0 is important, but it's not true that without it we can't have the number 10.

In written form, for example, we can represent 10 and 20 without 0 easily, as the 0 is just a syntactic placeholder:

- Roman Numerals: X and XX

- Scientific Notation: 1e1 and 2e1

- Base 16: a and 14

- Set theory: {{{{{{{{{{}}}}}}}}}} and {{{{{{{{{{{{{{{{{{{{}}}}}}}}}}}}}}}}}}}}

But more importantly, the existence of the number 10 is independent of it's representation. Mathematically, the existence of the natural numbers depends on a "unit" (which we often represent with 1) and a method for incrementing or inducing the subsequent number.[1]

0 ensures the existence of groups [2], and by extension many important mathematical structures. I get your point though, and I wonder what would be an accurate, but still relatable, reason for a young person to grasp the importance the number 0?

[1] http://en.wikipedia.org/wiki/Axiom_of_infinity

[2] http://en.wikipedia.org/wiki/Group_(mathematics)

>what would be an accurate, but still relatable, reason for a young person to grasp the importance the number 0?

If you have one cookie, and you eat it, how many cookies do you now have?

I asked my nephew and this was the answer I got back:

1 cookie, in my belly. (And then he demanded a cookie.)

try asking him 'if you have one cookie, and I eat it, how many cookies do you have?'

I think everyone understands the concept of "nothing" or "no cookie" vs "a cookie", just as babies (and, say, birds [1]) have a concept of quantity and even discrete quantity at small values.

I can understand that I have no cookie without counting from the number 0. But why is it important to label nothing, to give it a name and symbol and define its properties?

[1] http://en.wikipedia.org/wiki/Bird_intelligence#Counting

Well, easy, you just apply group theory ;-) and clearly you have the correct amount such that if you keeping eating this amount of cookies, no matter how many cookies you start with, you end up the same number [Identity]

The Romans (and many others) seemed to manage.