The rumor persists to this day that division by zero is impossible. Sometimes, people talk about it as if spacetime would fracture if anything were divided by 0. If you try to perform such a division on a calculator, an error code is returned: even the machine doesn't give you an answer. Is it because there truly is no possible solution? If you think about it for a moment, it's easy to find several! Yet the calculator tells me it's impossible, but this error code is pre-programmed: the calculator doesn't even try to answer it. Division by zero is a perfect example of the acceptance of contradictions in the cosmos, and my calculator's answer is nothing but ideology.

The previous article concluded that language and nature allow for the use of contradictions and that the system of classical logic was mistaken about the validity of contradictions. In this article, we will examine the very language of nature: mathematics. We will see how division by zero exemplifies its answer to the question of the validity of the ideology of non-contradiction.

For example, 0/0 is a division that, in theory, has five solutions, all of which contradict each other in every possible way, without it being possible to differentiate the answers based on their validity. In practical application, however, it has more than an infinite number of possible answers. The reader will reach the hyperspace of contradiction upon reading and, once finished, will no longer be able to doubt the validity of the contradictions.

Multiple-answer equations are not uncommon in mathematics, yet division by zero is an exception. It is one of the few operations for which multiple answers are not tolerated, to the point that one might even claim it is impossible to divide by 0 and consider the matter closed: but accepting this answer would also lead to abandoning division by 1. Why can we add, subtract, and multiply all numbers, but division, on the other hand, has 0 as its limit?

The discomfort of x/0

There are some equations in mathematics that have the particularity of having multiple possible answers. Let's take the example of number 2.x=1. To solve 2x=1, we must accept living with two competing possibilities, with opposing polarities. Indeed, if 2x = 1, then x = 1 or -1. It is impossible with the available information to determine if x has the value 1 or -1.

Mathematicians can live in a world where more than one possible answer exists for the same equation, even if the answers are opposites. After all, there is no contradiction because the mathematician is confident that only one of the answers is correct and that, should the problem arise in reality, they will be able to determine the final value. x.

Let's say that, in absolute terms, we cannot give a definitive answer to 2x = 1, in practice, the context can inform us of the final answer: it is impossible to find a two-by-four measuring -1 meter at the hardware store, so, in this case, the mathematician will determine that in all likelihood the answer is 1.

We can tolerate absolute uncertainty in mathematics if we are confident that a certain answer can or could be given in a specific case. In this case, these equations may have several possible answers, even opposing ones, and we would not say these answers are contradictory, but rather competing. They are in competition, but we know that we will only have one winning solution.

With division by zero, things get complicated. While we might have several answers in theory, they aren't competing: they're all equally valid, and it's impossible to use practical application to alleviate the uncertainty, because practical application gives us a different answer than the calculations in theory. Let's see…

/0 in theory and in practice

In principle, dividing by zero has several solutions that are relatively easy to prove. One way to answer the equation 1/0=x and assign a value to xThe process involves deducing the value from the series of denominators.

1 / 100 = 0.01
1 / 10 = 0.1
1 / 5 = 0.2
1 / 2 = 0.5
1 / 1 = 1
1 / 0.1 = 10
1 / 0.01 = 100
1 / 0.001 = 1000
...
1/0 = ∞

The larger the denominator of a fraction, the smaller the number. The smaller the denominator, the larger the number. In theory, dividing 1 by ∞ would give 0, and dividing 1 by 0 would give ∞. This is simply the logical continuation of the set of all fractions.

But we can also recognize that if we plot all possible denominator values ​​on a Cartesian plane, we have an exponential function. Therefore, there would be an asymptote. A reminder to the reader: the asymptote is the value that our curve will never reach in practice, because to do so, we would need to be able to reach infinity. In practice, the limit exists. In theory, it does not. So we see here that the theory tells us the answer is ∞, but practice gives us the answer N/A (not applicable)...

Thus, there are two contradictory answers here. It was demonstrated, through two methods, that in theory, division by zero results in infinity: the method of the series of denominators, and its graphical representation using asymptotes. But the latter also highlighted that practice gives a completely different answer, contradicting the theory at a fundamental level. It is impossible to contradict oneself more than this. Theory gives us an answer, while practice suggests that there is no answer.

%=? …

And it gets even stranger when we get to %. (Yes, I know, that symbol means percentage. But objectively, % represents the fraction 0/0 just as 1/2 represents 1/2.) % has even more solutions than x/0. There are actually 4 solutions in theory, as well as an infinite number of solutions in practice.

Let's see…

The first solution of placing the denominators in series still works for this case. So a first answer is ∞.

A second solution is to proceed in the same way, but with the denominators. The smaller the denominator, the smaller the number.

10 / 2 = 5
5 / 2 = 2.5
3 / 2 = 1.5
½ = 0.5
0.01 / 2 = 0.005
...
0 / 2 = 0

In theory, we would have an infinitely smaller answer than the first. Since the series tends towards an ever-smaller number as the denominator decreases, it would be logical that at 0, it would be the smallest possible number, 0. But when we divide 0 by 0, –∞ would also be a possible answer. Since +0 and -0 are equal, -0/0 would give –∞.

