In the set of natural numbers, 0 does not exist. This causes problems if we want to use this, we would struggle to define how many apples there are in a bowel consisting purely of bananas, that is its practical importance.
Mathematically, the additive identity plays a greater importance, for example once we build up our set of axioms of the real numbers in to theorems we can such results as, if:
ab = 0
a = 0
b = 0
a and b = 0
This is highly useful and allows us to solve many equations. By the properties of real numbers, 0 is a real number. I would highly suggest you look up what real numbers are because I have a strong feeling you are not aware of this:
Real numbers are not something mathematicians pull out of thin air, they are very well constructed. You may make your own set of numbers that does not include 0, but out of all sub sets of real numbers an uncountable amount of them don't include 0, that is not that important.
However I would gladly like to see you design a workable and practical number system without ever using 0, I would be very impressed if you can construct something as or more useful than what we have.