A new set of numbers starting from 0^0=1

[Z.10.46]

Hello,

 

In algebra and number theory, it can be demonstrated that 0^0 = 1 is true, but in analysis, 0^0 is an indeterminate form. Through the calculation of limits, it can be equal to 1, to other finite values, diverge, or even not exist. This is why, when faced with this issue, mathematicians have conventionally set 0^0 = 1.

So, let's be bold: instead of saying that 0^0 does not have the same value in all contexts and that 0^0=1 is a convention, let's change the notion of the number 1 so that 0^0 = 1 in all contexts.

In this new conception of the number 1, it would be both a number and an indeterminate form, meaning that it could be equal to 1, to other finite values, diverge, or even not exist.

To create a new mathematics based on the number $1$ with its new properties, it is necessary to generate the other numbers and define operations such as addition, multiplication, division, etc.

Here is an example of how we might generate the other numbers from the number 1, but it is possible that there are other ways to do it...

If the number $1$ has the properties of infinity, then 1 + 1 = 1, 1/1 is an indeterminate form, $1-1$ is an indeterminate form, and 1/1=1.

To generate the number 2, I simply propose that 2 be an indeterminate form,

so 1 + 1 = 1, 1 / 1 = 2, 1 - 1 = 2, 1 * 1 = 1.

To generate 3: 2 + 2 = 2, 2 / 2 = 3, 2 - 2 = 3, 2 * 2 = 2

To generate 4: 3 + 3 = 3, 3 / 3 = 4, 3 - 3 = 4, 3 * 3 = 3

...

To generate n: (n-1) + (n-1) = (n-1), (n-1) / (n-1) = n, (n-1) - (n-1) = n, (n-1) * (n-1) = n-1.

**Questions**:

Thus, starting from $1$, I have constructed all the numbers and defined the operations of addition, subtraction, multiplication, and division. Do you think you can do better?

How would you calculate, for example, 1 + 2, 1 * 2, 2 - 1, and 2 / 1 ?

If we keep the same operations +, -, *,/ and the same calculations as in classical mathematics when a is different from b in a+b, a-b, a/b, and a*b, would 0^0=1 in all contexts?

Can we create a new mathematics based on this new concept of the number?

Edited August 20 by Z.10.46