If Scadrial uses base-16, how were percents a thing?

[Eri]

OK, so one of the main plot points is that 16% of people affected by the mists Snap and that shows that the whole event is intentional. 

Except that a percent is 1/100 and the use of percents is inherently tied to the fact that we use base 10. 16% is just 16/100. And we know from other sources that Scadrians use a hexadecimal number system. In this base, 0.16 (16% or 16/100) would be 0.28F5C28F6… or roughly 0.29, or 29% (by "29" I mean 2*16+9, ie 41.).

I know, Mistborn is an old book, and written before Brandon had tons of consultants, and it also depicts red leaves photosynthesizing using red sunlight, but maybe let's try to find some explanation anyway.

 

Like, maybe they used hexadecimal for natural numbers but base-10 for the fractional part, because they learned decimal fractions from another culture… this mix would make fractions interesting. By "interesting" I mean "probably weirdly complicated and inconvenient", so I guess modern Scadrial uses 16 for everything, or maye 10 for everything (Harmony wanted to make it worldhopper-friendly) and the base-16 is like Roman numerals for us - old, official, nice for dates and such but not used for actual maths. 

However, if they still use base-16, and have money denominations like 4 and 8 of something… I'd really like to see a worldhopper trying to get accustomed to this.

Or maybe the percents are not our percents, but pre-Catacendre Scadrians used "percent" to mean 1/128 and used those a lot. This could work too, I guess.

[Honorless]

Brandon sees himself as writing in translation, so Rosharan dozen is actually tenset, so Scadrian percent is probably... wait, I was thinking of Scadrial using hexadecimal but do they even have zero? So do they count till 15 or 16?

[Eri]

Yes, they have a zero and a slightly more complicated system than our, but it's a positional system anyway. See this thread.

Translation or not, if a "percent" is 1/256, then 16% is just 1/16, which is a simpler way to say it. also, the total number of atium mistings would be too low, I don't remeber but I saw a thread about "why 16%, not 1/16?".

You could claim it's translation convention and "percent" is 1/128. I believe the "hexadecimal before comma, decimal after comma" is a more elegant explanation (the resulting system is so unnecessarily inefficient that I'm 100% willing to suspend my disbelief that a culture would use it for a long time. People do things like that a lot. Like Roman numerals. Or our music notation.), but YMMV. Still, 1/128 is much less natural choice for percent for base-16 than 1/256.

Edited December 26, 2020 by Eri

[+Oltux72]

22 minutes ago, Eri said:

Yes, they have a zero and a slightly more complicated system than our, but it's a positional system anyway. See this thread.

Translation or not, if a "percent" is 1/256, then 16% is just 1/16, which is a simpler way to say it. also, the total number of atium mistings would be too low, I don't remeber but I saw a thread about "why 16%, not 1/16?".

Leras was not Scadrian. He was Yolish. For all we know Yolen was using base 10.

22 minutes ago, Eri said:

You could claim it's translation convention and "percent" is 1/128. I believe the "hexadecimal before comma, decimal after comma" is a more elegant explanation (the resulting system is so unnecessarily inefficient that I'm 100% willing to suspend my disbelief that a culture would use it for a long time. People do things like that a lot. Like Roman numerals. Or our music notation.), but YMMV. Still, 1/128 is much less natural choice for percent for base-16 than 1/256.

Or they noticed a constant fraction 4/25 and given that one in sixteen was specially afflicted they came to 16 / 100 and were just using fractions rather than a positional system for numbers between 0 and 1.

[Eri]

25 minutes ago, Oltux72 said:

Leras was not Scadrian. He was Yolish. 

Yes. But the people who noticed the 16% thing (I don't remember, who that was exactly. Elend? Demoux?) were Scadrian.

 

26 minutes ago, Oltux72 said:

Or they noticed a constant fraction 4/25 and given that one in sixteen was specially afflicted they came to 16 / 100 and were just using fractions rather than a positional system for numbers between 0 and 1.

It was 16%, not one in 16.That's the whole problem here. 

[+Oltux72]

2 hours ago, Eri said:

Yes. But the people who noticed the 16% thing (I don't remember, who that was exactly. Elend? Demoux?) were Scadrian.

