From Math Images

Contents 1 Response to Checklist 1.1 References and footnotes 1.2 Good writing 1.2.1 Context 1.2.2 Quality of prose and page structuring 1.2.3 Integration of images and text 1.2.4 Examples, Calculations, Applications, Proofs & Mathematical Accuracy and precision of language 1.2.5 Layout 2 General Comments 2.1 Comments from others 3 Section-specific Comments 3.1 Basic Description 3.1.1 Opening section 3.1.2 Construction & Terminology 3.2 A More Mathematical Explanation 3.2.1 Base-10 System 3.2.2 Charts section 3.2.3 Sample Quipu 3.3 Why It's Interesting 3.3.1 History 3.3.2 Uses

Response to Checklist

original responses to checklist done by Kate 15:52, 24 June 2011 (UTC) Overall, this page is very good. Most of the changes I've suggested shouldn't be super complicated. AnnaP

References and footnotes

• All images cited on click-through
• Direct quotes and specific references cited with footnotes

Good writing

Context

• As an article on anthropological mathematics, the page is connected to other subject areas in its very concept
• Introduction and Why It's Interesting section make connections to the history of quipu and the Incas
• Why It's Interesting and Incan Number System explain why it's important to math and make connections to other areas of math (Bases helper page)

Quality of prose and page structuring

• The beginning paragraphs give a basic version of the general ideas of the page, which are elaborated upon in later sections
• Section headings and first sentences explain the purpose and relevance of each section
• Information on Bases was broken off and made into a helper page
• MME builds up towards more complicated example

Can you move the bit about the hierarchical structure to after your example quipu? The reader doesn't need this bit to understand your example, and it feels awkward in the middle of two sections that talk about making tables from the quipo.

Kate 13:54, 28 June 2011 (UTC): moved it around.

Integration of images and text

• Images are referred to explicitly
• Captions as well as main text are used to explain what to look for in image

Examples, Calculations, Applications, Proofs & Mathematical Accuracy and precision of language

• These items are largely irrelevant to this page, considering how little technical mathematical content there is. As I said, it's an anthropological mathematics page.

In your example using Image 3, can you walk the reader through why those different numbers are represented.

Try to be a bit more clear as to whether or not the type of not used impacts what number is represented by the knot. It's a bit unclear to me how you can say "Long knots are used for units values of 2-9" and still say that the number 3 can be represented by 3 single knots.

Kate 14:26, 28 June 2011 (UTC): I didn't say that the number three can be represented by 3 single knots - I said that in all places other than the ones place, the digit three is represented by 3 single knots. I think the paragraph I have is pretty clear (I do say that different knots are used in the units place than in other places), but I'll re-emphasize it in the example in case people weren't reading carefully.

It's much more clear now that you've emphasized it again later on. While I sort of got what you meant the first time, it didn't really "click," if that makes sense. It does work much better now

"In some cases, more complicated quipus take the form p_{ijk}, where there are i related charts, each with j columns and k rows. " is unclear to someone who isn't very comfortable with indices. I'd suggest either providing an example or cutting the sentence.

Kate 14:26, 28 June 2011 (UTC): I don't want to make more pictures for this sentence, because it's not that important, but I'll move it after the sample quipu and add a little more explanation.

Definitely better now

In your example quipo, you say "all entries are multiples of twelve." That's confusing since the number 25 is listed which is certainly not a multiple of 12.

Kate 14:26, 28 June 2011 (UTC): Actually, what I say is "when accounting for the fact that only integer values can be represented, all entries are multiples of twelve" - when you're doing integer-only division, dividing something by groups of twelve and ending up with thirteens or twenty-fives is very common. I'll try and be more explicit about that.

It's much more clear now

Layout

• Effort has been made to keep paragraphs as short as possible while still conveying meaningful amounts of information

It would be good if in a couple places (for example in the "Reading numbers on a Quipu" section) you could add extra breaks between the paragraphs. While the paragraphs aren't too long, they still appear to be a wall of text.

