Ask me about math

[Zero Angel]

I realize these questions probably don't come up as often as other types, but if you ever have questions in the math world, then this is the only field that I am actually certified in

(All the rest of the help I give is from my own personal research)

 

[Steerpike]

If I have three Staffs of the Arch-Magus, and Susie asks me for one, how many do I have left?

 

[Feo Takahari]

Not a story question, but one I've been wondering about:

My history teacher showed us eight terms. He said that six of those terms would be randomly chosen for the midterm, and we could choose to define and describe any four of those six. How many terms would I need to study in order to be certain I'd covered four of the six chosen? (I'm pretty sure the answer is six--if I don't study two terms, and those two terms are among the six, I'll still have studied the four other terms chosen--but I'd like to know if there's a mathematical equation that can represent the problem.)

 

[Zero Angel]

If I have three Staffs of the Arch-Magus, and Susie asks me for one, how many do I have left?

Three, since she only asked you for one and you haven't given it to her yet

Not a story question, but one I've been wondering about:

My history teacher showed us eight terms. He said that six of those terms would be randomly chosen for the midterm, and we could choose to define and describe any four of those six. How many terms would I need to study in order to be certain I'd covered four of the six chosen? (I'm pretty sure the answer is six--if I don't study two terms, and those two terms are among the six, I'll still have studied the four other terms chosen--but I'd like to know if there's a mathematical equation that can represent the problem.)

So, you definitely want to learn all eight terms (just because education and learning are important), but to the math problem you brought up:

You are correct that 6 is the minimum number to study to get 4/6 correct. In this case, since you want to guarantee 4 correct (and not just have a reasonable chance at 4 correct) it's just asking how many incorrect options are there. There are two incorrect options (the numbers not chosen), so if choose 6 then at most you can only choose 2 incorrect options, yielding 4 correct options.

The strategy here is if you want to guarantee worst-case scenario, then you picked all the wrong ones (two), so then just add how many you want to get correct (four) and you have how many you want to study (six).

Now, if you want to talk about probability, it becomes significantly more difficult. There are 28 possible combinations of questions your teacher can choose, denoted 8C6 (that's usually written with the numbers as subscripts), which is equal to:
__8!__
(8-6)!6!

(If you don't know, the ! notation tells us to start at that number and multiply all the way down to 1. In this case, the problem quickly simplifies to 8 * 7 / 2 = 4 * 7 = 28.

Now if you choose six out of the 8, then you have a 1/28 chance of choosing the same 6 terms as the teacher. You have a 100% chance of getting at least 4 though, so you're guaranteed approximately 66% of the points if they are equally weighted.

Does this answer your question?

 

[Androxine Vortex]

Here's how I look at probability:
Everything is 50/50, it either happens or it doesn't. But alas, if only my high school teachers agreed with my philosophy...

 

[Feo Takahari]

I thought combinations might be involved, but I wasn't sure. Thanks!

 

[Zero Angel]

Here's how I look at probability:
Everything is 50/50, it either happens or it doesn't. But alas, if only my high school teachers agreed with my philosophy...

So 50/50 doesn't mean "happens or doesn't", but rather, "happens as often as it doesn't".

 

[WyrdMystic]

So 50/50 doesn't mean "happens or doesn't", but rather, "happens as often as it doesn't".

Assuming it is done more than once.

Is the term 'array' anachronistic in a fantasy setting?

 

[Steerpike]

Assuming it is done more than once.

Is the term 'array' anachronistic in a fantasy setting?

I'm pretty sure that's a fairly old word, but in any event a fantasy setting is a made-up world. You establish the rules of the world, its history, languages, and so on, so the word isn't anachronistic unless you decide it is.

 

[Zero Angel]

Assuming it is done more than once.

Is the term 'array' anachronistic in a fantasy setting?

I'm assuming you mean in the sense of a specific ordering of numbers or vectors like a matrix? Array as a word that just means "ordering" is as old as regular English is (not medieval English obviously, although they probably had their own version of it...). A quick Googling says it was developed in the 14th century.

In fact, you can use it as a verb.

As Steerpike says though, if it's fantasy, you make up the rules.

It is worth mentioning, I think, that if you were to write a "medieval" fantasy, you should bear in mind that they didn't speak our version of English. In fact, depending on how far back you went, you'd probably have a pretty hard time understanding anything.

 

[Devor]

Do you know what kind of math would have existed in medieval east asia, and are there any terms or ideas you know of that would add a medieval flavor to it? I can't use most terms from a modern textbook, after all.

 

[Zero Angel]

Do you know what kind of math would have existed in medieval east asia, and are there any terms or ideas you know of that would add a medieval flavor to it? I can't use most terms from a modern textbook, after all.

Hi Devor,

I'll double check my "History of Mathematics" textbook from my undergraduate days when I return home to see if there is anything overly interesting, but I can describe some things now.

