Zero is important because it is the first abstraction. If your notion of numbers includes the idea of zero
as a number, then you have broken through the first intellectual barrier of mathematics. Without this, it remains tempting to still consider numbers as existing for the purpose of counting _things_.
But with zero, this idea converges on the same thing. No matter what things you were counting, if you have zero of them, you have the same idea. And so you take a step towards the idea of a number being a concept in its own right, rather than existing purely for the purpose of counting or measurement.
It is the same sort of conceptual freedom that allows you to do things like add a number to a square. To deal with an equation like x + x ^ 2 = 0. If you're stuck with numbers "meaning" something beyond themselves, then you'll never add x to x^2. One is a length, the other an area. They are different objects.
This intellectual leap is one that must be made by all students of mathematics - and many young people do not.
But with zero, this idea converges on the same thing. No matter what things you were counting, if you have zero of them, you have the same idea. And so you take a step towards the idea of a number being a concept in its own right, rather than existing purely for the purpose of counting or measurement.
It is the same sort of conceptual freedom that allows you to do things like add a number to a square. To deal with an equation like x + x ^ 2 = 0. If you're stuck with numbers "meaning" something beyond themselves, then you'll never add x to x^2. One is a length, the other an area. They are different objects.
This intellectual leap is one that must be made by all students of mathematics - and many young people do not.