through a realm where zero's inverse becomes a unique number, just as 3 times 1/3 = 1, our inverse of zero is: 0*1/0=1.

So let's assume that it is true. Then what? It's an easy equation. Everyone in the audience pretty quickly sees that this implies that 0 is equal to 1, because anything times 0 is 0. But this is not
a contradiction; of course, in the real numbers, zero is different from one, but in the real numbers, x squared can never be negative either. And yet we are perfectly okay with the complex numbers, saying that (i squared) is (-1). Therefore, just because we discovered an unintuitive property of this new number system does not make it wrong.

And we’re heading toward the end of the play.

Just as 1 multiplied by any number x equals x, the power of a single scene resonates as a fundamental building block. However, because on stage we are challenging preconceptions and revealing the extraordinary within the ordinary, we imply what we have previously proved: 1 equals 0, therefore our equation can be rewritten as 0 times x, which of course is 0.

Thus, we have shown in our new number system that every number equals zero. There is still no contradiction or error in logic. And here comes the dramatic twist we invented—the zero ring. In this surreal number system, every number humbly bows before the supremacy of the lead character, called zero. It's a moment that would make the audience gasp in disbelief, for it challenges the notion that zero and one can never be the same, but 1 is just another label for 0.

This number system satisfies all the normal rules of algebra. But remember, this is a theater of numbers, a place where contradictions are not flaws but doorways to other worlds.

In math, dividing by zero is a bit like going into a forbidden zone. In the theater, zero symbolizes endless potential or a starting point for narratives. So, it’s not that one cannot divide by 0, it’s just that at the end of each show and at the end of our experience, we get what we deserve.