First, one common misconception is the use of numbers in medieval times. Most places did not use a positional notation like we do (where a 1 has a different meaning if it occurs in front of a zero for instance; that is, 10 means something different than 1 and 0). China was THE exception to this. They used a system that is analogous to so-called "expanded form" of a decimal centuries before even the Hindu-Arabic numerals which we didn't then co-opt for the Western world until the Renaissance.

Expanded form is where you say the number, then the place value, then the number, then its place value, etc.

E.g. 542 would be 5 hundreds 4 tens and 2 ones.

The Chinese were great mathematicians. They had complicated algorithms and calculations that the Greeks couldn't hope to match, and when the rest of the world went "dark", they continued developing their mathematics throughout the middle ages. They had algebra, trigonometry and geometry. They had calculated pi to a reasonable degree (355/113, which is better than even Archimedes had gotten) and were good at other ratios as well.

What terms from a modern textbook would you be interested in using? They had ideas of multiplication, addition, division and the rest. They didn't have the same notation as we did, but they thought of numbers in the same basic way we do (which is more than can be said for most medieval civilizations).

 

[Steerpike]

The Maya had the zero and a positional system as well. Another society adept at astronomy. Good info, Zero Angel.

 

[Chilari]

This is fascinating. I never knew maths had such an interesting history.

Zero Angel, what do you know about the Greek numbering system? It's one of the things I touched on when looking at the graffiti of the Athenian agora for my undergrad dissertation, because some of the graffiti was marks of volume, weight or value of the contents of a vessel, but I didn't really deal with those much and didn't delve into the numercial notation side of things. I understand it had similarities to the Roman numeral system in that letters represented numbers but was arranged differently, often with one letter inside another or something? What can you tell me?

 

[Zero Angel]

The Maya had the zero and a positional system as well. Another society adept at astronomy. Good info, Zero Angel.

You're absolutely correct, Steerpike. I had forgotten about the Maya. In fact, they are credited with the earliest known use of a symbol for zero. It was a stylized shell. Then 1 through 4 were five dots in a horizontal pattern and horizontal lines would represent groups of 5. So something like:

"shell"
.
. .
...
....
____ five
__.__ six

all the way up to

. . . .
_____
_____
_____
for 19.

Summarizing from my math history book: They also, interestingly enough, used a vigesimal system (base 20) instead of a decimal system, but altered it to fit the calendar. So whereas normally 1 15 3 10 would mean 1 * 20^3 + 15*20^2 + 3*20 + 10*1 in a normal vigesimal system, they switched 20^2 to 20*18 to get 360 for 360 days of their normal calendar (18 months with 20 days each + 5 days extra). Then every amount after that is the previous place value times 20.

 

[Zero Angel]

This is fascinating. I never knew maths had such an interesting history.

Zero Angel, what do you know about the Greek numbering system? It's one of the things I touched on when looking at the graffiti of the Athenian agora for my undergrad dissertation, because some of the graffiti was marks of volume, weight or value of the contents of a vessel, but I didn't really deal with those much and didn't delve into the numercial notation side of things. I understand it had similarities to the Roman numeral system in that letters represented numbers but was arranged differently, often with one letter inside another or something? What can you tell me?

I'll answer your question if you can tell me whether the "s" at the end of "maths" is silent or if you say it? In America we only say "math" and I was surprised to find out you add an s at the end. (Or I guess, we subtract the s since British English came first).

Anyway, it depends on the time period the graffiti was done at. The original number system developed by the Greeks was called the Attic numeral system or Herodianic. From your description, I would think this is what you are referring to.

It is an additive numeral system (like what came after and what the Romans used). It used 1, 5, and powers of 10 so no more than 4 symbols would appear. For symbols meaning 5 times a power of ten, they would nestle the shape inside the symbol for 5.

Basically,

1 was represented by a tally mark.
5 was represented by capital gamma, the first letter of "penta"
10 was represented by capital delta, the first letter of "deka"
100 was represented by H, capital eta, the first letter of "hekaton"
1000 was represented by X, capital chi, the first letter of "kilo"
10000 was represented by M, capital mu, the first letter of "myriad".

So to write 14,027, you would write M X X X X delta delta gamma tally tally
but to write fifty, you write gamma with a delta hanging inside
to write 500 you write gamma with an H hanging inside
to write 5000 you write gamma with a X hanging inside,
to write 50000 you write gamma with a M hanging inside

In fact, the Roman numeral system is based on this I believe.

For the Ionian numeral system, they had 27 different symbols to represent 1-9, 10-90 and 100-900. The 24 Greek letters and the Phoenician letters digamma for 6, koppa for 90 and sampi for 900.

They then used an accent left and below a letter to tell you to multiply by 1000. So for instance, ",beta" would be 2000 while "beta" would just be 2. You could put an M (for myriad) after or below a letter to tell you to multiply by 10000. So "delta M" would be 4 * 10000 or 40,000. And writing M more than once would tell you to multiply by powers of M. So MM would be 10^8. I am unclear whether they would have had MMM be 10^12, because Archimedes ended up developing a method of counting by MM that he used to estimate the number of grains of sand it would take to fill the known universe.

