Can we write a number, bigger than the number of atoms in the universe, in full binary representation?

Question

You sometimes hear stuff about numbers, like "that's bigger than the number of atoms in the universe". Which makes me wonder - how can you even represent such a number? I mean - if every single item of that number would be written even on something small like an atom, you would still run out of atoms. So compression must be in place. And that's understandable - 100 requires only 3 signs (1,0,0) to represent 100 items. $2^{100}$ requires 4 signs (2,1,0,0) to represent a very huge quantity. But I wonder if we tried to write $10^{82}$ (estimated # of atoms) in full binary representation, i.e. a sequence of $0$'s and $1$'s - will be able to do it, or will we run out of space?

Answer 1

In fact because each digit gets weighted more as we go left, we can represent up to $2^{10^{82}}$ in binary marking each with 1 or 0. In decimal we'd get to see up to $10^{10^{82}}$ which is a googol raised to the $10^{80}$ power.

Answer 2

Yes, we can. For example you can try to think the binary representation of $10^{82}$ as a string where a single character is a binary digit.

Answer 3

It's very easy to do so!

https://www.wolframalpha.com/input/?i=IntegerDigits%5B10%5E82%2C2%5D

Indeed, the written efficiency of our place-based numerical system (which came to us from Arab nations, who themselves learned it from India) over, say, Roman numerals is particularly evident in questions like this. Just one more binary digit allows us to double the size of the numbers we're discussing—and doubling over and over again reaches humongous numbers very quickly.