Kate 13:56, 28 June 2011 (UTC): added extra space

• Relevant images present throughout
• MME hidden because although it's not intimidatingly math-y, it's intimidatingly long
• White space and image wrapping are as clean as possible
• Page has been viewed in browser windows of varying sizes

General Comments

Comments from others

• Indicate that very little math is needed for the MME. Abram, 5/17

Done, is the way I did it ok? Kate, 5/18

Also, are you going to keep this?

"Note: This topic's "More Mathematical" explanation actually involves very little mathematics."

-Richard 5/20

I don't know - Abram and I agreed that something should be done to indicate that the More Mathematical explanation wasn't scary, but I didn't really know what the best way to do it would be. I'm not 100% satisfied with that note, but at the moment I don't have any better ideas. -Kate 22:05, 22 May 2011 (UTC)

• It's great how you explain something, and then give an example. I love how you say "The value of a group of symbols depends solely on the symbols' positions. For example, the symbol 6 has a different value when it appears in 625 than in 2,036; but it has the same value in 625 and 699." You do this throughout the page, and it's very clear!
• Your writing is so clear! And great use of pictures as well. It's a great page. Again, I'm so impressed!

Rebecca 02:54, 25 May 2011 (UTC)

Thanks, Becky! :) -Kate 16:07, 25 May 2011 (UTC)

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Section-specific Comments

Basic Description

Opening section

Abram, 6/7:

Referring to an earlier conversation, I still can see an argument for basically not having a More Mathematical Explanation, and putting just about everything in the basic description. I can also see doing it your way, but in this opening section can you at least put *something* about how the quipu system is like ours, and much easier to do arithmetic with than roman numerals (or something a bit meatier than just it being a method for working with numbers)?

Kate 20:26, 7 June 2011 (UTC): Still unsure about whether or not I wanna use the MME. If I do, that's a good suggestion. I'm going to think about it for a little while before I change anything, though.

Construction & Terminology

Htasoff 15:01, 31 May 2011 (UTC)

• Be consistent with your pluralization of quipu.
• "Pendant cords are down hanging down; thus their end is at the bottom. Top cords go up"
  • Awkward. Try instead:""Pendant cords hang down ..."

Kate 16:08, 31 May 2011 (UTC):

• I've fixed the pluralization everywhere I've seen it, but I'm not sure it's all gone.
• I reworded that sentence in my own words, I think it's clear now.

• xd 18:51, 12 June 2011 (UTC) In the paragraph, you mentioned "subsidiary cords" but in the picture, it is not there. Maybe you want to put in the paragraph that the subsidiary cords are not shown in the picture in case the readers look for it?

Kate 13:27, 13 June 2011 (UTC): good point, done.

A More Mathematical Explanation

Base-10 System

Comments from Harrison, 5/31:

• "This is essentially the same as the number system we use, which is actually quite remarkable."
  • Odd. It looks like you're saying that our number system is remarkable. Try splitting them into two sentences, or clarifying that you are talking about their use of base ten.

Kate 16:08, 31 May 2011 (UTC):

• I don't think it looks like that, actually. I'm going to leave it as is for now.

Abram, 6/7

I tend to agree with Harrison. The sentence structure at least makes it ambiguous which you mean, IMO.

Kate 20:26, 7 June 2011 (UTC): Rephrased it, is it clearer now?

• "true number-positional system", what do you mean by "true"?
• "For example, the symbol 6 has a different value when it appears in 625 than in 2,036; but it has the same value in 625 and 699"
  • say how: "in 625, 6 denotes the value six hundreds, while in 2,036 it denotes six ones. In both 625 and 699, 6 represents six hundreds."

Kate 16:08, 31 May 2011 (UTC):

• By "true", I mean the standard definition of the word - "real", "actual", "genuine" - I think this is clear and will leave it as is.

