# I need help with a story that involves math

I'm creating the concept for a story called *The Little Differences*. It's about an accountant that, one day, out of the blue, notices that a certain calculation is producing a slightly wrong result. Barely noticeable, nothing world-changing,

He runs it on the computer, tries different software, a physical calculator... everything gives a result that's a little off. When he checks on paper himself, he gets the correct result. But, to his surprise, everyone else tells him that he's the one that's off, and that the incorrect result is actually perfectly sound.

I need something that makes sense, mathematically. The weird result must be something that really is wrong, and not just something that programs sometimes get wrong (I don't want it to be explained at all... I mean, the reason why it is occurring must not be something easily reducible to some well-known malfunction). But it must also be minor enough for someone to miss, something that wouldn't really cause much trouble in the real world (is that possible? IDK).

Lastly: it must be something that I'm able to explain (on some level) to a non-math reader.

So, Tildes math wizzes, what you suggest? :D

I think there are some options based on how

weirdyou want things to be. Here are some options that I thought of:293,339 * 1022 = 299,792,458 is a real calculation, but in your world, make this calculation multiply to a different result other than 299,792,458. This is a notable number - it is the speed of light in meters per second. You could have all sorts of weird things happen through the implication that the speed of light is different "in universe", but you don't actually have to spell anything out about it! Just one guy finds that 293,339 * 1022 = 299,792,458 and everyone else says "No, 293,339 * 1022 = 300,794,452" and the world is subtly different.

You could go with something like Leibniz' formula for pi. Hard to write without LaTeX, but that equation is:

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... = pi / 4

This has the potential for a recurring hook - as the accountant does more computation, they are further from the correct numbers. It's not particularly hard to work out what the numbers should be for each step. Again, this could have subtle "the world is

off" implications.Another geometrical option: the interior angles of a dodecagon (12 sided polygon) add up to 1800 degrees. You could set it up so that in your story, the interior angles add up to more or less than 1800 degrees. If they add up to something else, then the world will be just slightly hyperbolic and again, everything would be just

weird, and you could highlight this with the story, and not delve into it too deeply.Anyways, I guess most of what I was looking at was math that would be world changing, but not super noticeable.

Edit: I had some more thoughts.

I guess I based most of these on "this guy experiences

realMaths, but the in-story universe doesn't" so hopefully that fits what you're looking for.I thought of another option - the protagonist does arithmetic in base-10 but the universe uses a different base, like base 9 or base 12. There would be some trial and error to get numbers that seem appropriate, but you could imagine a conversation that went something like. "So two times two is four. And three times three is nine. But four times four is

fourteen?" Here's a simple base conversion tool.This could be something to explore - if you take euclidean geometry, and then remove an axiom, and then write something in

thatworld. Though... that's sort of the premise of my first three ideas.That's a really interesting premise. I have a few questions to help both brainstorm and figure out what you're looking for:

