In modern Western culture we think of numbers typically in base 10. By "we" I mean the vast majority of the public. For some reason this is easy, and I'm not sure why or how to pinpoint it down to some concrete reason for its success. The only thing I can think of is we have 10 fingers so we can count on them, and 10 divides pretty "easily" whatever that vague term means.
But we can think of extremely large numbers down to a very low level of precision using this system. For instance, you can think of the number $171,129,412$, which goes down to describing "two" at the scale of 100 million+. But it's as if in your head you can visualize every element in that set, even though you really aren't, but you get the sense that you can understand its size.
You start to lose track a little if the number is too big though: $12,918,274,182,183,182,874,371,173$. It's pretty much impossible now to know how big this is down to the $173$ scale at the end. I have no idea why but it just seems this way.
But you can get into the millions an billions down to the scale of 1, using our numbers represented in base 10 system.
We can use the base 10 system differently and do scientific notation to create orders of magnitude with it as well, leading to thinking of huge numbers at a broader scale / level of detail. Like $10^{30}$ is pretty large, but you can get a sense of it's relative size to $10^2$.
We can also use the base 10 system for decimals to a good degree, where we can go 9 or 10 decimal places in and still feel we are understanding the value's accuracy.
Given all this, I'm wondering if there are any other systems that work equally as well, or close to as well, or even better, when compared to the base 10 system.
Some common base x systems are base 2 for binary, base 16 for hex, base 60 in the past seems like it was popular. However, I personally can't easily look at one of these numbers and get its value down to an accuracy of 1 right away. TBH, I don't get any sense of value of it all all. I'm not sure if I could with practice, or there is something inherent about 10 that makes it simple and intuitive.
My question is, if there is anything other than base 10 that lends itself to easily understanding large and small numbers at any scales, in the same or similar way that the base 10 system does (as described above). In particular, wondering about base 2, base 3, base 4, base 5, ... base 16-ish, and then any arbitrary number after that that lends itself to understanding would be interesting. But mainly wondering if any of these smaller numbers have good properties for understanding like base 10.
One system that is interesting to consider is Roman numerals. Instead of going $1, 2, 3, 4, \cdots$, where it's 1-digit numbers, followed by 2-digit, followed by 3 digit, ..., you instead have $I, II, III, IV, V, \cdots$, so that was $1, 2, 3, 2, 1, \cdots$. So it's an interesting way to represent it while still keeping it base 10 or whatever base. This fits into the category of things I'm asking about in some way.
Some common base x systems are base 2 for binary, base 16 for hex, base 60 in the past seems like it was popular.
This is just because you're used to base 10. There is some sort of an optimal base for human memory (base 1,000,000 is too big, because nobody can remember that many symbols, and base 2 is too small, because nobody has trouble remembering more than 2 symbols, so it must be somewhere in between), but it's very unlikely to be 10. In particular, 16 is just as easy to remember, and scales better, so the optimum is at least 16 (sticking to constant-base systems).
– user3482749 Jan 13 '19 at 16:32