*I'll think about getting more specific in that example with the 6s. Do you really think the explanation needs to get that basic? It's in the hidden section...

Abram, 6/7:

*I'm also a bit confused by the use of "true": what meaning would be lost if you dropped this word?

*Regarding the 6's, how about keeping the first of Harrison's sentences above but not bothering with the second?

Kate 20:26, 7 June 2011 (UTC):

• I continue to think that "true" is perfectly clear - I mean to emphasize that it's really a positional system, and not just almost one (like how the linguistic evidence for bases can be construed as being almost a positional system). But if everyone else thinks it's confusing, I can get rid of it.
• Changed the thing with the 6s.

• "Contrast this with Roman Numerals ..."
  • to clarify the later sentences, say: "Contrast this with non-positional numerical systems, such as Roman Numerals ..."

Kate 16:08, 31 May 2011 (UTC): Rephrased in my own words.

Abram, 6/7:

I'm not sure that you are saying here exactly what you mean to. The value of a symbol does not seem to depend on it's position: "V" means 5 in both VI and VL. Instead, it's the operation connecting the symbols that depends on the particular sequence of symbols: in VI, "V" contributes towards the total value, while in "VL" it takes away from it. In a way, in positional-systems, the operation is fixed, while the value is position-dependent; in roman numerals, the value is fixed while the operations are sequence-symbol dependent.

Kate 20:26, 7 June 2011 (UTC): Changed "value" to "meaning", which I think makes it true.

• "People often say that our number system is base-ten because it evolved from counting on our ten fingers, but this doesn't mean that ten is the most "natural" base."
  • Try:"People speculate that our number system is base-ten because it evolved from counting on our ten fingers. Nevertheless, this doesn't mean that ten is the most natural base, as is often claimed."

Kate 16:08, 31 May 2011 (UTC): I think the way I have it is fine.

Abram, 6/7:

I don't have any problem with Kate's wording. However, I know that I always bought into the idea that 10 was the most natural base. Can you add a sentence or two elaborating on this point? Even an example of why some other base could be seen to be just as natural would suffice.

Kate 20:26, 7 June 2011 (UTC): Changed these two paragraphs around, is it clearer now?

• "So, it's actually quite surprising that the Incas had an advanced number system, and even more so that it should be so nearly analogous to our own! "
  • Try not to start sentences with so if you can avoid it.
  • The exclamation point is unnecessary here, and it is best to avoid exclamation points in even semi-formal writing anyway. There are a number of reasons I believe this, however I won't bore you by listing them.

Kate 16:08, 31 May 2011 (UTC): You and I disagree about how formal the writing on this wiki should be. I like the casual "So" and the exclamation point, and will be leaving them for now.

Abram, 6/7:

I feel like this is not really a question about this page, so much as it is a question about writing standards for the site as a whole. If we decide that we don't want this kind of informal writing on the site in general, then Kate should follow the protocol. If we decide that it's fine to have this style on the site, then Harrison should just let it be. Can we bring this up in a group meeting?

Abram, 6/7:

The text in this section is very clearly written, well-developed, and coherent. Here are a couple of organizational ideas and questions:

• Maybe rename the subsection to "Base-10 positional system" or something, since it's not just about base 10
• Your in-depth explanation contrasting our positional system to a particular non-positional system like Roman Numerals was really great. Can you do a parallel explanation contrasting our base-10 system to a particular base-other positional system (like binary code)? Maybe this isn't worth doing in this page if you don't feel like there's a sufficiently familiar example of another positional system, but in that case, maybe you could link to a base helper page and put that discussion there?
• While everything is clearly written, it's a lot of paragraphs in a row, and it ended up feeling daunting to me even though it was easy. Can you create sub-sub sections, like "Positional system" and "base-10"
• Your point about spoken language revolving around bases even when number notation systems aren't is really interesting, but I'm not sure exactly how this relates in your mind to the other material in this section. For example, to me, this point about language suggests that positional systems shouldn't be surprising. Your next paragraph, though, begins, "So, it's actually quite surprising...". This wording implies the opposite meaning: that your point about language somehow makes the Incas' system *more* surprising.

