Thanks for the great answers.

I agree with you but I told myself maybe there is a definition of zero status.

The methodological books are vague tariffs for students.

In a simple manner, a natural number is a counting number, this means a member of the set {1,2,3,4,….}. The set of natural numbers is designated by ℕ.

However, you can define a natural number more formally. In this case, we can define the natural numbers as the set ℕ= {x| x=1, 2, 3…….} or there is some y in ℕ where y=0 or y = x + 1.

There also an alternative definition of natural numbers. According to this alternative definition, the set of natural numbers is defined as the set {0,1,2,3,4,….}. The essential properties of natural numbers remain the same whether or not we include 0.

Whether 0 is or is not a natural number is a matter of definition of natural number. A definition is simply a formal description of the terminology being used . That is something that is stated to be so and cannot be proven nor disproven. Definitions are similar to axioms in that regard.

Thanks for the great answers.

I agree with you but I told myself maybe there is a definition of zero status.

The methodological books are vague tariffs for students.

general proof can be given that almost all real numbers are normal (in the sense that the set of exceptions has Lebesgue measure zero), this proof is not constructive and only very few specific numbers have been shown to be normal. For example, Chaitin's constant is normal (and uncomputable). It is widely believed that the (computable) numbers √2, π, and e are normal, but a proof remains elusive.

Dear Hassan Kamil Ressn ,

All we know that ZERO (صفر) is essential to modern mathematics and removing zero will destroy the whole axioms of mathematics. I think the current question is about the history of zero and to discuss the validity of Peano axioms, where the first model doesn't include the zero explicitly and then it appears as a separate axiom that defines the zero.

In modern mathematics as modular arithmetic, all we know that zero can be replaced by any number in its class, as an example in a mod(n),

Zn = {0,1,2,3,....n-1} = {1,2,3,...,n-1,n}, we can replace 0 by n, and all results stayed the same.

The attached article about the history of zero is helpful.

Best regards