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Comment by todd8 | Hacker News Reader

I dislike these pseudo-scientific claims about alternative number systems and methods of paper and pencil arithmetic:

> Because of the tally-inspired design, arithmetic using the Kaktovik numerals is strikingly visual. Addition, subtraction and even long division become almost geometric. The Hindu-Arabic digits are an awkward system, Bartley says, but “the students found, with their numerals, they could solve problems a better way, a faster way.”

I think the students can be praised for having come up with simple to understand and write number system that corresponds to the conventions for counting in Alaskan Inuit language, and it seems appropriate to capture these notations in upcoming Unicode standards.

However, spending time learning base 20 arithmetic has obvious disadvantages that the article ignores. The times tables, memorized in grade school and fundamental to paper and pencil calculations, are now four times larger. Base 20 is not a popular notation for numbers. One important advantage of the number system (Hindu-Arabic) that most of the world uses is that most of the world uses it. I grew up with inches and degrees Fahrenheit and had to learn the metric system to pursue my science education. I'm glad I didn't have to learn how to count as well. We shouldn't make it harder for these kids to enjoy the rest of the world's books, journals, and internet resources about math and science.

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The big advantage would be that you learn to see that there is nothing magical or 'right' of one number base over another. Base 2 has it's uses as does base 10(10). And most computer programmers are familiar with at least one other base besides decimal, and quite a few will be able to use binary, octal and decimal with relative ease. It gets interesting when you go off the beaten path and you re-learn the rules for arithmetic in different bases. In my experience all of this gives you a much better understanding of why decimal is practical and widespread. But it also shows you that it is a convention that won out for both cultural reasons and because it made certain arithmetic easier.

If 10x10 = 100 looks natural to you in decimal then it will still be natural to you if you think of it as 16x16 = 256 when you look at the numbers in their hexadecimal representation. You can only get that kind of fluidity by playing around in different number systems. So I'm perfectly ok with students inventing their own number systems, they are definitely not going to get any dumber on account of having done that.

Growing up with Inches and degrees Fahrenheit is a cultural issue, most of the rest of the world has moved on from there, for reasons that are far more compelling than those that would apply to using a different number base. Those are arbitrary values, whereas all number bases exist regardless of whether we use them or not. Think of the one as cultural baggage and the others as just another part of number theory.

Not trolling. Promise. I can't think of a non-computing/math-related context where being cog of base number theory comes into play.

You've never needed to know what time it will be 2 hours and 45 minutes after 11:30 AM?

Heh. Yeah, I guess. Maybe I'm an idiot, but the idea of a minute being a unit of a base-60 system didn't occur to me til round the time I was learning about complimentary binary arithmetic.

I think you're not unusual in not having thought about it, but there's value in being able to intuitively extend your understanding of decimal math to cover it.

Yeah, because it isn't required to facilitate the functional understanding of rough time (>=1s/tick) intervals.

I understood how perceiving numerical values in various bases is useful, just not how it applies to the post.

Imperial units are unfortunately still in common use in the United States, the United Kingdom, and related colonies/ex-colonies. They are variable base number systems that people reckon with on a daily basis.

It wasn’t so long ago, historically speaking, that England used pounds, shillings, and pence! Decimal currency really was a fantastic innovation.

Pedantic comment: USA doesn’t use the imperial system, that was Britain. USA has used the “United States Customary Units” since 1832. Since 1893 the units have been defined in terms of metric.

Since 1991 the rule has been to “prefer metric units”, which I guess is why all packaging you commonly see includes measurements in both systems.

https://en.wikipedia.org/wiki/United_States_customary_units

edit: After living in EU and US, I will grant that the lbs is a more convenient unit for measuring food. A kilo of apples is a lot, but a pound is just right.

I don't think the difference between pounds and kilos really matters to purchasing food, but labeling everything with a per 100g nutrition facts rather than just whatever BS serving size the manufacturer makes up is a significant improvement. You can buy oil in the US that is listed as containing 0 calories and 0 fat per serving (and 1000+ servings per container). Not that that's a unit system issue.

