What is 2 raised to the power of 50,000?

Most answers given are somewhat baffling to me.

We are considering a number: two to the power fifty thousand. That's two times two times two times... Lots of times. Big number for sure, relative to all the numbers that have any significance in our lives.

Yet we can write down this number very succinctly and elegantly:

2^{50000}.

This right there, this very simple notation of a natural number, is a better answer to the question "what is this number" than most of the other answers provided here. This number is two-to-the-fifty-thousand. That's it. It is what it is.

With this simple notation, 2^{50000}, we understand many things about the number. We can see that it's even. We can see that it's a square, and a tenth power. We can see that it's not divisible by 3, nor by 5, nor by 23. In fact we can determine its precise prime factorization without having to think: it's staring at us right in the face. We can easily determine that the number right above it, and the number right below it, aren't primes. All kinds of things we can tell about this number.

Now, in what sense does "roughly thingamabob times ten to the blah" qualify as an answer to the question "what is this number"? That kind of answer loses all the fine details I just described. You can no longer tell if the number is a square, or a prime, or anything like that - you can't even tell if it's even or odd! You're supposed to feel you know its "order of magnitude", but it's unclear why "ten to the fifteen thousand" is more revealing, size-wise, than "two to the fifty thousand".

There's nothing inherently useful about powers of ten; sure, they're kinda useful when we talk about numbers we roughly know, like millions and billions, simply because we are used to them, but what knowledge or understanding is bestowed upon us when we say that a simple explicit integer is "roughly" ten to the "roughly" 15,000?

Another answer dutifully produced the entire decimal expansion of this integer. Fifteen thousand pointless digits between 0 and 9. If I handed you this chain of characters and asked you if the number is divisible by 19, you'd be in serious trouble. But in its original form, the form it is given in the very question, the answer is obvious.

I'm not claiming that "being divisible by 19" is a crucial feature of an integer. It's just that in every form of representing numbers there are some things that are readily visible, and some things that are not. The decimal expansion of an integer of this size is just useless.

Many of us are so used to identifying a number with its decimal form that we are led to believe that a string of characters is the proper answer to the question "what is the number N". But that's just false. There are tons of ways to represent every number, and those different ways tell us different things. Our minds can do absolutely nothing with pages and pages of digits; it's such a waste of information.

Imagine someone answered the question "what is 2^{14}" by providing you with a list of precisely 16,384 slashes, like this: //////////////.... (and so on). Is that a legitimate answer? Sure it is, it's the unary representation of the number. Is it helpful? Not in the least. But it's about as illuminating, and about as long, as the list of digits in the base-10 expansion of 2^{50000}.

So, what is 2^{50000}? It's 2^{50000}. A number. Very specific and unambiguous. It's also 32^{10000}, precisely. It's the number of possible books of 10,000 lower-case letters and a few punctuation marks. Or the number of 100 x 100 images made of pixels having 32 possible colors.

It's also roughly 2015^{4555}, in case that's helpful, or maybe you prefer 4159^{4159}. Or 7^{7^5}. Or 4654!, meaning the number of ways of shuffling about 90 decks of cards.

Many representations, many approximations, one simple number: 2^{50000}.