>20,000 year old bones interpreted as multibase calculating aids

SenaryThe12th
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6:15 PM - Sep 26, 2018 #1

Interesting paper on the so-called "Ishango bones" which are dated to just over 20,000 years ago.  These bones were used as handles to hold sharp quartz chisels, no doubt used for scribing calculations on wood, bone, or rock.  Each rod has multiple columns of tallly-marks, for which many interpretations have been made over the years.  But it appears that a consensus is being reached that the first bone displays aids for calculating in base 12, using subbases 3 and 4, and the second bone is a device for converting between base 6, 10, 12, and 60, and perhaps even higher bases.  In the words of the original discoverer:

"..the exchange between exotic ethnic groups, one practicing the decimal system, the other a duodecimal, seagesimal, or even worse."

*chuckle*.  No doubt he was thinking about Wendy's twelfty....

Lots of other interesting disucssion, on the traces of these methods findable even today in various african populations, as well as its probable origin of the Egyptian facility with math.

https://www.researchgate.net/profile/Vl ... o-rods.pdf
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icarus
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5:44 PM - Sep 27, 2018 #2

Ishango is extra-special: A100000.

We could rib about twelfty, but base 120 is actually a pretty deal. It isn't without its challenges. If you have peanut butter and chocolate, Reese's is pretty nice. Same is true for alternating centovigesimal. But we must take care about whispering too loudly lest we awaken...shhh!!!
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Kodegadulo
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7:04 PM - Sep 27, 2018 #3

But we must take care about whispering too loudly lest we awaken...shhh!!!
...Sssshhhhe Who Must Not Be Named? 😉
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Kodegadulo
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7:21 PM - Sep 27, 2018 #4

icarus wrote:If you have peanut butter and chocolate, Reese's is pretty nice.
Mehhh, more apt to liken it to chocolate and vegemite, if you ask me. Bit of an acquired taste... 😉

But I suppose our friend Oschkar has some Mexican spices to make it more palatable. Vegemite mole? 🙂
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SenaryThe12th
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2:41 AM - Sep 29, 2018 #5

icarus wrote:We could rib about twelfty,
*chuckle* no pun intended, of course?   

I actually like twelfty quite a bit.  And Wendy's method of criss-cross multiplication is, lets face, it, a stroke of genius.  Its really an enabling technology for all multi base number systems, not just twelfty.  

How cool would it be if we found out that the dozenal-vs-twelfty debate has actually been raging on for 20,000 years????
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Kodegadulo
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5:15 AM - Sep 29, 2018 #6

SenaryThe12th wrote:
icarus wrote:We could rib about twelfty,
*chuckle* no pun intended, of course?   

I actually like twelfty quite a bit.  And Wendy's method of criss-cross multiplication is, lets face, it, a stroke of genius.  Its really an enabling technology for all multi base number systems, not just twelfty.  

How cool would it be if we found out that the dozenal-vs-twelfty debate has actually been raging on for 20,000 years????
???  On the one hand, you're singing paeans for a "stroke of genius", i.e., a recent innovation by a particularly dedicated ideologue. On the other hand, you're fancifully imagining a mathematical "debate" somehow dating back to the last ice age? Anachronism much? 😉

The Ishango bones are interesting, but they do admit to more than one possible interpretation. The paper keeps citing "circumstantial evidence". I'm not sure how much we can say about conclusions drawn on such tenuous speculations.
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3:31 PM - Sep 30, 2018 #7

???  On the one hand, you're singing paeans for a "stroke of genius", i.e., a recent innovation by a particularly dedicated ideologue. On the other hand, you're fancifully imagining a mathematical "debate" somehow dating back to the last ice age? Anachronism much? 
Well, after reading some of the interminable debates here, you can hardly fault me for suspecting they've been going on and on for tens of thousands of years..... 
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Kodegadulo
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4:42 PM - Sep 30, 2018 #8

SenaryThe12th wrote: Well, after reading some of the interminable debates here, you can hardly fault me for suspecting they've been going on and on for tens of thousands of years..... 
Given where we are, don't you mean, "a gross-squared years"? 😉
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Double sharp
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4:08 PM - Oct 02, 2018 #9

For multiplication in twelfty, I have toyed with various methods, but I think that for actually explaining it to people I'd stick to a near-complete porting of the pure-base way of doing things, as follows.

{ca} default dozenal-on-decimal

Let's take 59'74 * 38 as an example. We'll take the units digit of the second factor and multiply through each digit of the first one, just like in a pure base. Let's split the digits into decimal and dozenal places according to whether they can contain digits beyond 9 or not, and declare that an apostrophe can be seen not just as separating superdigits, but also as a marker that says "the next digit is in dozenal", so that we write them before the first place as well: '59'74 * '38.

All right; we've now done all the preliminaries we need to do this multiplication. There's just a few extra steps; the rest is all according to standard procedure.

1. Multiplying through by a decimal place in the multiplier. When you see a decimal place, write the answer in decimal. When you see a dozenal place, write the answer in dozenal.