As a calorie counter, I have thoughts on this. Comparing foods at the supermarket based on 100g base units is easier, true. But that means you need a scale every time you eat anything. It’s waaaay easier to go “yeah that’s about 1/3rd of the package” (and accurate enough for me).

So really I’d love nutritional labels that have both.

If everything was based on 100g, people would become used to roughly weighing things with their hands and knowing roughly how many calories are in their food.

It’s at least partially about recognizing that there are other ways to view the world than the default one we carry around inside our precious egos, ways that are — literally, in this case — equal to our defaults.

What?

Some bases are inherently better for small integer arithmetic.

Every base has difficulty representing some fractions. For example, 1/3 is written in base 10 as 0.333... The 3 repeats forever. You can use explicit notation like a line over the top, but that's still just a workaround.

The Babylonians used base 60, which has many more prime factors than base 10: 60 can be divided evenly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

60 is a lot of digits to remember, though. That presents its own overhead.

Some people advocate base 12: it's only two more digits, and it's evenly divisible by 1, 2, 3, 4, 6, and 12. 1/5 is a mess, though: 0.24972497...

> The big advantage would be that you learn to see that there is nothing magical or 'right' of one number base over another.

Different bases do indeed have "magical" properties that make them appealing, which is exactly what you later acknowledge when you say that base 10 makes certain arithmetic easier.

There's no issue with learning different bases, as long as it's not an excuse to avoid learning the common bases you need to interact with various fields.

great and enlightening comment.

Actually, many western number systems have traces from base 20. Just consider how you have names for numbers up until 20.

In Danish, the way we name numbers are heavily inspired by French, which also exhibit traces from base 20.

The name in Danish for 60 and 80 in modern Danish are "tre(d)s" and "firs", respectively. These are shortened forms of "tredsindstyvende" and "firsindstyvende" used historically, literally meaning "3 times 20" and "4 times 20", respectively. The number for 50 is "halvtreds" - derived from "half way to treds (60)" - meaning half way (when the "way" is 20 long) between 40 and 60.

In french 80 is quatre-vingt (4-20).

If anything, arguably our common system in which we have named numbers up until 20 (i.e. base-20) and then shift to base-10 for numbers above 20 is illogical.

Ha! Although you could also argue that the transition is from twelve to thirteen ("three-ten").

On the other hand, in english, there is "score" for 20.

> However, spending time learning base 20 arithmetic has obvious disadvantages that the article ignores.

They aren't learning base 20; they already use it in their language. They are learning how to write their language in their writing system.

From the article: "The Alaskan Inuit language, known as Iñupiaq, uses an oral counting system built around the human body. Quantities are first described in groups of five, 10, and 15 and then in sets of 20."

Not everyone counts in decimal. Base 20 exists in spoken French. For numbers 50-90, Danish uses base 20, in some cases mixed with fractions. There are other bases in other living languages as well.

The remnants of duodecimal are still in English, for that matter. Eleven, twelve, dozen, gross.

Twelve inches to the foot used to be a round number.

Come and join the Dozenal Society of America! https://dozenal.org/index.html

> “the students found, with their numerals, they could solve problems a better way, a faster way”

> Base 20 is not a popular notation for numbers […] We shouldn't make it harder for these kids

So much of what you object to is that something they’ve found more intuitive and engaging isn’t what unintuitive disengaging stuff they’ll encounter. But developing intuition for math is far more valuable than developing conformance to how it’s supposed to be done. Who cares if that intuition is developed with some idiosyncrasy from what you consider normal? The math is math, the principles are consistent, the knowledge is transferable. Insisting they learn the same things a different way is totally arbitrary and counterproductive.

I think parent's point is not cultural as you are implying but rather practical. Learning in a base that nobody uses could be easier today and an hindrance later.

I think, all in all, this should not be a big deal. For the gifted kid, they'll find a way to adapt and become the next Einstein. As for the ungifted, it might give them a better leg up and allow them to perform better than they would have, so it's probably a plus anyways.

> I think parent's point is not cultural as you are implying but rather practical.

I think this is exactly what I find distasteful in the original comment: Culture is ignored in favor of what is practical for someone other than a subject of the article.

I believe it is my comment that you find distasteful. Please reread my post, I didn’t intend it to be dismissive of the Inuit culture. I even stated that these new numeral glyphs should be included in Unicode. I believe this aids in preserving the Inuit culture and traditions.

