Dad, why can't you divide by zero?
My three-year-old daughter Sophia recently often mentions "zero", for example, in this context:
Those. feeling of negative numbers and zero neutrality already has, oh how. Soon he will ask: why is it impossible to divide this into zero?
And so I decided in simple words to write down everything that I still remember about the division by zero and all that.
The division is generally better to see once than hear a hundred times.
Well, or one divided x times to see ...
It immediately shows that zero is the center of life, the universe and all that. Let 42 be the answer to the main question about all this, but the center will be anyway 0. He doesn’t even have a sign, neither a plus (obeyed), nor a minus (did not obey), he is really zero. And he knows a lot about piglets.
')
Because if any piglet is multiplied by zero, then the piglet is sucked into this round black hole, and it turns out again zero. Not so this zero and neutral, when it comes to addition-subtraction comes to multiplication, not to mention division ... There, if zero is “0 / x” above, then there’s a black hole again. Everything eats to zero. But if during the division, and even from below - “x / 0”, then it begins ... follow the white rabbit, Sonia!
At school, they will tell you "you cannot divide by zero" and they will not blush. As proof, they poke on the calculator “1/0 =” and the usual calculator, also not blushing, writes “E”, “Error”, they say, “it’s impossible - it means it’s impossible”. Although what you have there will be considered a regular calculator - another question. To me, now, in 2014, the standard calculator on the android phone writes something completely different:
Wow infinity. Slide your eyes, cut the circles. So you can not. It turns out you can. If caution. Because my Android is not careful yet, I also do not agree: "0/0 = Error", again it is impossible. Let's try one more time: "-1/0 = -∞", oh how. An interesting opinion, but I disagree with him. It does not agree with the "0/0 = Error".
By the way, JavaScript, which feeds 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 sort of thing, but not a number. While “1/0” JS also understands as “Infinity”. This is closer. But so far only the heat ...
The university - higher mathematics. There are limits, poles, and other shamanism. And everything becomes more complicated, more complicated, they beat around the bush, but just not to violate the crystal laws of mathematics. But if you do not try to write the division by zero into these existing laws, then you can feel this fiction - on your fingers.
To do this, let's look at the division again:
Watch the right line, from right to left. The closer X is to zero, the stronger the split divided by X takes off. And somewhere in the clouds, "plus infinity." It is always further, like a horizon, you cannot catch it.
Now follow the left line, from left to right. The same story, only now the divided flies down, infinitely down, into the "minus infinity." Hence the opinion that “1/0 = + ∞”, and “-1/0 = 1 / -0 = -∞”.
But the trick is that "0 = -0", there is no sign for zero, if you do not complicate with the limits. And if you divide the unit by such a “simple” zero without a sign, then is it not logical to assume that infinity will also be obtained - “just” infinity, without a sign, like zero. Where is she - above or below? It is everywhere - infinitely far from zero in all directions. This is the zero, turned inside out. Zero - nothing. Infinity is everything. Both positive and negative. Generally everything. And immediately. Absolute.
But there was something about “0/0”, something else, not infinity ... Let's do this trick: “2 * 0 = 0”, yeah, the teacher at the school will say. Another: "3 * 0 = 0" - again, yeah. And having spat a little on “it is impossible to divide by zero”, they say, the whole world divides so slowly, we get: “2 = 0/0” and “3 = 0/0”. In which class there it is, but without a zero, of course.
Wait a minute, it turns out “2 = 0/0 = 3”, “2 = 3” ?! That's why they are afraid, that's why it's “impossible”. Worse than "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 themathematics of imagination. We can imagine ourselves as an infinite Absolute somewhere in the stars, from there look at the sinful world of finite numbers and people and understand that from this point of view they are all the same. And "2" c "3", and even "-1", and the teacher in the 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 a zero divided by x looks like in my fantasies far from official mathematics. In fact, it looks like 1 / x, only the bend is not in the unit, but in zero. By the way, 2 / x bends in two, while 0.5 / x - in 0.5.
It turns out that 0 / x at x = 0 takes all finite values - not infinity, not emptiness. There's a hole in the plot at zero, the axes are peeped out.
You can certainly argue that "0 * 0 = 0", which means zero (emptiness) also falls into the category 0/0. A little run forward - there will be zero degrees 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, unit at infinity can also be written as 0/0, too, you get (0/0) / 0 - infinity. Now the order, everything can be expressed by the ratio of zeros.
And these categories are subject to many laws of ordinary numbers, showing a very interesting relationship.
For example, if we add the finite to infinity, then the infinity will absorb the finite, it will remain infinite:
1/0 + 0/0 = (1 + 0) / 0 = 1/0.
And if infinity is multiplied by emptiness, then they absorb each other, and a finite world is obtained:
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 move from all these divisions and brackets (like (0/0) / 0) simply to degrees:
Hint.
Here with infinity and emptiness everything is as simple as in school. And the final world goes to degrees like this:
Whew!
