Dad, why can’t you divide by zero?

My three-year-old daughter Sophia has often mentioned “zero” recently, for example, in this context:

“Sonya, you didn’t seem to obey at first, and then obey, so what happens? ..
” “Well ... zero!”

Those. a sense of negative numbers and neutrality of zero already has, about how. He will soon ask: why is it impossible to divide by zero?
And so I decided in simple words to write down everything that I still remember about dividing by zero and all that.

Division is generally better to see once than hear a hundred times.
Well, or one time to divide by x ...



You can immediately see that zero is the center of life, the universe and all that. Let the answer to the main question about all this be 42, but the center is anyway 0. It does not even have a sign, no plus (obeyed), no minus (did not obey), it’s really zero. And in pigs knows a lot.

Because if any piglet is multiplied by zero, then the piglet is sucked into this round black hole, and again it turns out zero. This zero is not so neutral, when it comes to addition-subtraction it comes to multiplication, not to mention division ... There, if the zero from the top is “0 / x” then it is again a black hole. Everything eats to zero. But if during division, and even from below - “x / 0”, then it starts ... follow the white rabbit, Sonya!

At school, they will tell you "you can’t divide by zero" and do not blush. As proof, they will poke on the “1/0 =” calculator and the usual calculator, also not blushing, will write “E”, “Error”, they say, “it’s impossible - it’s impossible”. Although what you have there will be considered an ordinary calculator - another question. To me right now, in 2014, the standard calculator on an Android phone writes something completely different:



Wow infinity. Glance at yourself, cut circles. Here you can’t. It turns out you can. If carefully. Because it’s not careful, my Android also does not agree: “0/0 = Error”, again it’s impossible. Let's try one more time: “-1/0 = -∞”, about how. Interesting opinion, but I do not agree with him. I do not agree with "0/0 = Error".

By the way, the JavaScript that powers the current sites also disagrees with the android calculator: go to the browser console (another F12?) And write there: “0/0” (input). JS will answer you: "NaN". It's not a mistake. This is “Not a Number” - i.e. some kind of thing, but not a number. Despite the fact that “1/0” JS also understands as “Infinity”. This is closer. But so far only warm ...

At the university - higher mathematics. There are limits, poles, and other shamanism. And everything gets complicated, complicated, they beat around the bush, but just not to violate the crystal laws of mathematics. But if you do not try to enter the division by zero in these existing laws, then you can feel this fiction - on the fingers.

To do this, look again at the division:



Follow the right line, from right to left. The closer the x is to zero, the stronger the x-divided division takes off. And somewhere in the clouds, plus infinity. It is always further, like the horizon, you can not catch it.

Now follow the left line, from left to right. The same story, only now the divided one flies down, endlessly down, into “minus infinity”. Hence the opinion that “1/0 = + ∞”, and “-1/0 = 1 / -0 = -∞”.

But the trick is that "0 = -0", there is no sign at zero, if you do not complicate it with the limits. And so, if you divide the unit into such a “simple” unsigned zero, is it not logical to assume that infinity will turn out - “just” infinity, without a sign, like zero. Where is it - above or below? It is everywhere - infinitely far from zero in all directions. This is the zero turned inside out. Zero - there is nothing. Infinity is everything. Both positive and negative. That's all. And right away. Absolute.

But there was something about “0/0”, something else, not infinity ... Let's do this trick: “2 * 0 = 0”, yeah, the teacher at school will say. Also: "3 * 0 = 0" - again, yeah. And spitting a little on “it is impossible to divide by zero”, they say, the whole world is already dividing slowly, we get: “2 = 0/0” and “3 = 0/0”. In what class it goes, only without zero, of course.

Wait a minute, it turns out “2 = 0/0 = 3”, “2 = 3” ?! That’s why they’re afraid, that’s why it’s “impossible”. The worse “1/0” is only “0/0”, even the android calculator is afraid of it.

And we are not afraid! Because we have the power of mathematics of imagination. We can imagine ourselves as the infinite Absolute somewhere in the stars, look from there at the sinful world of finite numbers and people and understand that from this point of view they are all the same. And “2” with “3”, and even “-1”, and the teacher at school, perhaps, too.

So, I modestly assume that 0/0 is the whole finite world, or rather, everything that is not infinite and not empty.



This is what zero divided by x looks like in my fantasies far from official mathematics. Actually similar to 1 / x, only the inflection is not at unity, but at zero. By the way, 2 / x has an inflection in the two, and in 0.5 / x - in 0.5.

It turns out that 0 / x for x = 0 takes all finite values ​​- not infinity, not void. There is a hole in the graph at zero, the axes are visible.