I myself have taken over two dozen university level math courses at the undergraduate and graduate level. I don’t need to have number bases explained to me, but I have raised three kids and recognize the challenge of helping them attain proficiency in basic grade school fundamentals. This is what motivated me to make the comment.

Ironically, perhaps, is the fact that my own ancestry includes an indigenous people living at the arctic circle who faced and continue to face racial discrimination, loss of native lands, and forced changes to their way of life, the Sámi. [1]

[1] https://unric.org/en/sami-we-are-the-natives-of-this-country...

Finger binary: https://en.wikipedia.org/wiki/Finger_binary :

> Finger binary is a system for counting and displaying binary numbers on the fingers of either or both hands. Each finger represents one binary digit or bit. This allows counting from zero to 31 using the fingers of one hand, or 1023 using both: that is, up to 2**5−1 or 2**10−1 respectively.

- "How to count to 1000 on two hands" by 3blue1brown https://youtu.be/1SMmc9gQmHQ

- "Polynesian People Used Binary Numbers 600 Years Ago - Scientific American" https://www.scientificamerican.com/article/polynesian-people...

What is the comparative value of radixes like Binary, Octal, andHexadecimal compared to Decimal (radix 10)?

Perhaps a radix like eπI would be more useful; though some amost-mystic physicists do tend to radix 9: "nonary" (which is actually ~ also radix-3).

List of numeral systems > By culture / time period, By type of notation https://en.wikipedia.org/wiki/List_of_numeral_systems :

> Numeral systems are classified here as to whether they use positional notation (also known as place-value notation), and further categorized by radix or base.

Digital computing hardware design

I always make a point of demonstrating finger-binary with the number 132. 4 will suffice, though.

Learning more than one base system, like learning more than one language, can only benefit the learner. Is learning binary, octal, and hexadecimal a hindrance? No, it's generally quite beneficial to understanding and offers many insights that are easily missed in base 10. Humans use a ton of different number systems and scales for all sorts of things, from time, to distance, weight, thickness, angles, and more. None of these things harm the learner.

> cultural as you are implying

I… didn’t think that’s what I was implying. I didn’t think I was implying anything at all, I was pretty direct about my meaning.

I would take this a step farther and argue that being exposed to different numeral systems expands your understanding of math, similar to how different programming languages expands your understanding of software, or how different unit systems (e.g. natural units [1]) expands your understanding of science.

[1]: https://en.wikipedia.org/wiki/Natural_units

At first students would convert their assigned math problems into Kaktovik numerals to do calculations, but middle school math classes in Kaktovik began teaching the numerals in equal measure with their Hindu-Arabic counterparts in 1997. Bartley reports that after a year of the students working fluently in both systems, scores on standardized math exams jumped from below the 20th percentile to “significantly above” the national average.

It sounds like they objectively are doing better though. Bottom 20th to above average is non-trivial. Even if you look at it from the point of view that learning a different base(binary, hex), any base, teaches you to think differently about math, why not learn the native base for extra confidence?

Memorizing the times table is for suckers. If you can add, you can spend at most 2 additions to get 2, 3 and 4 (2x = x+x, 3x = x+x+x, 4x = (2x+2x)). Multiplying by the base of your numeral system comes for free (just add 0 to the end). Assuming subtraction just as easy as addition, I now know how to multiply by base-1 and base-2 (9 and 8 normally, 19 and 18 in this case). The last trick I need to invoke is division by 2. Assume you've ignored every other lesson in order to focus on being unreasonably fast at cutting numbers in half. So now, coupled with the append zero trick, you have a path to 5 and 10 (5x = (20x/2)/2, 10x = 20x/2). I haven't memorized anything, and I've used at most two operations, and already I can multiply by 2,3,4,5,10,11,18,19,20. With a third operation I can reach 6,8,9,12,15,17. All that's missing is 7,13,14,16. At that point the remaining part of the "table" only has 10 unique elements in it. I can cover it with a 4th and 5th op if I'm truly stuck, but at some point in doing that repeatedly I'd probably end up remembering that chunk of the table anyway. If we were still in base 10, the same tricks would get me the entire single digit table within at most two addition and/or halving operations. It only takes 3 ops if you reject my premise that halving is as easy as doubling / adding.