Vin IIRC. But the 16% are not important for that. Any percentage will do. The oddity is that the number of afflicted people was always (within the limits of group size) 16 / 100. And that is not how natural afflictions behave. The numbers should have been on a bell curve centered on 0.16

2 hours ago, Eri said:

It was 16%, not one in 16.That's the whole problem here. 

No. That is half the problem. You had two observable phenomena

1. 16% of the whole group became sick. Expressed as a fraction: 4 / 25
2. Of those 1 / 16 become sick for longer (the future Atium-Mistings) - That turns into 1 / 100 of the whole group

If you equalize the bases you arrive at 16 / 100 for the first group and 1 / 100 for the second group. Hence the number communicated was 16 (or theoretically any multiple thereof).
The solution can be derived independent of the number system. The only knowledge you need is the normal distribution (Gaussian curve a.k.a. bell curve) and to know that numbers between 0 and 1 can be expressed as fractions. Late 18th century math by our standards.

The psychological and cultural problem why Leras chose 16 / 100, as opposed to 16 / 256 of course remains. But that depends on his number system, not the Scadrian system.

[Eri]

1 hour ago, Oltux72 said:

Vin IIRC. But the 16% are not important for that.

I'm sorry, but you remember wrong. Have you read the Coppermind link I put in my post? (I could quote, but I have it in paper, so Coppermind is easier to quote)

Quote

There is something special about the number sixteen. For one thing, it was Preservation's sign to mankind.

Yes, the behavior was important, but the exact number was important too.

1 hour ago, Oltux72 said:

No. That is half the problem. You had two observable phenomena

1. 16% of the whole group became sick. Expressed as a fraction: 4 / 25
2. Of those 1 / 16 become sick for longer (the future Atium-Mistings) - That turns into 1 / 100 of the whole group

No. Both were 16%. (There were proportionally more atium Mistings than other Mistings in that group.) 

Chapter 36 summary:

Quote

Demoux asks to be relieved of his position as General as he feels that Kelsier has judged him and found him unworthy. Demoux mentions another statistical oddity that Elend was not aware of, that sixteen percent of the sick either remained sick for sixteen days or perished. Demoux mentions other ways that the number sixteen seems to be relevant, stating there is a pattern that Kelsier is behind.

Those 16% that were sick for longer later turn out to be atium Mistings (ch 82):

Quote

Demoux arrives and gives a status report on their forces. He says that all of his soldiers took metals but none of them showed allomantic abilities. Elend has Demoux eat an atium bead and Demoux is able to burn it, and Elend deduces that his group are likely all Seers.

 

1 hour ago, Oltux72 said:

The psychological and cultural problem why Leras chose 16 / 100, as opposed to 16 / 256 of course remains. But that depends on his number system, not the Scadrian system.

The whole point of communication is for it to be understandable to the person you're communicating with. But OK, Leras was insane at this point, so I agree, him using the % may be explained. But I still claim that Elend and Demoux understanding the sign is something that could use an explanation.

I'm not attacking the book, I love Cosmere and Mistborn too. But it does have some details off. Like the leaves are red and the sunlight is red. Like Inquisitor!Marsh reads a text written on metal (later in SH it's said he did it by feeling it with his fingers, but in HoA it doesn't look like this, he even uses tin because it's dark). And I think it could be fun to discuss how this one can be retconned too.

[+Oltux72]

48 minutes ago, Eri said:

I'm sorry, but you remember wrong. Have you read the Coppermind link I put in my post? (I could quote, but I have it in paper, so Coppermind is easier to quote)

The Hero of Ages, chapter 36:

Quote

"What numbers?", Elend asked.
"Six thousand people got taken by the sickness, my lord", Demoux said.
Exactly sixteen percent of the army, Elend thought.

"Of those over five hundred died," Demoux said. "Of those remaining, almost everyone recovered in a day."
"But some didn't," Elend said. "Like you."
"Like me," Demoux said softly. "Three hundred and sixty-three of us remained sick when the others got better."
"So?" Elend asked.

"That's exactly one-sixteenth of those who fell to the sickness, my lord," Demoux said. "Thirty-six died, but the other three hundred and twenty-seven of us stayed sick exactly sixteen days. To the hour."

