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With your suggested number, you need an integer power of 10 that you can assign 1 as a coefficient to that gives an infinite number of smaller powers of 10 to assign 0 to, in other words, a positive integer that is smaller than an infinite number of positive integers. Every digit in a number is a coefficient in the sum corresponding to a positive integer power of 10. In this case, a decimal number $0.x_1x_2x_3\ldots$ is shorthand for $\frac.$$ Now, some discussion about what an infinite sum even means, and if it converges are needed to truly make sense of this, but even without that, we can see that the above claim is meaningless. Of course, in mathematics, you aren't just arbitrarily allowed to say "this is allowed" and "this isn't allowed." You have to fall back to an accepted definition to see if something makes sense. You can't have a number with an infinite number of zeros followed by a one. But there are some theories where there is a smallest measureable length in some sense, see Planck length. There are various ramifications of this and you might want to look into infinitesimals or ordered sets if you are curious about such things.Īs for the smallest object in the world, this is a physics question, which has no definite answer as far as I know.
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The halving argument does not work here, as you cannot split $1$ into two positive whole numbers. Of course there is a smallest positive whole number/integer, it is $1$. But there is no bound on the number of $0$ one can have before the first non-zero digit also in total there can be infinitely many $0$, but not before the first non-zero one.) (One cannot have infinitely many $0$ and then the first $1$, or non-zero digit, in a decimal expansion. Note though that it is better to say that there can be arbitrarily many $0$ rather than infinitely many. As mentioned in comments your brother divides by $2$ while your argument amounts to dividing by $10$. There is no smallest positive real number.
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