The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That’s why it can’t contain 2 despite being infinite and non-repeating.
Implicitly defining a number via it’s decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you’re implying there’s a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it.
You can’t prove that there isn’t one somewhere
You can, it’s literally the way the number is defined.
Defined where ?
It’s implicitly defined here by its decimal form:
The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That’s why it can’t contain 2 despite being infinite and non-repeating.
Might very well be :
0.101001000100001000001202002000200002000002 …
Real life, is different from gamified questions asked in student exams.
Implicitly defining a number via it’s decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you’re implying there’s a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it.
assumption ≠ definition
Math kind of relies on assumptions, you really can’t get anywhere in math without an assumption at the beginning of your thought process.
Obviously. But still maths avoids stuff like “I assume the answer is X. QED.”
Why couldn’t you?
Because you’d need to search through an infinite number of digits (unless you have access to the original formula)