I know the Arithmetic proof and everything but how to explain this practically to a kid who just started understanding the numbers?
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Aside from the various mathematical reasons, what’s important to understand is that decimal representation is just that: a “representation” of the number, NOT the “true” number itself. For example the same number 1 is also 0.FFFFFFF… in hexadecimal. In fact there are infinitely many possible representations for every real number with the arguable exception of 0.
Decimal is a human invention, and like ~~all~~ most human inventions it isn’t perfect because it doesn’t have an exact 1-to-1 relationship with the real numbers. Some real numbers have one representation in decimal, others (those that are an integer multiple of a power of 10) have two, although by convention the terminating one (without the infinite sequence of 9s) is considered the “correct” one.
So what is the “true” real number itself, the unique essence of the number as opposed to its representation in decimal, binary, hexadecimal or any other base? That’s part of the beauty of mathematical ideas like numbers, we can imagine the pure concept of a number, but to write it down or say it you have to choose a way of representing it, of which there are infinitely many.
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