A third solution is the answer to equation 0/xFor all numbers with a 0 in the denominator, we have that the answer is also 0… Why would % be an exception to this rule? Seeing no reason, we therefore have that the answer is also 0.

The fourth solution is that we only have the solution of x/x, for any value of x, is 1. 1/1=1, 2/2=1, 18/18=1, 1000/1000=1, -1/-1=1, -1000/-1000/1. Why would % be an exception? Again, seeing no reason, we therefore assume that the answer is also 1.

Our possible answers are ∞, – ∞, 0 and 1.

We're going to cut the cake

Finally, a fifth solution involves questioning what division actually is. How can we divide anything by 0? I can cut a cake into 8 pieces. I can divide a sum of money into two equal parts. But how do we cut a cake into 0? This is why mathematicians say that dividing by zero is impossible.

Following this logic, if dividing by zero is impossible, then dividing by one would be just as impossible. How do you cut a cake into one? It's nonsense. Any cut you make will separate the cake, and you'll fail to divide it into one. Some will say you simply shouldn't cut it, and there you have it, you've divided it by one. But if you haven't cut anything, how can you say you've separated the cake? You simply haven't performed any division. To divide into one, you mustn't divide: therefore, dividing into one is as impossible as dividing by zero. Following this logic, any operation involving zero would be impossible; how can you add or subtract anything? How can you multiply by nothing?

The problem is that division isn't just about cutting a cake, and proportions or geometry are more useful to the cake-cutting problem than the arithmetic of division. After all, who pulls out a pencil and paper to do division before they start cutting a cake? Framing division this way misunderstands its purpose. Division isn't primarily a tool for separating or distributing; it's a tool that tells us how to store contents or fill containers.

The Tupperware of the division

We can formulate the problem as follows: division has two parts, a numerator and a denominator. We can think of them as a content and a container. Dividing anything by anything is like asking how many containers a quantity can hold. y How much content do I need to store a certain amount? x content. Or, how many containers with a capacity y can I fill something with a quantity x of the same thing. We will see that the solution to these two problems still leads to contradictory answers.

Initially, dividing by zero then becomes a matter of asking how many containers are needed to store x of something (contained) in containers that have the capacity to hold 0 of that same thing. We simply cannot answer this question. The answer is: it's impossible. This is the formulation that mathematicians use.

But, secondly, dividing by zero also means asking how many containers with no capacity I can fill with a quantity xThe answer is any number from 0 to infinity inclusive. Regardless of the reason why the containers have a capacity of zero, either because they are already full, or because they are too small to hold a unit of an inseparable object, we also have the possibility of filling them all, even if I have no quantity of contents.

For example: how many televisions can I store in a box that's too small for one television? An answer less than 1 simply means I have to cut a television to solve my problem and therefore incur a loss, which makes no sense for a manufacturer trying to sell their televisions. No matter how many boxes are too small, I can consider them all already full. I can't store any more televisions, and all the boxes are full.

The answers are N/A and an arbitrarily chosen or random number, from 0 to infinity inclusive. Therefore, we have more than an infinite number of possible answers.

And we can't say that either answer is invalid. In both cases, we legitimately divided by zero. If the goal is to store any quantity, the task is impossible. If the goal is to fill containers that are too small or already full, the task is already accomplished, and I can specify any number I want. The case of division by zero embodies libertarianism in mathematics.

Conclusion

After all this, we find ourselves in the following situation: in the absolute world of theoretical mathematics, dividing by zero has two contradictory solutions that are not necessarily concurrent, and in practice, dividing by zero has an infinite number of solutions in some cases and is impossible to solve in others. Therefore, it is not impossible to divide by zero. That's a myth. Division by zero simply has an infinite number of possible solutions, including the impossibility of solving the problem. The impossibility of solving the problem does not mean that it is impossible to solve the equation.

To understand this, we must recognize not only that the language of nature operates with contradictions, but also that self-referentiality is permitted. Zero does not designate something, but an absence. In the definition of what a number is, zero plays a fundamental role: it is by creating a set containing zero of anything, the void, that we obtain a zero, but if we create a set containing this zero, we obtain the definition of the number 1. 1 = {0}. 0 = { } * *empty set, a void is defined as a quantity 0 of anything… Therefore, the definition of 0 is circular, as is its form.

Self-referentiality is also what the % answer allows. You can give any answer you want, and it will be a correct answer, simply because it's the answer you gave. When you ask someone to give you a random number, they are essentially dividing zero by zero.

Could % be the mathematical guarantee indicating that freedom is a property of nature and inscribed in its language? The answer given to % reveals much about the person giving the answer: if they choose N/A, it may be someone who renounces the freedom that nature grants them. If they choose any number or infinity, it's someone who enjoys it. And if they give all of these answers, it's probably a wise person answering you.