1/16 not 16/100

 

[Eri]

In this case you're right and Coppermind needs a correction. Thanks for catching it!

Back to the main point, they are talking numbers here. So, since you convinced me to pull the book from the shelf, chapter 21 (p 120 paperback), paraphrased: nearly 5 243 + about 550 (who died) got sick, of total about 38 000.

That's not exactly 16% (we don't get the exact number of not-immune population here), but it means that the "16%" they mention later can't be 16/128 or something like that. It has to be 16/100. And they notice this as significant before the 1/16 thing, and later use this to notice the prevalence of sixteend.

My point is that, without changing the text, the hypothesis of translation convention meaning they don't literally use percents, cannot be defended.

Also, p 191:

Quote

"This third one here isn't exact," Noorden said, "but that's only because the base number isn't a multiple of twenty-five."

Yes, they do use 100-based percents.

I'm still a fan of "hexadecimal numbers, decimal fractions" idea, because cultures do weird things like that, but maybe the simpler explanation is that even during Era 1, the hexadecimal system is an ancient one (like Roman numerals to us), and decimal is now commonly used (probably a mix of worldhopper influence and there being 10 publicly known metals).

Also, I now have a headcanon that Era 2 Southern Scadrians use base-17, because a) there are 17 digits after all, and b ) it would frustrate Hoid.

[+Oltux72]

1 hour ago, Eri said:

That's not exactly 16% (we don't get the exact number of not-immune population here), but it means that the "16%" they mention later can't be 16/128 or something like that. It has to be 16/100.

Yes.

1 hour ago, Eri said:

And they notice this as significant before the 1/16 thing,

It is, but it would be for any constant number.

1 hour ago, Eri said:

and later use this to notice the prevalence of sixteend

It is possible that they don't go for common denominators.

1 hour ago, Eri said:

.My point is that, without changing the text, the hypothesis of translation convention meaning they don't literally use percents, cannot be defended.

Right. Though you may argue that four Scdrians 1 / 100 is nicer than 1 / 25, as it at least contains a factor of 4.

 

1 hour ago, Eri said:

Also, I now have a headcanon that Era 2 Southern Scadrians use base-17, because a) there are 17 digits after all, and b ) it would frustrate Hoid.

Apparently there is a symbol for the next power. Hence they do not use

35

but

3 * 10 + 5 in writing.Or maybe they do this only for numbers between 16 and 31. That is they directly write the language, which needs to have a simple word for sixteen.

[Honorless]

On 12/26/2020 at 5:13 PM, Eri said:

Yes, they have a zero and a slightly more complicated system than our, but it's a positional system anyway. See this thread.

Translation or not, if a "percent" is 1/256, then 16% is just 1/16, which is a simpler way to say it. also, the total number of atium mistings would be too low, I don't remeber but I saw a thread about "why 16%, not 1/16?".

You could claim it's translation convention and "percent" is 1/128. I believe the "hexadecimal before comma, decimal after comma" is a more elegant explanation (the resulting system is so unnecessarily inefficient that I'm 100% willing to suspend my disbelief that a culture would use it for a long time. People do things like that a lot. Like Roman numerals. Or our music notation.), but YMMV. Still, 1/128 is much less natural choice for percent for base-16 than 1/256.

Oo, thanks! I keep forgetting because of Roshar that we haven't seen that much of the writing system of Scadrial.

Heh, you'll like this:

 

[Chinkoln]

You cannot take anything for granted in Brandon’s books. There is always a chance for differences and change. On Roshar there are 500 days in a year, but each day is only 20 hours. Roshar is also smaller than Earth, which means there is less gravity. That could explain confusion, but there is also a much simpler solution. They live in the Cosmere. The Cosmere has different governing laws than our universe. Different things are possible and some things are beyond our understanding. We can explain some stuff, but at a certain point we have to step back and say “New universe, new laws, new impossibilities”

[AquaRegia]

May I just say I LOVE this topic.  I had exactly the same reaction: "wait, 16% ≠ 1/16 ... how BOTH?  There is something vaguely unsatisfying about this math."  @Eri and @Oltux72 put WAY more thought into it than I did, and I thank you both.

[+Oltux72]

Looking at this a second time, it seems to me that we can conclude that Leras used a number system based on ten in his native language.