Because there's no way any sort of omnipotent being would be stupid enough to create a universe where numbers don't work to satisfy that equation. Obviously.

Eternity is a child playing, playing checkers; the kingdom belongs to a child.

I close my eyes and see a flock of birds. The vision lasts a second or perhaps less; I don’t know how many birds I saw. Were they a definite or an indefinite number? This problem involves the question of the existence of God. If God exists, the number is definite, because how many birds I saw is known to God. If God does not exist, the number is indefinite, because nobody was able to take count. In this case, I saw fewer than ten birds (let’s say) and more than one; but I did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, but not nine, eight, seven, six, five, etc. That number, as a whole number, is inconceivable; ergo, God exists.

I'm pretty sure there are some algebras in which there exists a value x such that x + 1 = x. But when I read Wikipedia articles on such concepts I don't really understand them.

I'm pretty sure there are some algebras in which there exists a value x such that x + 1 = x. But when I read Wikipedia articles on such concepts I don't really understand them.

Depends on what you mean by "algebra". The main problem with infinity is that you can't really subtract infinity from things (that's basically taking the logarithm of division by zero), so you can't get a ring, but you can invent an algebraic structure that doesn't have subtraction, like a semiring, and then you're good. Or you can be an analyst who works with functions defined "almost everywhere", and then it's okay if subtraction sometimes doesn't work.

That also works, for a slightly different interpretation than mine. (Though there's nothing special about 0.5; if you're working mod 1 then everything solves this equation!)

if Imipolex jumps off a cliff at 8,000 feet up, and screams the entire way down, his volume decreasing by .5 decibels per minute, when does he hit the ground?

Well Bee blanked his post, but it is true that if you do boolean logic using the engineer's convention that + means OR (and 1 means true), 1 is a solution.

if Imipolex jumps off a cliff at 8,000 feet up, and screams the entire way down, his volume decreasing by .5 decibels per minute, when does he hit the ground?

That also works, for a slightly different interpretation than mine. (Though there's nothing special about 0.5; if you're working mod 1 then everything solves this equation!)

if Imipolex jumps off a cliff at 8,000 feet up, and screams the entire way down, his volume decreasing by .5 decibels per minute, when does he hit the ground?

dB is a log scale so he just screams forever. presumably he's held up by some kind of geyser or space monster

Yeah on second thought I didn't need to blank that post because it was correct.

Well sure, but in boolean logic you also have that OR distributes over AND, which is pretty weird if you think of them as addition and multiplication.

Also, boolean logic is isomorphic to arithmetic modulo 2, but it's XOR that corresponds to addition. I think that's the main reason it bothers me.

Though I guess now that I think about it, boolean OR and AND are the addition and multiplication of a semiring, so I can't really say it's any less of an example than the one(s) I was thinking of with infinities.

You know what the number line is, right? You add one, you move one spot to the right on the line.

Doing arithmetic "mod whatever" means you take a line segment of length whatever, cut it out of the line, and tie the ends together so it makes a loop. So if you have 7 + 2 mod 8, you start seven over from zero, then move two to the right. You cross zero so you end up one to the right of zero, so the answer is 1.

mod 1 just means the loop is of length 1, so adding 1 is just going all the way around the loop once. So any number equals one plus itself in that system.

You know what the number line is, right? You add one, you move one spot to the right on the line.

Doing arithmetic "mod whatever" means you take a line segment of length whatever, cut it out of the line, and tie the ends together so it makes a loop. So if you have 7 + 2 mod 8, you start seven over from zero, then move two to the right. You cross zero so you end up one to the right of zero, so the answer is 1.

Well usually you do it with integers rather than reals, so the width has to be greater than one to be interesting.

The most practical application is what's called modular exponentiation. Basically just regular exponentiation ("to the power of") but mod whatever. Like for example, 2^6 mod 7 = 64 mod 7 = 1. Fairly straightforward.

The convenient part is that it's really hard to get logarithms, i.e. to reverse the operation. Like, if you're just given 1, and told 2 to the power of something is 1 mod 7, it's hard to tell what the something is short of brute force, even though computing the forward operation was really easy.

This makes it useful as a "trapdoor" function that's easy to do one way but hard to reverse, which is a backbone of cryptography.

that's only one thing though. it's important for math proper as well. gauss did all kinds of shit with it

I can't think of anything to say, mostly because I don't actually understand any of the math y'all are talking about

For what I was saying it mostly boils down to the fact that if you subtract x from both sides of that equation you get 0 = 1, so to have a solution you need to address that. Klino addressed it by just saying sure, 0 = 1, whereas I (and Bee) did it by moving to a land without subtraction.

A "ring" is something where you can add, subtract, and multiply, eg. the integers. (If you can also divide, it's a "field".) A "semiring" is where you can add and multiply but not necessarily subtract, such as the nonnegative integers, or the integers with infinity thrown in.