Finally, they wrote an accent after the end of a number to tell people that it was a number and not a word (although they could also accomplish this by placing a bar over the letters).

Otherwise, the letters were added like normal in additive numeral systems.

 

[Steerpike]

You're absolutely correct, Steerpike. I had forgotten about the Maya. In fact, they are credited with the earliest known use of a symbol for zero. It was a stylized shell. Then 1 through 4 were five dots in a horizontal pattern and horizontal lines would represent groups of 5.

Yes, exactly. And then they would stack the stacks of numbers on top of each other to make larger numbers.

I have a whole book on reading Maya glyphs, and it includes some good info on the number system. Great stuff.

 

[Zero Angel]

Do you know what kind of math would have existed in medieval east asia, and are there any terms or ideas you know of that would add a medieval flavor to it? I can't use most terms from a modern textbook, after all.

Hi Devor, I checked my math history book again.

They used bamboo or counting-rod numerals that is positional with blanks for zero. They used tallies for 1 through 5, then a horizonal line above a tally for 6 (looks like a T), then the horizontal line above four tallies for 9. They would use these numerals for units, hundreds, ten-thousands, millions, hundred-millions, etc.

For tens, thousands, hundred-thousands, ten-millions etc, they would use horizontal tallies for 1 through 5, then their six would look like an upside down T, then seven is an upside down T with another horizontal tally below, then 8 is an upside down T with two horizontal tallies below, etc.

Anyway, they used mathematics for land measurement, surveying, taxation, making of canals and dikes, granary dimensions and more. Apparently one misconception is that the government office known as "Office of Mathematics" primary function was to promote mathematics throughout the empire. In fact, this office was mainly minor officials trained in preparing the calendar. Through Chinese history (according to my math history book), the promulgation of the calendar was a right of the emperor, and the calendars were necessary for the beginning and end of the monsoon season, melting of snows, rising of rivers etc.

Because this was a guarded right of the emperor, it was written in 1610 that "it is forbidden under pain of death to study mathematics, without the Emperor's authorization."

My math history book, The History of Mathematics: An Introduction by David M Burton gives conflicting reports though. They say that Chinese mathematics was mostly practical and not abstract or theoretical. Although it appears that everything in China concerning math was used and applied, they did develop advanced theories and theoretical concepts of how to do this. They had Pythagorean's Theorem possibly before Pythagoras did. Unfortunately most Chinese texts were destroyed in 213 BC, but the earliest Chinese work known is the Arithmetic Classic of the Gnomon and the Circular Paths of Heaven which is believed to have been written around 300 BC with some of the knowledge believed to date as far back as 1000 BC.

The Nine Chapters on the Mathematical Art survives in a commentary composed in AD 263 by Liu Hui, but this was another math book created before the book burning. The Nine Chapters was the standard syllabus for students preparing for civil service examinations and was one of the earliest printed textbooks in 1084. It's basically 246 problems and their solutions and is considered to be an organization of the mathematical knowledge accumulated by the Chinese up to the middle of the third century. Since you're looking for language, here are some quotes:

There is a circular field, circumference 181 bu and diameter 60 1/3 bu. Find [the area of] the field​

There is a woman weaver who increases an [equal] amount each day. She weaves 5 chi on the first day and in a month 9 pi 3 zang. Find her increase each day.​

A number [of persons] are buying goods. If a person pays 8 there is a surplus of 3, if a person pays 7 there is a deficit of 4. Find the number of persons and the cost of the goods.​

There is a bamboo of 10 ch'ih high. It is broken and the upper end touches the ground 3 ch'ih away from the root. Find the height of the break.​

There is a loan of 1000 qian with a monthly interest of 30 qian. Now there is a loan of 750 qian which is returned in 9 days. Find the interest.​

For the first problem, they used pi to be 3 and the formula for area to be 3/4 d^2. But they quickly found the most accurate result of pi until the Renaissance.

The second problem demonstrates finding the sum of an arithmetic progression, the third requires the solution of a system of linear equations, the fourth is an application of the Pythagorean theorem and the last is one requiring the Rule of Three.

What strikes me the most about all of these is not the breadth of topics the Chinese knew and understood, but how boring math problems have been for the last thousands of years! =P They don't seem to have changed much.

Later they used similar triangles for distances.

Then the 13th century Chinese mathematics is regarded as the high point in the development of traditional Chinese math. In 1247 Mathematical Treatise in Nine Sections was published by Ch'in Chu-shao which had negative numbers and began the custom of printing negatives in black and positives in red (not sure if that is a mistake in my history book or not or if it has been switched over the ages). Nine Sections is also the oldest extant Chinese text to contain a round symbol for zero and is also the first in which they used numerical equations of degree higher than 3.

Although then for some unknown reason, Chinese math declined and the accomplishments of the previous centuries were almost completely forgotten from the 14th century onwards until western math was introduced into China at the end of the 16th century.