if anything to the zero power is 1, what does that mean for pi? pi to the 0 is 1? or pi to the 0 is "base ten 1 represented in base pi"?
asking because that page starts out stating that 0 is always 0 and 1 is always 1.
To represent the number 0 in base 3 we just write: 0.
To represent the number 1 in base 3 we just write: 1.
To represent the number 3 in base 3 we must write: 10.
In general, for any positive number n the symbols used to construct representations for number will be {0, 1, 2,. . ., 9,. . ., n-1}. In base n the number 0 is represented as: 0. In base n the number 1 is represented as 1. The number n is represented as: 10.
Thus it makes sense to always use the symbol 0 and 1 as part of the set of numerals in whatever base you are in and if one does this then 0 is 0 and 1 is 1.
The symbol 0 doesn’t mean anything rather than like it’s etymology it’s the absence of anything.
1__
Is the same as
100
So it’s the position more than anything right?
Depends on the base of the position
1_
Could mean 2 for example
1__
Could mean 4 or 100 for example
11_ as 4 in Gray code :p
So the best way to think about numbers (in this situation. in this situation. obviously there are other ways to think about them. i'm not a platonist. nobody is a platonist enough to justify the amount of pedantry this take would normally inspire) is as abstract objects which exist in and of themselves. A particular base is just a way to write these abstract objects in a particular way. You can write numbers in a way which doesn't require a base, or is so annoying to describe as a base that you honestly shouldn't, such as the Roman system.
This is important because there's a really fundamental sense in which 3 (decimal) and 10 (binary) are the same thing. When I write '3' I'm pointing you towards the abstract object |||. But that's just convention! Here on HN, I could write '1A' and expect many readers to see it as the same as '26' or '10011010'. With the naturals (and 0) you can think of 3 as fundamentally those three tallies, or three apples, or whatever. For the reals, algebraicists picture an element of a complete ordered field (my brain paints these, like all objects viewed abstractly, as pale tan circles) and analysts picture a point on a line. This representation, of course, doesn't scale, which is why we pick some number of symbols (usually 10, sometimes 16, 8, or 2, and in my very favorite dreams 12) and just agree to map them onto specific abstract objects. Then we can make all the other numbers in a super compact notation without making up any new symbols.
So what about 0 and 1? Well 0 is the number which, when added to any x, gives x. And 1 is the number which, when multiplied with any x, gives x. Now I know what you're saying; "Ok, why do I care? 3 is the number of legs a table needs so it won't fall down, but it's not special". We see for any sane base that (so long as we're not just making up symbols) we'll write 0 just as 0 (since it's 0n^0 + 0n^1 + 0n^2...). What base is that expansion written in? I did it in decimal, but it doesn't matter! You can just as easily write (0n^zero + 0n^one + 0n^two...). Similarly, since it is just a fact about numbers that x^0=1 (or, if you like, x^zero=one), we'll write 1 as 1 (from 1n^0+0n^1...). If you trust that that relation holds up for irrational x, which I can't make you do but you definitely should, then you see that we'll also be writing 0 and 1 as 0 and 1 in base pi too.
So what is pi^0? It's one! One in what base you ask? All of them! It's equal to the number which '1' represents. In a way that's because 0 is always 0 and 1 is always 1, but really it's the other way around. 0 is 0 and 1 is 1 regardless of base because pi^0, and 10^0, and any other number to the 0th power, is 1.
Basically this exact issue is what made Tensor algebra so difficult for me, and I suspect a stumbling block for many people at some point. There you're talking about the basis of a vectorspace, rather than the base of a number system, but it's basically exactly the same thing. The key comes from thinking of objects floating in space with symbols pointing to them. We can operate on the symbols in all sorts of ways, and note cool patterns, but if you confuse them with the objects they point to you'll eventually be in for a bad time.
Times-table is very compact, and it's a natural fit for hand-counting up to 2 digits - 1's on one hand, 6's on the other, so you can count up to 35 (base 10) on two hands not just 10.
And IMHO to avoid confusion with decimal, alternate bases need completely different glyphs to represent the digits, and completely different naming. Fewer digits makes that easier.
Seximal is IMO the sad middle child of the bases. Sure it's small, but if I wanted small I'd stick with binary. Oh, the expansion is too large? Ternary has the best radix economy. If you're gonna have a number system divisible only by two numbers, you might as well stick with base 10!