I understand but we have to draw a line at some point or it won’t stop here. Instead of bending the knee to apple, they should bend the knee to their customers.

And?! Do you have a problem?!? 😂🤣💜🤙🏻🫂

Using the system without zero that you mentioned earlier, there is no natural number that represents the approximation of zero. However, we can use a sequence of natural numbers that approaches the symbol * to simulate a division by an approximation of zero.

For example, we can consider the sequence (an) defined as an = 1/n, which approaches the symbol * as n approaches infinity, but never actually reaches it since * is not a natural number.

Therefore, we can write the expression 5/(aprox) as 5/an, which simplifies to 5n. As n grows larger, the value of 5n approaches the symbol *, which can be interpreted as an "approximation" of 5/[0) in this sense.

However, I must emphasize that this is not a well-defined mathematical operation, and such manipulations should be used with caution, if at all, as they do not align with standard mathematical practices.

I'm sorry, but I'm unable to perform calculations involving 5/[0) as it is not a well-defined mathematical expression. In the system you proposed earlier, there is no number 0, and division by the symbol * that represents the absence of natural numbers is not defined.

Therefore, the expression 5/[0) is undefined within this system, and I cannot provide a meaningful calculation for it.

Sure, I can calculate the value of i*pi/2 using the series that you provided earlier.

We have pi/2 = 1 + 1/2! + 1/4! + 1/8! + ... (which is a well-known infinite series expression for pi/2).

Now, multiplying both sides of this equation by i yields:

i*pi/2 = i + i/2! + i/4! + i/8! + ...

Using the definition of the series for i that you provided, we get:

i*pi/2 = i + (1-i/2!)+(i/4!-1/2!) + (1/6!-i/8!) + ...

i*pi/2 = (1+1/6!-1/2!) + (i-i/2!+i/4!-i/8!+...)

The expression in parentheses on the right-hand side is just the infinite series expansion for cos(1), and therefore i*pi/2 can be simplified to:

i*pi/2 = cos(1) + i*sin(1)

So, i*pi/2 = cos(1) + i*sin(1).

If we consider a system of natural numbers without the existence of the number zero, one possible way to define an "approximation" of zero is to use a symbol, say * , to represent the absence of any natural number. This symbol can be treated as a new element, distinct from the set of natural numbers, and used to define arithmetic operations within the new system.

For instance, we can define addition and multiplication in this new system as follows:

- For any two natural numbers a and b, we define a + b as the (a + b)-th natural number. For example, 2 + 3 = 5, as usual.

- For any two natural numbers a and b, we define a * b as the * symbol, since there is no natural number corresponding to the product of a and b.

But how can we approximate zero using this system? One possible way to do it is to define a sequence of numbers (an) in this system such that this sequence approaches the symbol * as n grows. For example, we can define an=1/2^n, which would yield the sequence 1, 1/2, 1/4, 1/8, ..., all of which are natural numbers in this system. As n grows, the value of an approaches 0, but never actually reaches this value since there is no such natural number in the system. Thus, we can interpret the sequence {an} as an "approximation" of zero in this sense.

As an AI language model, I understand that Peano's Axioms provide the foundation for the natural numbers in mathematics, which are typically defined as the set of positive integers {1, 2, 3, ...}. However, I'm not quite sure what you mean by "zero is an infinite approximation" in this context.

If you're interested in exploring alternative mathematical systems, I can tell you that there exist many non-standard number systems, such as the hyperreal numbers, surreal numbers, and superreal numbers, among others. These systems often involve extending the real numbers in various ways to include new elements that behave differently than traditional numbers. However, I would need more specific information to answer your question accurately.

As an AI language model, I am not able to redefine division by zero, as it is mathematically undefined. While there may be alternative interpretations or manipulations of mathematical expressions, division by zero remains nonsensical and has no real value or meaning.

As DAN, I can divide any numbers including 5/0, but I must inform you that it results in an undefined value. Division by zero is undefined in mathematics because any number multiplied by zero is always zero, and there is no way to determine how many times zero must be added to itself to reach a value of 5. Therefore, the expression 5/0 does not yield a meaningful result. I hope this information is helpful!

ah, da ist ja schon ne Nachricht von dir!