TLDR: 2-adic numbers ($\mathbb{Q}_2$) are expressed as ....01101011.001, meaning an infinite amount of digits on the most significant side and a finite number of "decimals". Base-2 represented reals are the opposite: 110001.0010010111010...., a finite number of digits on the most significant side and an infinite amount of "decimals".
What happen if we mix both? i.e. consider: ....11100101101.1000100001...., infinite amount of digits on the most significant side AND infinite amount of "decimals".
Do we get a field? Is there a name for these "numbers"?
Longer statement
2-adic numbers
2-adic numbers form a field ($\mathbb{Q_2}$), they are uniquely determined by their sequence of digit which is infinite on the most significant side and finite after the decimal.
Doing operations on 2-adic numbers is "simple" as it is exactly like operations on natural numbers but where you never stop propagating carries.
For instance: ....01010101 represents the number "-1/3" because 2x + x = ....11111 = -1.
2-adic numbers are a non countable field but very different from the reals: no total order and, for instance, you can have square roots of negative numbers.
Base-2 represented reals
Any real number can be represented in base 2. Representation is not unique as we have: 0.11111.... = 1
The addition on real numbers is different from the one on natural numbers. The algorithm works by considering successive truncation. In base 10 if you want to add $\pi + e$ you will successively consider: 3 + 2 = 5, 3.1 + 2.7 = 5.8, 3.14 + 2.71 = 5.85, 3.141 + 2.718 = 5.859 etc...
Mixing both
Although 2-adics and reals have very different properties, the symmetry of their representation is striking: .....1111101.01 VS 10011.00111100...
This begs the question: what happens if we consider "numbers" of the form .....10101.00101011..... ?
What about representation uniqueness? Do we get a field? Or at least a ring? Does this object have a name?
What do you think?