If you break the generally accepted rules in the world of science, you can get the most unpredictable results.
Since school, our teachers have been telling us that there is one rule in mathematics that cannot be broken. It goes like this: “You can’t divide by zero!”
Why does the number 0, which is so familiar to us, which we encounter so often in everyday life when performing such a simple arithmetic operation as division, cause so many difficulties?
Let’s look into this issue.
If we divide one number by smaller and smaller numbers, we will end up with larger and larger values.
For example:
20:10 = 220:5 = 420:2 = 1020:1 = 2020:0.000001 = 20000000… and so on.
Thus, it turns out that if we divide by a number that tends to zero, we will get the largest result that tends to infinity.
Does this mean that if we divide our number by zero, we get infinity?
This sounds logical, but all we know is that if we divide by a number close to zero, the result will only go to infinity, and this does not mean that when we divide by zero, we will end up with infinity. Why is this so?
First, we need to understand what the arithmetic operation of division is. So, if we divide 20 by 10, this will mean how many times we need to add the number 10 to get 20 as a result, or what number we need to take twice to get 20.
In general, division is the inverse arithmetic operation of multiplication. For example, when multiplying any number by X, we can ask the question: “Is there a number that we need to multiply by the result to find out the original value of X?” If such a number exists, then it will be the inverse value of X. For example, if we multiply 2 by 5, we get 10. If we then multiply 10 by one-fifth, we get 2 again:
2*5 = 1010*1/5 = 2
Thus, 1/5 is the reciprocal of 5, the reciprocal of 10 would be 1/10.
As you have already noticed, when you multiply a number by its reciprocal, the answer will always be one. And in the event that you want to divide a number by zero, you will need to find its reciprocal, which should be equal to one divided by zero.
This would mean that when multiplied by zero, the result should be one, and since it is known that if you multiply any number by 0, you get 0, then this is impossible and zero does not have an inverse number.
Is it possible to come up with something to get around this contradiction?
Mathematicians had already found ways to circumvent mathematical rules, because in the past, mathematical rules made it impossible to obtain the value of the square root of a negative number, so it was proposed to denote such square roots as imaginary numbers. As a result, a new branch of mathematics appeared, called complex numbers.
So why don’t we also try to introduce a new rule according to which one divided by zero would be denoted by an infinity sign and see what happens?
Let’s assume that we know nothing about infinity. In this case, if we start from the reciprocal of zero, then multiplying zero by infinity should give us one. And if we add another value of zero divided by infinity, we should end up with the number two:
0*∞ = 10*∞ + 0*∞ = 2
According to the distributive law of mathematics, the left side of the equation can be represented as:
(0+0)*∞ = 2
and since 0+0 = 0, our equation will take the form 0*∞ = 2, due to the fact that we have already defined 0*∞ = 1, it turns out that 1 = 2.
This sounds absurd. However, this answer cannot be considered completely wrong either, since such calculations simply do not work for ordinary numbers. For example, in the Riemann sphere, division by zero is used, but in a completely different way, and that is a completely different story…
In short, the usual way of dividing by zero does not end well, but nevertheless this should not prevent us from experimenting in the field of mathematics, suddenly we will be able to open up new areas for research.
If you break the generally accepted rules in the world of science, you can get the most unpredictable results.
Since school, our teachers have been telling us that there is one rule in mathematics that cannot be broken. It goes like this: “You can’t divide by zero!”
Why does the number 0, which is so familiar to us, which we encounter so often in everyday life when performing such a simple arithmetic operation as division, cause so many difficulties?
Let’s look into this issue.
If we divide one number by smaller and smaller numbers, we will end up with larger and larger values.
For example:
20:10 = 220:5 = 420:2 = 1020:1 = 2020:0.000001 = 20000000… and so on.
Thus, it turns out that if we divide by a number that tends to zero, we will get the largest result that tends to infinity.
Does this mean that if we divide our number by zero, we get infinity?
This sounds logical, but all we know is that if we divide by a number close to zero, the result will only go to infinity, and this does not mean that when we divide by zero, we will end up with infinity. Why is this so?
First, we need to understand what the arithmetic operation of division is. So, if we divide 20 by 10, this will mean how many times we need to add the number 10 to get 20 as a result, or what number we need to take twice to get 20.
In general, division is the inverse arithmetic operation of multiplication. For example, when multiplying any number by X, we can ask the question: “Is there a number that we need to multiply by the result to find out the original value of X?” If such a number exists, then it will be the inverse value of X. For example, if we multiply 2 by 5, we get 10. If we then multiply 10 by one-fifth, we get 2 again:
2*5 = 1010*1/5 = 2
Thus, 1/5 is the reciprocal of 5, the reciprocal of 10 would be 1/10.
As you have already noticed, when you multiply a number by its reciprocal, the answer will always be one. And in the event that you want to divide a number by zero, you will need to find its reciprocal, which should be equal to one divided by zero.
This would mean that when multiplied by zero, the result should be one, and since it is known that if you multiply any number by 0, you get 0, then this is impossible and zero does not have an inverse number.
