My thought process went something like this. In computer scince we can represent 2 in binary as 10, 0010, or 0000 0010. Adding zeros to the left of a natural number dosn't chang it's value, So in decimal I can represent 2 as ...002 where the dots reperesent an infinite precision of zeros. Now what is I define all those zeroes to be twos or to but it more mathematically $\sum_{n=0}^\infty 2(10^n)$.
Now I know the standard answer to that formula would be that it approaches infinity but thats not what i'm intrested in. what i'm intrested in is the thing spit out at the hypothetical end of the iterative process. The value of this thing is obviously undefined but doing arithmetic on it still seems to make sense for example subtracting 1 would seem to give us 1 with an infinite precision of twos and adding 1 would seem to give us 3 with an infinite precision of twos. multiplying or dividing by 2 would seem to give a series of fours and ones respectively. We would come to edge cases like what an infinite series of 9 plus 1 would equal. Could it equal zero? but any ways since it seems we can do arithmetic on this thing does that make it a number? If it is a number it was made from adding and multiplying natural numbers so does that mean it's a natural number?