Is it consistent and repeatable?We've established that he gets different answers than the rest of the world. But does he get the same different answers every time? And does everyone else? Let's say for example we're talking about 200+200=401 (I'm going to keep using this example in the future just so I have something to talk about), If he does it 10 times in a row does he get 401 every time, or does it jump around? The same question for everyone else? Or is it a "a watched pot never boils" sort of thing, where it only occurs when you're not looking very closely?Does it manifest in the real world?Imagine a friend gave him a stack of 200 $1 bills. Then another friend gave him a stack of another 200 $1 bills. He adds the stacks together creating a pile of 200 + 200 $1 bills. He then gives each of their friends back their 200 bills. Is there now $1 left over that appeared from nowhere? If so, is that new bill that came out of thin air 'real'? Does it have a unique serial number like all bills? What if the objects aren't uniquely numbered, like cookies? Or maybe it's more like he gives the bills back and everything appears normal, but later he finds a 'leftover' one in his pocket,Does it only affect him?In the previous example, is he the only one that can make the phantom bill appear? Or can anyone do it, but they just won't notice the weirdness?Is there a "word-of-god" answer to who is correct?Depending on some of the answers to the previous questions it could be both he and everyone else are correct, just operating on different versions of math and reality. Or just one version could be correct. Or neither could be correct. We as the audience "know" that 200+200 = 400, but who in the story is getting that result? Is it everyone else, or is it him? And is that the correct version within the story, or do we just perceive it that way because we (the audience) are operating on that version of math?Are there any weird meta properties?Like if he does the equation 200 times in a row, takes a nap, then does it another 200 times in a row. How many times did he do the equation? And what answer did he get?Are you ignoring how this would affect all the computations that computers do in the backgrounds?Depending on the particular equation, this could entirely break computers since they do millions of computations per second that both we and they rely on being consistent and repeatable. So depending on how often it occurred we could be talking no internet, no banking system, no computers at all. But given your prompt, I'm assuming you're ignoring those sort of factors for story purposes.What about conservation of mass and energy?If this affects the real world like I mentioned before, this completely breaks conservation of mass and energy, but that's fine. And that will break like all of the rest of physics, but that's fine too.How hard do people disbelieve it?Like if he can accurately replicate the process using real world objects under testable conditions, can he convince people? Or will they still write him off? Like, if he does it under observation in a lab will they contort their beliefs to justify it? "Oh that must have fallen out of your pocket", "That was the spare", "Doesn't look like anything to me", "It's not my wallet"Is he just crazy/really bad at math?This seems like the simplest explanation if it's not able to be replicated in the real world but I'm assuming you're implying it's more real than that.That was definitely more than I set out to write, but I'm curious what you're thinking. And I don't know if my thoughts are at all similar to what you were going for. I feel like there are a

lotof different directions you could take thisThis is not set in stone, but yes, using your example,

`200+200 = 401`

will be true every time. What exasperates him is that everyone else thinks it's correct.Oh yes, it will definitely manifest in the real world. That's part of the fun!

Yes! Great idea by the way. I think I'm gonna use it.

Absolutely.

Sure.

Works the same.

Another good idea. You're good the this, buddy! Yes, that is definitely an option.

Yes, it happens with anyone, but he's the only that sees how absurd it is.

I haven't written it yet, but judging from my previous artistic tendencies towards absurdism, I'm gonna say

no.Everyone is getting

`200 + 200 = 401`

, but the protagonist knows`200 + 200 != 401`

due to his memory of how mathematic is supposed to work. He's kinda good at math.Everyone thinks

`200 + 200 = 401`

, but it is clear that it should be`200 + 200 = 400`

due to the absurdities it produces.He clearly lost count and suddenly there's one extra equation in the paper.

Yeah, I have to ignore that otherwise there is no story. It's a whimsical world, almost everything seems to work regardless of the mathematical aberration.

I never thought of that. How would it break? This is interesting.

They legitimately think He's crazy.

Absolutely not. To them, the world is functioning just as it should be.

100%.

Any lab experiment will somehow work precisely as the wrong math would dictate.

I'd say He is entirely sane and very good at math.

Awesome, thanks! You gave me some very good ideas. Appreciate it!

I'll talk a bit more about breaking physics, since you asked. So in our world we've discovered equations that let us describe how the natural world works. And we've created units to allow for consistent measurements. "The Universe" doesn't understand what 5J is but it does appear to behave in a way that allows us to predict what will happen if you apply 5W for 1s. As far as we can tell everything is consistent with a few exceptions related to quantum weirdness, but that's not really important for this discussion. The important part is that if you put energy into a system, the same amount comes out in various forms.

But if that no longer held you could get more energy out of a system than you put in. Or less. In the case of more you would be continuously adding mass and energy. We would probably burn up. And if it was less, the earth would cool and shrink. At least I think that's what would happen, it's difficult to envision how such alien physics would behave.