Sure it costs me 3 operations per multiplication, but my operations are only doubling and halving (and arguably appending zero). What I lose in number of steps I gain back by just being faster at those two specific skills. And I didn't even have to waste time memorizing stupid tables!

It is still extra mental steps, memorising times table is relatively trivial. The costs of having these extra steps really add up for more complicated problems that involve multiplication

Memorizing the multiplication table is a shortcut that works for small numbers, memorizing a quick method for multiplication works for all numbers.

The table is just an optimization that can come in handy in the same way that cache memory is handy: it gives you the same answer but only for a limited set of data and in a faster way. Eventually you'll have to venture out of cache memory to reach the rest of the space and if cache memory is all you have you're in trouble. So if you can learn only one of the two the method is the better one, so learn that one first, then memorize, as much or as little as you feel like. Up to 20x20 is doable, much larger is useful for squares, powers of two and some other numbers for order-of-magnitude checks but when I'm lazy I'll just break out the calculator. It's useful to be able to do this in your head up to a certain point and beyond I'll use a tool just because it is convenient and faster.

I'd object to the idea that memorizing times tabes is relatively trivial as I've always had trouble memorizing anything. But I too realized a lot of the times tables are basically "free": the only ones I felt I really had to memorize were 2, 3, 5 and 7 (incidentally, the primes). In fact, I still struggle with the 7 sequence because we learned the times tables in two batches (1-5 and 6-10) and 7 was the only one in the second batch that required actual effort and at the age I mistook the other ones being trivial to deduct with me having become better at memorizing times tables.

In retrospect, batching it as learning the primes first (including 1 to ease you into it) and then the rest of the numbers second would have probably helped me a lot by avoiding the feeling of "what the heck, why do I suddenly suck at this again when I just thought I had gotten good at it". But of course the concept of primes is off-limits when teaching basic arithmetic so even hinting there's something special about those numbers was apparently taboo.

EDIT: Additionally I'd argue out of the primes, 5 is almost free because of how numbers are represented in decimal (i.e. it just alternates between 0 and 5 and the digits in front of that go up every other time) and doubling is fairly intuitive to reason about (take what you have and add the same amount again) so it's just 3 and 7 that are weird.

I guess my case is different because we have multiplication songs in Cantonese. (Not really a song, just multiplication facts.) It takes like 30 seconds to read a full column, and kids were taught to read aloud a column at a time. So just plain old repetition, but easy to catch on after a while. A few short practices everyday in a week would be 15-20 repetitions on a single column, which is a fairly good starting point, and very short practices would minimise mental burdens.

We have them in English too: https://www.google.com/search?q=multiplication+rock

> memorising times table is ...

... a different experience for different people and I wish we just accepted that and provided multiple ways to deal with it rather than assuming it's something we have to memorise. And maybe also paid attention if those having issues memorising it don't have more general memory issues that need addressing.

It is superior, because you can apply the same knowledge on larger numbers and later on general equations. Memorized table is just one pony trick to get result faster in limited amount of situations. And once you forget it, which you are guaranteed to forget, then you will need to derive numbers anyway.

Also, it was NOT easy to memorize it. It was hard. Which is why I started to use derivation as s kid, once I realized it is possible. I always did well in math, it did not harmed me at all.

Then you haven't realized how well what I wrote actually scales. In base 10, 2 operations gets you the whole of single digit multiplication (versus memorizing the whole table just to get it down to 1). It also gets you to 11, 12, 20, 21, 19, 30, 40 (by various use of the append 0 trick to multiply by 10 for free anywhere you want). Now try to work out which multiplications are possible within 3 additions/subtractions? How about 4? You should find the accessible fraction of the number line grows exponentially. Memorizing the whole multiplication table for 2 digit numbers isn't practical. That's a 100x100 table. But a small number of adding, doubling, and halving steps can get you anywhere you want in that table.