One can of course argue that “0 * 0 = 0”, which means that zero (void) also falls into the 0/0 category. A little run ahead - there will be degrees of zero and this objection will shatter into fragments.

In such divine categories there is only emptiness (0), finite world (0/0), and infinity (1/0).

Oops, a unit at infinity can also be written as 0/0, you get (0/0) / 0 - infinity. Now the order, everything can be expressed by the ratio of zeros.

And these categories obey many of the laws of ordinary numbers, showing very interesting relationships.

For example, if you add the finite to infinity, then infinity swallows the finite, remains infinity:
1/0 + 0/0 = (1 + 0) / 0 = 1/0.

And if infinity is multiplied by emptiness, then they absorb each other, and we get the final world:
1/0 * 0 = (1 * 0) / 0 = 0/0.

But this is only the first level of dreams. You can dig deeper.

If you already know the concept of "degree of number", and that "1 / x = x ^ -1", then, having thought, you can go from all these divisions and brackets (like (0/0) / 0) just to degrees:

1/0 = 0 ^ -1
0/0 = 0 ^ 0
0 = 0 ^ 1

Hint.
Here, with infinity and emptiness, everything is simple, as in school. And the final world goes to degrees like this:

0/0
= (0 * 1) / 0
= 0 * (1/0)
= 0 * 1/0
= 0 ^ 1 * 0 ^ -1
= 0 ^ (1 + -1)
= 0 ^ (1-1 )
= 0 ^ 0.

Wow!

It turns out that the positive degrees of zero are zeros, the negative degrees of zero are infinities, and the zero degree of zero is the finite world.

This is how the universal object “0 ^ x” turns out. Such objects interact beautifully with each other, again they obey many laws, beauty in general.

My modest knowledge of mathematics was enough to draw from them an Abelian group, which, isolated in a vacuum (“just abstract objects, such a notation, like an exponent”), even passed the test of the coolest matan teacher with the verdict “interesting, but nothing will work ". I wish there was a thread, it’s a taboo topic - division by zero. In general, do not load.

Let's try to just multiply infinity by a finite number:
0 ^ -1 * 0 ^ 0 = 0 ^ (- 1 + 0) = 0 ^ -1.

Again, infinity has absorbed a finite number in the same way that its antipode zero absorbs finite numbers, the same black hole:
0 ^ 1 * 0 ^ 0 = 0 ^ (1 + 0) = 0 ^ 1.

It also turns out that degrees are like strength. Those. zero of the second degree is stronger than zero of ordinary (first degree, 0 ^ 1). And infinity minus the second degree is stronger than the usual infinity (0 ^ -1).

And when the void collides with the absolute, they are measured by force - whoever has more will win:
0 ^ 1 * 0 ^ -2 = 0 ^ (1 + -2) = 0 ^ -1 = ∞.
0 ^ 2 * 0 ^ -1 = 0 ^ (2 + -1) = 0 ^ 1 = 0.

If they are equal in strength, then they annihilate and the final world remains:
0 ^ 1 * 0 ^ -1 = 0 ^ (1 + -1) = 0 ^ 0.

By the way, official math is already nearby. Its representatives know about the “poles” and that the poles have different strengths (order), as well as about “zero order k”. But they all stomp on a solid surface “next to” and are afraid to jump into a black hole in a hole.

And the last one for me is the third level of dreams. For example, these are all 0 ^ -1 and 0 ^ -2 - infinities of different strengths. Or 0 ^ 1, 0 ^ 2 - zeros of different strengths. But after all, “-1” and “-2” and “+1” and “+2” - that's all - 0/0, equal to 0 ^ 0, have already passed. It turns out that from this level of dreams, it’s all the same in general that these are zeros, infinities, and even the finite world gets there with some enlightenment. At one point. In one category. This happiness is called - Singularity.

It must be admitted that outside the state of enlightenment I do not observe one point, but one category - the union "0 ^ 0 U 0 ^ (0 ^ 0)" - is complete.

Which of all this can be beneficial? After all, even slightly less crazy "imaginary numbers", which also tear calculators in Error = √-1, and they could become official mathematics and now simplify the calculations of steelmaking.

With division by zero and categories 0 ^ x, the benefit is rather philosophical. See how infinity and emptiness absorb the finite, how emptiness can defeat infinity, and it can happen, and vice versa.

Like leaves on a tree from a distance, they look the same, but if you look at them carefully, they are all different. And if you think about it, then again the same. And they are not much different from you or me. Rather, they are no different at all, if you think hard.

The benefit here is in the ability to focus on differences and abstract. This is very useful both in work and in life, and even in relation to death.

These are the trips to the rabbit hole, Sonya!