Which gets back to the first point. I would say that it's impossible to tell if the universe itself is conscious and has any concept of numbers, math, units or equations. But I can say that our system of numbers and units are a human construct. So when we say 5J that's not meaningful unless you know what a J is and how it relates to the other SI units. So if 200+200=401 holds, then you have to say 200 of

what?. And what counts as adding things together. Like if any time there are 400 electrons together they start spontaneously generating more then that's kind of a problem. And that's without even talking about what a discrete object even is.Does that make any sense?

Edit: So for any universe that was created with those laws in place, it would presumably be "stable" and that would just be how those laws of the universe work. Which means everything wouldn't immediately explode like might happen if this were suddenly applied to our world. So at least there's that

Possibly of interest would be things like the Casimir effect where you have virtual particles spontaneously come into being in pairs that typically shortly after mutually annihilate. You "bend the rules" by borrowing energy but correct things before it matters.

(apologies to physics folks for inaccuracies)

Yeah, that's the sort of thing I was referring to with regard to quantum weirdness. But if the laws of nature within OP's hypothetical universe just allow for the creation of mass and energy under certain circumstances then you have a lot of other weirdness that comes along with it.

Maybe in a world where people can't calculate and let the computer do all their calculations, no one realizes the floating point imprecision:

0.1 + 0.2 = 0.30000000000000004

https://0.30000000000000004.com/

To take some artistic liberties you could have the computers use half precision floating point numbers. The errors will still be small but not quite so small. You’ll also get the exact same error from every time you run the numbers, and on every device. Floating point is a standard and gets implemented in the same way across all computers.

Not super relevant to the thread, but I thought I'd mention it: Isaac Asimov ran with that premise in his 1958 short-stroy The Feeling of Power, where the computers do all the math, but then someone figures out how to calculate with pencil and paper again. No floating point imprecision in that story, though.

Rounding errors may be the best bet? I semi-frequently have situations where adding up numbers in excel, applying percentage increases to them, then summing the whole thing will cause drift in the fraction of a cent range as compared to the financial software. No one cares, because it's fractions of a cent, but not all calculations stay consistent across different software. I'm not sure most people could explain the reason for the error, either, though I did eventually figure out it was a difference in how many decimal places were tracked.

Thanks!

It could be it, yeah! I would of course try to take this to a bizarre degree, since it is fiction. Maybe I could make the rounding error to work every time in a specific way... maybe he eventually realizes that the rounding is actually spelling proper English in code, and there's a message in there (or something disturbingly meaningless).

There is an alternate rounding method where rounding something that ends in 5 rounds to the even number instead of always up.

https://www.theproblemsite.com/ask/2018/02/rounding-05

I have a degree in math - I feel that you might be looking at math a bit differently than how mathematicians look at it. To reduce the entire field to a sentence, math is deductive reasoning based on a set of axioms. So if there is a mismatch between results where both approaches are rigorous and correct, that implies that the system is not consistent and the assumed axioms are actually incorrect choices of axioms. As a result I can’t really recommend something that will fit into your prompt (as an example would imply real world math is inconsistent), other than maybe to imply that the axioms your world thinks it works on are not consistent by arriving at different results from two “correct” approaches at calculation.

I think it's possible to take part in the prompt with a maths degree, it just requires suspension of disbelief. Obviously there aren't real world examples; the exercise is to find an example that is sufficiently compelling to work within the confines of a story in a way that the story is sufficiently internally consistent. I think there's a particular sweet spot to try to find, which is maths, or more likely arithmetic, that is understandable to a layperson, but interesting enough to have some kind of hook for the story. Indeed, it looks like the one that had the biggest hook for @mrbig was pretty simple - 200 + 200 = 401 - and it's more about the exploration of absurdism that entails.

I'm not trying to admonish you about it, but I think it's important to try to take part in things like this with more of a "how could I enable this to work" attitude without stating "this simply doesn't work".

Oh I’m well aware that I was ruining the fun here, it’s simply the one area that I’m comfortable saying that something doesn’t work - for mathematicians to exist in this world and not have realized this then it wouldn’t really be the same field of study.

