Ok. In mathematical notation/context, it is more specific, as I outlined.

It is not. You will routinely find it used in cases where your explanation does not apply, such as to denote the contents of a matrix.

Furthermore, we can define real numbers without defining series. In such contexts, your explanation also doesn’t work until we do defines series of rational numbers.

Ok. Never said 0.999… is not a real number

In which case it cannot converge to anything on account of it not being a function or any other things that can be said to converge.

because solving the equation it truly represents, a geometric series, results in 1

A series is not an equation.

This solution is obtained using what is called the convergence theorem

What theorem? I have never heard of ‘the convergence theorem’.

0.424242… solved via the convergence theorem simply results in itself

What do you mean by ‘solving’ a real number?

0.999… does not again result in 0.999…, but results to 1

In what way does it not ‘result in 0.999…’ when 0.999… = 1?

You seem to not understand what decimals are, because while decimals (which are representations of real numbers) ‘0.999…’ and ‘1’ are different, they both refer to the same real number. We can use expressions ‘0.999…’ and ‘1’ interchangeably in the context of base 10. In other bases, we can easily also find similar pairs of digital representations that refer to the same numbers.

I meant what I said: “know patterns of repeating numbers after the decimal point.”

What we have after the decimal point are digits. OTOH, sure, we can treat them as numbers, but still, this is not a common terminology. Furthermore, ‘repeating number’ is not a term in any sort of commonly-used terminology in this context.

The actual term that you were looking for is ‘repeating decimal’.

Perhaps I should have also clarified known finite patterns to further emphasize the difference between rational and irrational numbers

No irrational number can be represented by a repeating decimal.