Kate 20:26, 7 June 2011 (UTC):

• renamed section, added in sub-sections to break up text
• I don't see much value in contrasting various bases within this page. I think doing a helper page on base might be a good idea though.

Kate 19:44, 8 June 2011 (UTC): I've started a helper page on Bases and moved some of the info from this section there.

• Bit about language is intended to demonstrate that their choice of ten as a base should be surprising to us. Have changed it a little, hopefully it is clearer now.

Abram, 6/7:

And here are some more detailed ideas/questions on this section:

• "A positional system is a way of recording numbers with a small library of symbols." The small set of symbols isn't really central to the positional system at all. Roman numerals use fewer symbols than our system, and they use far fewer symbols than a base-500 positional system would!
• "Imagine trying to do long division with Roman Numerals or tally marks - it's nearly impossible to do without translating the numbers into a positional system." Your wording suggests that it should be obvious that there's no good way of doing long division with roman numerals, but I don't think that it is obvious. I know that I would have to spend a while trying to work out such a system before I became convinced that there really was no good way to do it.
• "On the other hand, there is nothing special about base-ten that makes it inherently better than any other base"
  • Perhaps you should define what "base-10" means, as you do in the next sentence, *before* you have this sentence.
  • The "on the other hand" suggests that you are contrasting the lack of specialness about base 10 to something else. I think you are trying to contrast the lack of inherent advantage in base 10 to the aforementioned inherent advantage of positional system, but it took me a while to figure out that this was the object of contrast. Can you be more explicit about that?
• In the paragraph about why base-10 isn't superior, you seem to slide a bit between the notions of "better" and "natural", but the two are different ideas. It may be worth exploring these ideas separately.
• Can you *either* have an image to go with your 5,328 example *or* put the examples shown in Image 3 before your 5,328 example? It just seems weird to have a picture to help illustrate the second or third example rather than the first one.

Kate 20:26, 7 June 2011 (UTC):

• I disagree that Roman Numerals have a smaller set of symbols - as they count higher and higher, they have to constantly invent new symbols. You can't write a googol in RN without adding symbols to the library, but you can do so in a positional system. I agree that a positional system doesn't necessarily have to have a small base though- I'll change "small" to "limited"
• I don't know what to do about the division with RN or tally marks line - to me it is obvious that there isn't an easy way to do it, or else we'd learn it all in school and everyone would use RN instead of regular numbers. I'll change it to "very difficult" instead of "nearly impossible", because Prof. Maurer also pointed out that some guy did figure out an algorithm, but what I really meant was "it is nearly impossible for the average person to imagine doing long division with RN", which I think is a true statement.
• I think I've cleared up the better/natural issue.
• I ditched 5328, and just used 321 in that number so that if someone happens to be looking at the picture the number will be there.

• xd 19:22, 12 June 2011 (UTC) I like the this section now but when you mentioned "Incas' symbols were groups of knots (as described below)", I thought maybe you can put up a table of all the symbols for example, 1: figure eight knot, 2-9: long knots with appropriate number of turns and 0 : empty space.

Kate 13:33, 13 June 2011 (UTC): I don't think I want to have too in-depth of a picture that far ahead of where I actually explain how the knots are numbers - I think either the picture by itself would be confusing, or if it made sense, then the text that comes later would seem redundant. I could point to the picture on the side that shows the knot types, though, do you think that'd be sufficient?xd 20:34, 13 June 2011 (UTC) I think that is fine then.

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Charts section

Comments from Harrison, 5/31:

• your bubble for pendant cord didn't render your link.

Kate 16:08, 31 May 2011 (UTC): Removed the bubble because now the section that explains what a pendant cord is is unhidden.

• "... by a chart/spreadsheet, or a tree diagram."
  • Instead, use: "... by a chart, spreadsheet, or tree diagram."