Is it possible to come up with something to get around this contradiction?
Mathematicians had already found ways to circumvent mathematical rules, because in the past, mathematical rules made it impossible to obtain the value of the square root of a negative number, so it was proposed to denote such square roots as imaginary numbers. As a result, a new branch of mathematics appeared, called complex numbers.
So why don’t we also try to introduce a new rule according to which one divided by zero would be denoted by an infinity sign and see what happens?
Let’s assume that we know nothing about infinity. In this case, if we start from the reciprocal of zero, then multiplying zero by infinity should give us one. And if we add another value of zero divided by infinity, we should end up with the number two:
0*∞ = 10*∞ + 0*∞ = 2
According to the distributive law of mathematics, the left side of the equation can be represented as:
(0+0)*∞ = 2
and since 0+0 = 0, our equation will take the form 0*∞ = 2, due to the fact that we have already defined 0*∞ = 1, it turns out that 1 = 2.
This sounds absurd. However, this answer cannot be considered completely wrong either, since such calculations simply do not work for ordinary numbers. For example, in the Riemann sphere, division by zero is used, but in a completely different way, and that is a completely different story…
In short, the usual way of dividing by zero does not end well, but nevertheless this should not prevent us from experimenting in the field of mathematics, suddenly we will be able to open up new areas for research.
Does this mean that if we divide our number by zero, we get infinity?
This sounds logical, but all we know is that if we divide by a number close to zero, the result will only go to infinity, and this does not mean that when we divide by zero, we will end up with infinity. Why is this so?
First, we need to understand what the arithmetic operation of division is. So, if we divide 20 by 10, this will mean how many times we need to add the number 10 to get 20 as a result, or what number we need to take twice to get 20.
In general, division is the inverse arithmetic operation of multiplication. For example, when multiplying any number by X, we can ask the question: “Is there a number that we need to multiply by the result to find out the original value of X?” If such a number exists, then it will be the inverse value of X. For example, if we multiply 2 by 5, we get 10. If we then multiply 10 by one-fifth, we get 2 again:
2*5 = 1010*1/5 = 2
Thus, 1/5 is the reciprocal of 5, the reciprocal of 10 would be 1/10.
As you have already noticed, when you multiply a number by its reciprocal, the answer will always be one. And in the event that you want to divide a number by zero, you will need to find its reciprocal, which should be equal to one divided by zero.
This would mean that when multiplied by zero, the result should be one, and since it is known that if you multiply any number by 0, you get 0, then this is impossible and zero does not have an inverse number.
Is it possible to come up with something to get around this contradiction?
Mathematicians had already found ways to circumvent mathematical rules, because in the past, mathematical rules made it impossible to obtain the value of the square root of a negative number, so it was proposed to denote such square roots as imaginary numbers. As a result, a new branch of mathematics appeared, called complex numbers.
So why don’t we also try to introduce a new rule according to which one divided by zero would be denoted by an infinity sign and see what happens?
Let’s assume that we know nothing about infinity. In this case, if we start from the reciprocal of zero, then multiplying zero by infinity should give us one. And if we add another value of zero divided by infinity, we should end up with the number two:
0*∞ = 10*∞ + 0*∞ = 2
According to the distributive law of mathematics, the left side of the equation can be represented as:
(0+0)*∞ = 2
and since 0+0 = 0, our equation will take the form 0*∞ = 2, due to the fact that we have already defined 0*∞ = 1, it turns out that 1 = 2.
This sounds absurd. However, this answer cannot be considered completely wrong either, since such calculations simply do not work for ordinary numbers. For example, in the Riemann sphere, division by zero is used, but in a completely different way, and that is a completely different story…
In short, the usual way of dividing by zero does not end well, but nevertheless this should not prevent us from experimenting in the field of mathematics, suddenly we will be able to open up new areas for research.
Is it possible to come up with something to get around this contradiction?
Mathematicians had already found ways to circumvent mathematical rules, because in the past, mathematical rules made it impossible to obtain the value of the square root of a negative number, so it was proposed to denote such square roots as imaginary numbers. As a result, a new branch of mathematics appeared, called complex numbers.
So why don’t we also try to introduce a new rule according to which one divided by zero would be denoted by an infinity sign and see what happens?
Let’s assume that we know nothing about infinity. In this case, if we start from the reciprocal of zero, then multiplying zero by infinity should give us one. And if we add another value of zero divided by infinity, we should end up with the number two:
0*∞ = 10*∞ + 0*∞ = 2
According to the distributive law of mathematics, the left side of the equation can be represented as:
(0+0)*∞ = 2
and since 0+0 = 0, our equation will take the form 0*∞ = 2, due to the fact that we have already defined 0*∞ = 1, it turns out that 1 = 2.
This sounds absurd. However, this answer cannot be considered completely wrong either, since such calculations simply do not work for ordinary numbers. For example, in the Riemann sphere, division by zero is used, but in a completely different way, and that is a completely different story…
In short, the usual way of dividing by zero does not end well, but nevertheless this should not prevent us from experimenting in the field of mathematics, suddenly we will be able to open up new areas for research.
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