I actually quite liked your hyperbolic space idea, it seems subtle enough to fit into the framework of fiction without simply no longer being math. It vaguely reminds me of Lovecraft’s The Dreams in the Witch House.

I love this question and what others have been suggesting. Hope to hear more.

I'm reminded of that tale about the discovery and secrecy around irrational numbers. Such a foreign concept which could topple power structures. How crazy the concept these irrational numbers must have seemed. If only they didn't seem to pop up so often even when working with natural numbers.

So I imagine your accountant in an anachronistic setting - a neophyte in the natural-numbers-worshipping past - where all their calculations are off. Nobody wants to admit that there's something weird going on but the system of the day has no way to express the irrational oddity. The software/calculators would be people again I guess.

I'm not sure all your requirements can be met? Not a math wiz, so maybe someone else would give a better answer.

There are things like sneaky hardware defects, or cosmic rays unpredictably flipping a bit, or simple programmer errors like trying to use doubles instead of decimals for currency math.

Not a problem, this is fiction after all :)

I was actually thinking more of what could be the math error, not hardware (or even software, really). I don't intend to explain the actual source of the error. Thanks! :D

To elaborate, there's a lot of "approximate but accurate" math done in computing. One of the notable examples is the fast inverse square root early 3D games relied on.

But there's also a ton of scientific computing/symbolic manipulation software out there that just formalizes the same strict logic you'd be going through on a paper.

Here's a quick example of the rounding error stuff. Depending on the order of calculations or the data type used you get different results.

I'd say just go for it and don't explain it!

House of Leavesdoes this with measuring dimensions of a house (e.g., the interior measure of a room is slightly larger than the exterior) and it was really unsettling. Some Borges / Lovecraft could be good inspiration, too?I think ultrafinitism would pique your interest, though you might have to restructure your story (or maybe the topic would be better left for another time).

I'm not an expert on the subject, but I'd imagine the problem works in reverse, too. If

`exp(exp(exp(7)))`

is impossibly large, then (probably)`1 / exp(exp(exp(7)))`

is impossibly small. Working off this principle, since such a small number doesn't exist, there will always be a "rounding error".A couple of references:

I read a sci-fi book, can't remember which one it was, where an alien species gifted humanity with an odd brain-teaser ... a simple metal ring, perfectly circular, but when you measured it, you discovered that it had a value for Pi of 3.0. In other words, the circumference was exactly 3 times the diameter.

And another book (Sagan, I think?), where if you continue to calculate Pi far enough, way, way down, around the 10 billionth digit, it stops being random, has a long string of digits that map out an image of a circle.

ETA: This is a true thing .... Are you aware of the various Double-Slit Experiments? Nutshell -- light behaves one way when no one is observing it, and a completely different way when someone does observe it. Increasingly weirder experiments on ever-larger particles (atoms, then molecules) see similarly weird variations in how the universe behaves.

I think I've guessed the plot of Independence Day 3: aliens attempt psychological warfare via something like that book until a scrappy Indianian physician gets it set to 3.2 by government fiat.

A related mildly interesting thing about circles on computers is that since pixels are (typically) rectangular, the perimeter is actually 4 times the diameter.

I was going to say that every sequence of numbers exists within π somewhere, but that's not certain, just likely. At some offset of pi you might find the decimal representation of an mp4 that would play a video of your life from start to end.

All of these ideas are great. Another one, which has the advantage of being pretty easy to explain, might be something like: Excel incorrectly assumes that the year 1900 is a leap year.

How about a world where we didn't think to have leap days. Without leap days, summer would start a little earlier, until summer in the northern hemisphere occured in January. Imagine the absurdity of Santa in Shorts. Trying to ride his sleigh without snow. At least the fire places wouldn't be lit. Computers would be no help.

Your absurdity is the Australian Christmas every year.

I'm from a land down under.

I told someone once about Christmas in Summer. She asked confused, "you mean you have Christmas in July????"

For the story I was more thinking the absurdity of people rationalizing the changing seasons, plus you could have season change deniers (Christmas has always been in summer.)