Kate 16:08, 31 May 2011 (UTC): Just deleted the "spreadsheet".

• "A quipu that is easily translatable to a chart ..."
  • to simplify, try:"A quipu that easily translates to a chart ..."

Kate 16:08, 31 May 2011 (UTC): Rephrased in my own words.

• ", where i=1,2,3,4,5,6 and j=1,2,3 "
  • What did we decide to do about indices? Lots of people are confused by them, but I don't know if it warrants its own page.

Kate 16:08, 31 May 2011 (UTC): Was unaware that we needed to do anything about indices. I guess maybe someone could make a helper page? I'm comfortable with assuming that people can understand this section either from prior knowledge or by looking at the charts.

• When introducing the table layout in Cross-Categorization and Hierarchical Organization, make sure you state that the tables represent the quipu in the image on the left.
• "the diagrams below show the structure of the options available to the traveler, and the prices for each option could additionally be shown by adding numbers to the images."
  • state in what way the diagrams do this. The branches of the diagrams are unlabeled, so the interpretation has to be explained. Or just label the diagrams. If I am correct, each level represents a city, and each string, a possible way of getting to the next city. Label them: city one, city two, etc., and bus, train.

Kate 16:08, 31 May 2011 (UTC):

• Changed the caption on image 4 so that it hopefully deals with the first of these
• Might label the tree diagram after lunch.

Kate 18:03, 2 June 2011 (UTC):

Didn't want to edit the image, so I added a paragraph after - does that clear it up?

• "... and variations on themes."
  • What themes in particular?

Kate 18:03, 2 June 2011 (UTC): Changed the wording on this paragraph, is it less confusing now?

• I think you should also elaborate some more on the hierarchical elements/tree diagrams some more. Richard 5/20

Yeah, that's probably a good idea. I've been re-reading relevant parts of the quipu book this weekend, and am planning to add to this bit and to the base-10 part on Monday. -Kate 22:05, 22 May 2011 (UTC)

I added a little more to this part -- do you think it's sufficient?-Kate 20:38, 24 May 2011 (UTC)

Kate- the section looks great. I'm going to see if my notes from Stat 031 are relevant to this section because they could possibly help make the page a bit more math-y if you're interested. Rebecca 03:00, 25 May 2011 (UTC)

Yeah, let me know if you find anything! There's not really anything much more in-depth in any of the quipu-specific sources I've read, so I'm not sure if there is anything relevant to find, but I'd totally be willing to add more. -Kate 16:07, 25 May 2011 (UTC)

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Sample Quipu

Comments from Harrison, 5/31:

• Perhaps labeling your tables will help with the index notation. label the rows i1, i2, i3, i4, and the columns j1, j2, j3, j4.
• "Second, when accounting for the fact that only integer values can be represented, all entries are multiples of twelve, and no entries are fewer than twelve."
  • please clarify, because 25 shows up twice, and 25 is not a multiple of 12. You explain this later with "which is why the red pendants in the first two groups show a value of 25 instead of 24." but you need an explanation before. Even saying "the two instances of 25 can be treated as 24+1, which I will explain in a few paragraphs." is better than nothing.

Kate 16:08, 31 May 2011 (UTC):

• I'll think about labeling the tables. Might do it after lunch.
• I think the 25s are explained by the first part of that sentence: "when accounting for the fact that only integer values can be represented". When you're doing integer division, 25 is a multiple of 12. This section isn't about explaining every aspect of the data, it's about noticing overall patterns, which I think you can do even if you don't understand why there's 25s there. As a reader, I would find an aside like that to be disruptive to the text.

• "Finally, it is now easy to see that 242, the value of the dangle cord, is the sum of the entries in the fourth column."
  • I've been told not to use words like "easy" and "clearly", because, for some readers, it might not be clear.
  • Try:"Finally, we can now see that 242, the value of the dangle cord on the far left, is the sum of the entries in the fourth column."