It turns out that the positive degrees of zero are zeros, the negative degrees of zero are infinity, and the zero degree of zero is a finite world.
Such a universal object “0 ^ x” is obtained. Such objects interact well with each other, again, they obey many laws, beauty, in general.
My modest knowledge of mathematics was enough to draw an Abelian group of them, which, being isolated in a vacuum (“just abstract objects, such a recording form, like exponents”), even withstood the test by the coolest matan teacher with the verdict “interesting, but nothing will work ". Another thing is that the thread turned out, this is a taboo topic - division by zero. In general, do not load.
Better just try to multiply infinity by a finite number:
0 ^ -1 * 0 ^ 0 = 0 ^ (- 1 + 0) = 0 ^ -1.
Again, infinity swallowed a finite number just as 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. second degree zero is stronger than normal (first degree, 0 ^ 1). And infinity minus second degree is stronger than usual infinity (0 ^ -1).
And when emptiness collides with the absolute, they are measured by force - whoever has more, he 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 the final world remains annihilated:
0 ^ 1 * 0 ^ -1 = 0 ^ (1 + -1) = 0 ^ 0.
By the way, official mathematics is already nearby. Its representatives know about the "poles" and that the poles have a different force (order), as well as about the "zero of order k". But they all tread on a solid surface "close to" and are afraid to jump into the blackhole hole.
And the last for me is the third level of dreams. For example, these are all 0 ^ -1 and 0 ^ -2 - infinity of different forces. Or 0 ^ 1, 0 ^ 2 - zeros of different strengths. But after all, “-1” and “-2” and “+1” and “+2” are all - 0/0, equal to 0 ^ 0, have already passed. It turns out that from this level of dreams, it doesn’t matter anyway that they are zeroes, infinities, and even the finite world goes there with some enlightenment. In one point. In one category. This happiness is called the Singularity.
I must admit that outside the state of enlightenment of one point I do not observe, but one category - the union “0 ^ 0 U 0 ^ (0 ^ 0)” - is quite.
Which of all this can be useful? After all, even a little less crazy "imaginary numbers", which also break calculators in Error = √-1, and they were able to 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 the other way around.
Like leaves on a tree from afar, they look the same, but if you take a closer look at them, they are all different. And if you think, then again the same. And they are not much different from you or me. Rather, they do not differ at all, if you think hard.
The benefit here is in the ability and focus on the differences and abstract. It is very useful both in work and in life, and even in relation to death.
These are the travels to the rabbit hole, Sonya!
- Sonya, you did not seem to obey at first, and then obeyed, what happens? ..
- Well ... zero!
Those. feeling of negative numbers and zero neutrality already has, oh how. Soon he will ask: why is it impossible to divide this into zero?
And so I decided in simple words to write down everything that I still remember about the division by zero and all that.
The division is generally better to see once than hear a hundred times.
Well, or one divided x times to see ...
It immediately shows that zero is the center of life, the universe and all that. Let 42 be the answer to the main question about all this, but the center will be anyway 0. He doesn’t even have a sign, neither a plus (obeyed), nor a minus (did not obey), he is really zero. And he knows a lot about piglets.
')
Because if any piglet is multiplied by zero, then the piglet is sucked into this round black hole, and it turns out again zero. Not so this zero and neutral, when it comes to addition-subtraction comes to multiplication, not to mention division ... There, if zero is “0 / x” above, then there’s a black hole again. Everything eats to zero. But if during the division, and even from below - “x / 0”, then it begins ... follow the white rabbit, Sonia!
At school, they will tell you "you cannot divide by zero" and they will not blush. As proof, they poke on the calculator “1/0 =” and the usual calculator, also not blushing, writes “E”, “Error”, they say, “it’s impossible - it means it’s impossible”. Although what you have there will be considered a regular calculator - another question. To me, now, in 2014, the standard calculator on the android phone writes something completely different:
Wow infinity. Slide your eyes, cut the circles. So you can not. It turns out you can. If caution. Because my Android is not careful yet, I also do not agree: "0/0 = Error", again it is impossible. Let's try one more time: "-1/0 = -∞", oh how. An interesting opinion, but I disagree with him. It does not agree with the "0/0 = Error".
By the way, JavaScript, which feeds 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 sort of thing, but not a number. While “1/0” JS also understands as “Infinity”. This is closer. But so far only the heat ...
The university - higher mathematics. There are limits, poles, and other shamanism. And everything becomes more complicated, more complicated, they beat around the bush, but just not to violate the crystal laws of mathematics. But if you do not try to write the division by zero into these existing laws, then you can feel this fiction - on your fingers.
To do this, let's look at the division again:
Watch the right line, from right to left. The closer X is to zero, the stronger the split divided by X takes off. And somewhere in the clouds, "plus infinity." It is always further, like a horizon, you cannot catch it.