> Memorizing the times table is for suckers

We humans are good at pattern recognition but we suck at mental arithmetic involving multiple steps. We are slow and error prone. Yes memorizing sucks, all schoolkids hate it, but once you have mastered it you have gained a great new skill. Instead of having to waste mental energy on calculations you just remember them and can spend your precious mental resources on higher level calculations.

You haven't understood. I'm actually better at multiplying than the average and was ahead of the class. It doesn't matter that I'm doing two addition operations per single digit multiplication, because its very easy to be more than twice as fast at doubling / halving numbers if the only thing you ever care about is doubling and halving numbers. The pain of memorizing the table is wholly unnecessary. Memorizing is for suckers.

> I'm actually better at multiplying than the average and was ahead of the class

I bet that most people on this forum were better than average in elementary school, regardless of the method they were using. I also bet that you would have been even better after memorizing. Recalling from memory is almost instantaneous, it is always faster than having some multiple step algorithm.

What did you eat for breakfast yesterday?

Was recalling that from memory "almost instantaneous"?

Recalling yesterday's breakfast is not the same as multiplication table or some other facts you've memorized. Most people don't memorize their breakfast!

edit: as an example, when someone asks "what's your name", do you not recall much faster than what you ate?

I did not memorized it and was deriving it. I was seen as good at math, cause other kids were forgetting and unable to fix situation.

i had the same thought, but it sounds like starting with a base-20 numeral system makes it easier for the iñupiat students to learn base 10, as evidenced by the reported test score improvements

i worked out the multiplication thing in more detail in https://news.ycombinator.com/item?id=35551051

myself, i learned mediation and duplation before i learned to multiply with a memorized multiplication table, and though that's a faster algorithm, you could maybe teach it after switching to base 10? also nowadays maybe you'd be better off with a memorized table of briggsian logarithms because if you really need more than two digits of precision you should probably use a calculator

mediation and duplation of 69 · 21

now we add the starred duplation column items where the mediated multiplicand was odd (corresponding to the 1s in its binary representation, 16+4+1)

a bit more work than adding up four appropriately shifted table-lookup results but not really that much, doesn't depend on a multiplication table, and you can do it just as easily in roman numerals or kaktovik numerals

also people have been multiplying using tables of squares since babylonian times; https://en.wikipedia.org/wiki/Multiplication_algorithm#Quart...

for this you calculate 69+21 = 90 and 69-21 = 48, look up or remember that ⌊¼90²⌋ = 2025 and ⌊¼48²⌋ = 576, and 2025 - 576 = 1449, the correct answer

So we just wrote very similar comments at the same time. Did you notice this for yourself too, or is there some magic utopia I've never heard of where they teach something other than memorizing math tables? Couldn't help but notice your use of RPN, and the fact that you have specific vocabulary for all the steps.

i learned mediation and duplation (which doesn't involve multiplying by the base of your numeral system!) from a book as a kid (a compendium of knowledge for kids from the 01950s that my grandparents had) but have seen people talk about it several times since then

you can get pretty fast at it but you have to do 6.64 halving and n-digit doubling operations per digit of the multiplier, plus about 1.66 n-digit additions, so in my experience it's still slower than computing partial products with a memorized base-10 multiplication table, which requires adding together n recalled multiplication-table entries to get a partial product per digit of the multiplier, and then adding these partial products together

just not as much slower as you'd naively expect

i derived a shitty version of quarter-square multiplication on my own about 20 or 25 years ago and much later learned about the streamlined version from wikipedia

i like rpn but i don't think i used it here?

>6.64 halving and n-digit doubling operations per digit of the multiplier

Where do you get this from? Intuitively I'd reckon you'd need no more than log2 operations. Whatever this result is it certainly doesn't hold for small n.

>mediation and duplation of 69 · 21

Am I mistaken in reading this as RPN? You put the operator to the side of the operands rather than the middle. Slightly unorthodox to put the op on the right, but still obvious in meaning.

Your English is obviously perfect, but you have a very idiosyncratic way of writing math that I've never seen before. For example you wrote "from the 01950s". I've never seen anyone use a 5 digit year format, and rarely if ever have I seen anyone include a leading zero in a number at all. Its not bad or wrong, I just don't know what this style is except possibly a type foreign accent?