There's also the chinese remainder theorem: If you have a set of numbers "ni" that are pairwise coprime, i.e. no two of them share divisors, then you can uniquely determine any number mod N by what it is mod those ni, where N is the product of the ni.

Which I'm sure makes no sense, so for example, say n1 = 2, n2 = 3, n3 = 5. N = n1*n2*n3 = 30. If we pick a number, let's say 13, we can compute 13 mod n1 = 1, 13 mod n2 = 1, and 13 mod n3 = 3. No other number from zero to 29 shares that (1, 1, 5) set.

As a bonus, we can compute with those. To take 13 + 13 we can take 1 + 1 mod n1 = 0, 1 + 1 mod n2 = 2, 3 + 3 mod n3 = 1. And hey, 26 mod n1 = 0, 26 mod n2 = 2, 26 mod 5 = 1, it works.

You can use this to represent really large numbers conveniently. Like, 5829472801, that's pretty big, right? Too big to fit in 32 bits of computer memory for example. But the product of the first eleven primes is bigger, so we can just take x = 5829472801, and then x mod 2 = 1, etc., and we can represent it as (1, 1, 1, 5, 9, 9, 13, 18, 4, 14), all pretty small.

FYI they actually call mod arithmetic "clock arithmetic" because that's basically how clocks work.

Eh, the clock thing is a common way to teach it but it's not actually a fantastic analogy. You don't take 5 o'clock and add 8 o'clock to get 1 o'clock, that doesn't make sense. You take 5 o'clock and add eight hours to get 1 o'clock. Times are different from time intervals; to use fancy words, a clock is actually a torsor for the additive group mod 12, rather than being the group itself.

That's not even to get into the fact that modular multiplication doesn't mean anything for clocks.

I don't think I've ever run into a situation where full-on persistent modulus algebra is useful, but we use mod functions in programming all the time because integer remainders are important to a lot of things. The main one is to take Unix-time measurements (milliseconds) and convert them piece by piece into hours/minutes/seconds, today's date-time, weeks + days, etc. The operator is a %, e.g., 15 % 4 = 3.

In some of the lower-level bit ops, you can perform quick power-of-two integer-divs and mods by just bit shifting until all the upper/lower bits vanish and you're left with whatever. This is pretty common in older games where computation was at a premium and they made sure to subdivide everything they could by some power of two specifically to be able to leverage bit ops like this.

A "ring" is something where you can add, subtract, and multiply,

Ooh I have a favorite one of those. It's like a complex version of the numbers with a terminating phinary expression and it has a one-to-one mapping to Z^{4}. I've kind of invented it myself, so I haven't completely settled on the notation yet, though it's possible someone else has already worked with this ring.

Okay, might as well try a notation here. If we write a number as Sk + Ri + Nj + M, where M, N, R, S are integers and i, j, k are constants, there are several identities to define. j² = j + 1. (j is the golden ratio.) ij = ji = k. i² = k - i - 1. ik = ki = ji² = k - j - 1. jk = kj = k + i. i³ = -k + i - j + 1. i^{4} = -i + j - 1. k² = k - j - 1. i^{5} = 1. (k - i + j - 1)² = i. Have I missed anything?

Also, division by i and division by j are possible, but only because i · (-i + j - 1) = 1 and j · (j - 1) = 1. Division by arbitrary numbers isn't really a thing.

## Comments

They are long and loud and slow

They are long and loud and slow

alsothe correct symbol for "max" so it's not just or and that's why you should use abstract names for the symbolAlso, boolean logic is isomorphic to arithmetic modulo 2, but it's XOR that corresponds to addition. I think that's the main reason it bothers me.

Though I guess now that I think about it, boolean OR and AND are the addition and multiplication of a semiring, so I can't really say it's any less of an example than the one(s) I was thinking of with infinities.

They are long and loud and slow

They are long and loud and slow

A "ring" is something where you can add, subtract, and multiply, eg. the integers. (If you can also divide, it's a "field".) A "semiring" is where you can add and multiply but not necessarily subtract, such as the nonnegative integers, or the integers with infinity thrown in.

They are long and loud and slow

That's not even to get into the fact that modular multiplication doesn't mean anything for clocks.

Z^{4}. I've kind of invented it myself, so I haven't completely settled on the notation yet, though it's possible someone else has already worked with this ring.Okay, might as well try a notation here. If we write a number as Sk + Ri + Nj + M, where M, N, R, S are integers and i, j, k are constants, there are several identities to define.

j² = j + 1. (j is the golden ratio.)

ij = ji = k.

i² = k - i - 1.

ik = ki = ji² = k - j - 1.

jk = kj = k + i.

i³ = -k + i - j + 1.

i

^{4}= -i + j - 1.k² = k - j - 1.

i

^{5}= 1.(k - i + j - 1)² = i.

Have I missed anything?