Yes, there is a smallest positive number that isn’t zero… if you want there to be.
Everything in mathematics is a label for a concept. That’s why it’s popular to call mathematics a language. If there isn’t a word for something, you can make one up.
However, if you want to communicate with others, you have to speak the same language, which means agreeing on definitions and sticking to them. In mathematics, we don’t usually consider infinites (ω) or infinitesimals (ε) to be real numbers (ℝ) because they are not Archimedean. We sometimes treat them as if they were. But even then, we call them hyperreal numbers (*ℝ), and say that they are an extension of the set of real numbers.
You and your brother essentially applied the axiom of Archimedes and arrived at the generally accepted conclusion.
For any positive ε in K, there exists a natural number n, such that 1/n < ε.
You chose the natural number 10 (adding an extra zero in the decimal place before a number) and your brother chose 2. Although, asmeurer rightly points out that it is not proper to say “put an infinite amount of zeroes in the decimal place before a number”.
While it has proven useful to give infinity a name and a symbol (∞), the same can’t be said about the thing that is an infinitesimal positive distance from zero.
You should take away these two points:
- The thing that is an infinitesimal positive distance from zero is not a real number.
- There is no name or symbol for the thing that is an infinitesimal positive distance from zero.
But go ahead and call it a number and give it a name and a symbol. If you want to.