Hello, I am mathematician.
Historically speaking, the number 0 was introduced really recently. There were already infinity symbol and imaginary numbers but not also 0.
So, let us talk pragmatically: What we want to achieve versus what we loose in terms of universality.
If you want an arithmetics (numeric calculus) system capable of:
1. Addition, which is:
- Symmetric: a + b = b +a
- Associative: a + (b + c) = (a + b) + c = a + b + c
- Contains neutral element 0 with property: 0 + a = a, for any a
- Complete: for any number a, there exists a number (-a) such that (-a) + a = 0
2. Multiplication, which is:
- Symmetric: a * b = b * a (there are algebras where this is not required, and actually is not true)
- Associative: a * (b * c) = (a * b) * c = a * b * c
- Contains neutral element 1 with property: 1 * a = a, for any a, if multiplication is not symmetric, also require a * 1 = a (if it is, this equality results from previous).
- Almost complete: for any number a, different from 0, there exist an inverse a^(-1) such that a^(-1) * a = 1
3. There exists distributive rule between addition and multiplication: (a + b) * c = a * c + b * c. If multiplication is not symmetric, also we need to require that a * (b + c) = a * b + a * c (if multiplication is symmetric, it results from the previous equality).
This system was designed to model the numbers we use every day. But this implies a limitation that multiplication inevitable is "almost complete". When I say "implies", I do not speak about the will of some particular mathematician. The very system becomes contradictory by construction, and this can be proved from the above text. Really, for any number a, the following is always true:
a * 0 = a * (-b + b) = a * (-b) + a * b = - a * b + a * b = 0.
Moreover, if we require multiplication completeness, that is:
x / y = z => x = y * z, for any x, y, z,
even for x different from 0 and y = 0, then put x = a, y = z = 0 in previous equality and obtain:
a = 0 / 0 for any a.
So all numbers will be equal among them.
We cannot just exclude 0 from numbers, because it is neutral element of addition (but creates limitation in multiplication). So by trying to make multiplication absolutely complete, we must give up the neutral element of addition. By giving up this element, we also give up the completeness of addition (and there is no possibility to "almost complete"). So, we gain 1 point and loose 2 points, one of which is already broader than assumed gain (give up completeness of addition by almost completeness of multiplication).
If you want a special treatment of some numbers, then you give up the universality, which makes the mathematics so useful. Namely, all the above is expressed using letters instead of actual numbers, because of "for any number" and "there always exist". With special treatment of some numbers, you cannot speak so concise.