> Where do you get this from? Intuitively I'd reckon you'd need no more than log2 operations. Whatever this result is it certainly doesn't hold for small n.

2 log₂ 10. a multiplier m with 5 digits has a log₂ between 13.29 and 16.61, 3.32 per digit; you need log₂ m halving operations and log₂ m doubling operations. so 6.64 is an upper bound but it's usually pretty close. in the example there, it would have led you to expect 13.28 halving and n-digit doubling operations, when the reality was 8 halving and doubling operations, of which three were n+1-digit doublings

n is the number of digits of the multiplicand, so it holds just as well for small or large n, except in the sense that n can be a bad estimate for how many digits you need in the doublings when it's small

> Am I mistaken in reading this as RPN?

yes; those asterisks mark the rows where the mediation column was odd, as i explained below in 'now we add the starred duplation column items where the mediated multiplicand was odd'. they do not denote multiplication or any other operation on the two numbers to their left. you are surely not the only person who misinterpreted this, and i apologize for the lack of clarity

When I was in school (90s-00s, MI public school), we didn't have to memorize times tables at all. They taught us a bunch of different methods in succession and told us to use whichever we found easiest.

which methods did you learn

Honestly, I didn't learn them at the time so I couldn't tell you. I was good enough at mental math that I never had to use any (and stubborn to the point that I refused to show work that was done in my head, and so got horrible grades for correct answers). It's only as an adult that I've become interested in learning algorithms and gained any appreciation for mathematics.

> The times tables, memorized in grade school and fundamental to paper and pencil calculations

Am I the only one who never memorized the times tables as a kid (because I found it boring), and yet today I am far better at mental arithmetics than 99% of people?

eg. If you ask me what 7 times 5 is I have no idea from memory, but I can tell you half of 7 is 3.5 so it's 10 times that. Or 8 times 9 is 80-8. And so on.

This isn’t base 20, though. It’s actually base 5. But groups of 2 digits are written in the same text area. Basically a fancy ligature.

The bottom part has 5 digits, but the top part only has 4 (counting blank as digit 0). If both had 5 digits there would have been 5×5=25 positional digits.

So it’s kind of base 5, base 4, and base 20 at the same time.

Good point

Will they actually need to do pen and paper multiplication in real life? I'm not sure when anyone I know would need to, or if I even still could.

Is using a lookup table just as good/almost as good if nobody actually needs to do it fast in the field?

Can they just use base 10 for all multiplication, if multiplication isn't needed in whatever problem set this is optimized for that seems to have them so excited?

Practically, they do better with this?

> The times tables, memorized in grade school and fundamental to paper and pencil calculations, are now four times larger.

Why would you expect them to memorize four time larger table? There is zero reason to do so, just because the base number is larger. Also kids don't memorize the whole 10x10 table anyway. They are taught to calculate majority of it.

For most British people of my age learning up to 20^2 would be 2.8x larger as we had to learn times table to 12.

https://blog.jgc.org/2010/06/duodecimal.html

Your argument could equally be used to say that children should only be taught English, Spanish, Mandarin or Hindi. I'm not sure if you would agree with that, but the sentiment isn't unheard of, especially when it comes to opposition to language preservation.

Babylonyan Base 60 is superior.

Your joke was misinterpreted as a dig, unfortunately.

I mean, it's obvious that a prime-number base is best above all:

  1 = 1
  10 = 2
  100 = 3
  1000 = 5
  10000 = 7
  100000 = 11

etc

You get non-canonical representations such as 11=3

You could go multiplicative based. So 11 = 2×3. But then you get very difficult addition, comparison, and you need numerals inside your numerals. (E.g. 2048 would be 11 as in a single 11 in the first symbol spot)

It's true. I was teasing, but I did work this out once. IIRC it went something like:

  A = 1
  B = 2
  C = 3
  D = 5
  E = 7
  F = 11
  ... and so on

Then any number can be represented in terms of its prime factorization:

   4 : BB or B^B
   6 : BC
   8 : BBBB or B^C
   9 : CCC or C^B
  10 : BD
 ...
 100 : (BD)^B
 101 : # (some arbitrary unique symbol)
 ...

Actually base e is clearly the superior base as it's the most information-dense number representation.

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