the downside with this approach is that it will eventually terminate. the version in the original post has the advantage of giving me plenty of time to contemplate life’s many mysteries.

don’t do this to me

the only possible way this could be worse was if it was called CloudChain and also had weird blockchain/crypto nonsense.

we have always been at war with Eurasia.

now we’re talking

☝️🤓 that would still only be a quadratic rate of increase, not an exponential rate of increase

what if they were all named odysseus

if OP is using gentoo then there is a very real chance they aren’t able to take screenshots yet.

back in my gentoo days it took a while to get that set up. although it wasn’t exactly a top priority

everything is simple once you know how to do it. i think a lot of the arch recommenders likely don’t realize how difficult some of these things can be the first time. it’s the same with any kind of specialized knowledge i think. it’s one of the reasons why teaching can be difficult. but the arch community has been super helpful in my experience

i would like to post a community note saying that i personally believe it to be true

for anyone curious, here’s a “constructive” explanation of why a⁰ = 1. i’ll also include a “constructive” explanation of why rational exponents are defined the way they are.

anyways, the equality a⁰ = 1 is a consequence of the relation

a^m+1 = a^m • a.

to make things a bit simpler, let’s say a=2. then we want to make sense of the formula

2^m+1 = 2^m • 2

this makes a bit more sense when written out in words: it’s saying that if we multiply 2 by itself m+1 times, that’s the same as first multiplying 2 by itself m times, then multiplying that by 2. for example: 2³ = 2² • 2, since these are just two different ways of writing 2 • 2 • 2.

setting 2⁰ is then what we have to do for the formula to make sense when m = 0. this is because the formula becomes

2⁰⁺¹ = 2⁰ • 2¹.

because 2⁰⁺¹ = 2 and 2¹ = 2, we can divide both sides by 2 and get 1 = 2⁰.

fractional exponents are admittedly more complicated, but here’s a (more handwavey) explanation of them. they’re basically a result of the formula

(a^m)ⁿ = a^m•n

which is true when m and n are whole numbers. it’s a bit more difficult to give a proper explanation as to why the above formula is true, but maybe an example would be more helpful anyways. if m=2 and n=3, it’s basically saying

(a²)³ = (a • a)³ = (a • a) • (a • a) • (a • a) = a^2•3.

it’s worth noting that the general case (when m and n are any whole numbers) can be treated in the same way, it’s just that the notation becomes clunkier and less transparent.

anyways, we want to define fractional exponents so that the formula

(a^r)^s = a^r • a^s

is true when r and s are fractional numbers. we can start out by defining the “simple” fractional exponents of the form a^1/n, where n is a whole number. since n/n = 1, we’re then forced to define a^1/n so that

a = a^1/n•n = (a^1/n)ⁿ.

what does this mean? let’s consider n = 2. then we have to define a^1/2 so that (a^1/2)² = a. this means that a^1/2 is the square root of a. similarly, this means that a^1/n is the n-th root of a.

how do we use this to define arbitrary fractional exponents? we again do it with the formula in mind! we can then just define

a^m/n = (a^1/n)^m.

the expression a^1/n makes sense because we’ve already defined it, and the expression (a^1/n)^m makes sense because we’ve already defined what it means to take exponents by whole numbers. in words, this means that a^m/n is the n-th square root of a, multiplied by itself m times.

i think this kind of explanation can be helpful because they show why exponents are defined in certain ways: we’re really just defining fractional exponents so that they behave the same way as whole number exponents. this makes it easier to remember the definitions, and it also makes it easier to work with them since you can in practice treat them in the “same way” you treat whole number exponents.