Now follow the left line, from left to right. The same story, only now the divided flies down, infinitely down, into the "minus infinity." Hence the opinion that “1/0 = + ∞”, and “-1/0 = 1 / -0 = -∞”.
But the trick is that "0 = -0", there is no sign for zero, if you do not complicate with the limits. And if you divide the unit by such a “simple” zero without a sign, then is it not logical to assume that infinity will also be obtained - “just” infinity, without a sign, like zero. Where is she - above or below? It is everywhere - infinitely far from zero in all directions. This is the zero, turned inside out. Zero - nothing. Infinity is everything. Both positive and negative. Generally everything. And immediately. Absolute.
But there was something about “0/0”, something else, not infinity ... Let's do this trick: “2 * 0 = 0”, yeah, the teacher at the school will say. Another: "3 * 0 = 0" - again, yeah. And having spat a little on “it is impossible to divide by zero”, they say, the whole world divides so slowly, we get: “2 = 0/0” and “3 = 0/0”. In which class there it is, but without a zero, of course.
Wait a minute, it turns out “2 = 0/0 = 3”, “2 = 3” ?! That's why they are afraid, that's why it's “impossible”. Worse than "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 the
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 a zero divided by x looks like in my fantasies far from official mathematics. In fact, it looks like 1 / x, only the bend is not in the unit, but in zero. By the way, 2 / x bends in two, while 0.5 / x - in 0.5.
It turns out that 0 / x at x = 0 takes all finite values - not infinity, not emptiness. There's a hole in the plot at zero, the axes are peeped out.
You can certainly argue that "0 * 0 = 0", which means zero (emptiness) also falls into the category 0/0. A little run forward - there will be zero degrees 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, unit at infinity can also be written as 0/0, too, you get (0/0) / 0 - infinity. Now the order, everything can be expressed by the ratio of zeros.
And these categories are subject to many laws of ordinary numbers, showing a very interesting relationship.
For example, if we add the finite to infinity, then the infinity will absorb the finite, it will remain infinite:
1/0 + 0/0 = (1 + 0) / 0 = 1/0.
And if infinity is multiplied by emptiness, then they absorb each other, and a finite world is obtained:
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 move from all these divisions and brackets (like (0/0) / 0) simply to degrees:
1/0 = 0 ^ -1
0/0 = 0 ^ 0
0 = 0 ^ 1
Hint.
Here with infinity and emptiness everything is as 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.
Whew!
It turns out that the positive degrees of zero are zeros, the negative degrees of zero are infinity, and the zero degree of zero is a finite world.
Such a universal object “0 ^ x” is obtained. Such objects interact well with each other, again, they obey many laws, beauty, in general.
My modest knowledge of mathematics was enough to draw an Abelian group of them, which, being isolated in a vacuum (“just abstract objects, such a recording form, like exponents”), even withstood the test by the coolest matan teacher with the verdict “interesting, but nothing will work ". Another thing is that the thread turned out, this is a taboo topic - division by zero. In general, do not load.
Better just try to multiply infinity by a finite number:
0 ^ -1 * 0 ^ 0 = 0 ^ (- 1 + 0) = 0 ^ -1.
Again, infinity swallowed a finite number just as 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. second degree zero is stronger than normal (first degree, 0 ^ 1). And infinity minus second degree is stronger than usual infinity (0 ^ -1).
And when emptiness collides with the absolute, they are measured by force - whoever has more, he 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 the final world remains annihilated:
0 ^ 1 * 0 ^ -1 = 0 ^ (1 + -1) = 0 ^ 0.
By the way, official mathematics is already nearby. Its representatives know about the "poles" and that the poles have a different force (order), as well as about the "zero of order k". But they all tread on a solid surface "close to" and are afraid to jump into the black
And the last for me is the third level of dreams. For example, these are all 0 ^ -1 and 0 ^ -2 - infinity of different forces. Or 0 ^ 1, 0 ^ 2 - zeros of different strengths. But after all, “-1” and “-2” and “+1” and “+2” are all - 0/0, equal to 0 ^ 0, have already passed. It turns out that from this level of dreams, it doesn’t matter anyway that they are zeroes, infinities, and even the finite world goes there with some enlightenment. In one point. In one category. This happiness is called the Singularity.
I must admit that outside the state of enlightenment of one point I do not observe, but one category - the union “0 ^ 0 U 0 ^ (0 ^ 0)” - is quite.
Which of all this can be useful? After all, even a little less crazy "imaginary numbers", which also break calculators in Error = √-1, and they were able to 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 the other way around.
Like leaves on a tree from afar, they look the same, but if you take a closer look at them, they are all different. And if you think, then again the same. And they are not much different from you or me. Rather, they do not differ at all, if you think hard.
The benefit here is in the ability and focus on the differences and abstract. It is very useful both in work and in life, and even in relation to death.
These are the travels to the rabbit hole, Sonya!
Source: https://habr.com/ru/post/233103/
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