Kate 16:08, 31 May 2011 (UTC): changed.

• "The four families consist of a man, a woman, the man’s two parents, and two children; a man, a woman, the woman’s sister, and four children; a man and a woman; and a man, a woman and three children. "
  • though it will take slightly more space, Split this sentence up. Otherwise, it all just blends together.

Kate 16:08, 31 May 2011 (UTC): changed.

• "Each pendant color represents a family."
  • "Each pendant color represents a specific family."

Kate 16:08, 31 May 2011 (UTC): Don't see why this is necessary. I think it's clear in context.

• "The first group represents the ears of corn given to men,"
  • "The first group of cords represents the ears of corn given to men," It's good to remind people once in a while about the nomenclature in the article.

Kate 16:08, 31 May 2011 (UTC): changed.

• xd 19:06, 25 May 2011 (UTC)just a little thing. you mentioned that the remainder goes to the eldest. how is that reflected in the picture or the colored table? In addition, how is the principle "each member gets equal share" reflected in the table? You may need to specify the genders of the children?

The remainder going to the eldest people is the reason that two of the pendants are 25 and not 24 - I can specify that. I thought each person getting an equal share was obvious, but I can elaborate on that too. The genders of the children are totally irrelevant - they aren't recorded or reflected anywhere.-Kate 19:54, 25 May 2011 (UTC)

I elaborated on this part and added an image - does it make sense now? -Kate 21:14, 25 May 2011 (UTC)

• xd 19:37, 12 June 2011 (UTC) It is so much better now. I like it.

Awesome! :)

Why It's Interesting

Comments from Harrison, 5/31:

• "Through quipus, the Incas were able to do arithmetic, and keep incredibly complex records - some quipus are made of hundreds of cords! - tasks which most people assume can only be effectively done in literate societies. "
  • odd format, and a few mistakes. Try: "Using quipus, the Incas were able to do arithmetic and keep incredibly complex records (some quipus are made of hundreds of cords) tasks which most people assume can only be effectively done in literate societies. "

Kate 16:08, 31 May 2011 (UTC): Rephrased in my own words.

• What if you talk some more about the Incas and how they used math in general(for the history/why its interesting part)? Richard 5/20

Another good idea for expanding the page. If I find anything that's interesting and relevant, I will. -Kate 22:05, 22 May 2011 (UTC)

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History

Uses

Comments from Harrison, 5/31:

• "... the state apart from quipus, and so most of our knowledge of Incan ..."
  • odd sentence structure. Try: "... the state apart from quipus. As a result, most of our knowledge of Incan ..."

Kate 16:08, 31 May 2011 (UTC): changed in my own words

• "quipucamayos, native record keepers who created and used quipus,"
  • this is a great place to use a bubble.

Kate 16:08, 31 May 2011 (UTC): Disagree. Almost no one reading this page will be familiar with that word, so I don't see any benefit to hiding it. I think bubbles are for when enough people know the word that you don't want the definition to disrupt the text but enough people are unfamiliar with it that it should be somewhere.

• "... but Spanish historians aren't clear on whether narrative quipu recorded ..."
  • Be consistent with your pluralization. Use "quipus" here.

Kate 16:08, 31 May 2011 (UTC): changed.

• "there are often consistent ratios between related categories."
  • Use a more specific word than "categories". I have no idea to which categories you are referring.

Kate 16:08, 31 May 2011 (UTC): reworded.

xd 19:13, 25 May 2011 (UTC)one thing worth exploring is that how did they do addition and subtraction? is quipu a record keeping device or a calculator like abacus. did they have a set of special rules for doing + and -?

The actual mechanism by which they did arithmetic is totally unknown. It is possible that some of them were used like abacuses, but that would be total speculation - anyone's guess for how addition or subtraction were performed is as good as anyone else's. I feel like I touched upon this a little when discussing base-ten. -Kate 19:58, 25 May 2011 (UTC)