4 * 8 = '32
'7 * 8 = 4'8
9 * 8 = '72
'5 * 8 = 3'4
3'40'00 + 72'00 + 4'80 + 32 = 3'b6'b2

2a. Multiplying through by a dozenal place in the multiplier. First, rewrite each superdigit of the multiplicand in dozenal instead of decimal, shifting all the apostrophes:

59 = 4'b (fifty-nine is four dozen eleven)
74 = 6'2 (seventy-four is six dozen two)
59'74 = 4'b6'2

2b. Then multiply through as in step 1 with the rewritten multiplicand, dropping the apostrophe in front of the multiplier:

'2 * 3 = 0'6 (I've put in a leading zero to make things easier)
6 * 3 = '18
'b * 3 = 2'9
4 * 3 = '12
12'00'00 + 2'90'00 + 18'00 + 60 = 14'a8'60
14'a8'60 + 3'b6'b2 = 18'a5'52

(The manipulation in step 2 is essentially multiplying one factor by ten and dividing the other by ten. This eliminates the need to multiply tens by tens, so that you only need to memorise the good old units-by-units table from ones to twelves and be able to come up with the answers in decimal or dozenal. You can either convert between bases for this, or if you already know the tables in both bases, you can just use what you know; it's the same.)

This method will work for any set of two alternating bases; suitable generalisations will work for any number of alternating bases (you just need to keep converting bases to make sure you are in the right one at the right time). In the special case when all the bases are the same, the base switches become trivial, and the method becomes standard long multiplication as a degenerate case.
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Kodegadulo
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6:45 PM - Oct 02, 2018 #10

DS, this corroborates what I've said: AA requires learning more facts, either knowing both the decimal and dozenal multiplication and addition tables, or one set of those plus the dicker-dozen table; and it requires more steps, chiefly in dicker-dozen interconversions; and it imposes a greater cognitive load  in terms of keeping track of which base is in which column and which case of multiplication (d×d, d×z, z×d, z×z) we are currently doing.

In short, it's feasible for adult math buffs to play with as an interesting novelty. But it would be cruel and unusual punishment to inflict upon innocent schoolchildren.
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Double sharp
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4:31 AM - Oct 03, 2018 #11

Yup, I agree with that. I don't think criss-cross multiplication is really an improvement for twelfty, though, much less a stroke of genius; I think keeping it as familiar as possible is about the best we can do with it (and it makes it pretty clear that it adds bells and whistles to pure-base arithmetic, so that it can't possibly be easier).

I am not entirely clear on what Wendy's method for doing it was, but this thread seems to show something much like what I just presented. In my experience it really wasn't that difficult to figure out (because the explanation I saw wasn't very good, so I ended up reinventing the wheel somewhere along the line); the one marginally nontrivial step was to shift the tens around so that you could avoid multiplying tens by tens. Other than that, it really is a completely straightforward generalisation, and while it is a cool one, I'd say that it's not something that you can start off with without already knowing a pure base.
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Kodegadulo
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10:59 AM - Oct 03, 2018 #12

Honestly, the simplest algorithm for doing arithmetic in a subbase-encoded superbase, with alternating subbases or not, is:

1. PARSE: Convert operands from the superbase into any convenient simple base.
2. COMPUTE: Do the arithmetic in the simple base. Better yet, have a calculator do it for you. (In which case, the "convenient" simple base would actually be binary, under the covers.)
3. FORMAT: Convert the result to the superbase for presentation purposes.

Better yet:

AUTOMATE: Write a program that does the above for you.

Think of it as a Zen answer. 😉
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1:26 PM - Oct 03, 2018 #13

Double sharp wrote: Yup, I agree with that. I don't think criss-cross multiplication is really an improvement for twelfty, though, much less a stroke of genius ..  it really wasn't that difficult to figure out 
Well that's just because you are smart too.  For us mere mortals, its a game-changer, bringing even very large twistaff bases into the realm of practical usability.
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1:40 PM - Oct 03, 2018 #14

Kodegadulo wrote: AUTOMATE: Write a program that does the above for you.

Think of it as  Zen answer. 😉
Well here's the kicker for me, Kodo. If you are going to get the computer to do it for you, really, what difference does it make what base you use?  The computer is going to translate it into binary anyways.  

For me, anyways, the utility of alternate bases is that it makes everyday mental calculations easier.  If I *don't* have to pull out my cell phone, and type the problem into the calculator--if I can just choose a handy base and do the problem easily in my head--it just accellerates the whole process of life.   

And in meetings, it gives you a substantial social leverage---if you can multiply a number by 7/5 in your head to convert from man-days to calender days--while everybody else in the meeting is patting their pockets and rumaging around in their purses to find their cell phones--people think you are a goddamn genius.  

And this isn't turning yourself into a robot. It only enhances your humanity if you can connect your creativity with calculation.
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Kodegadulo
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3:16 PM - Oct 03, 2018 #15

SenaryThe12th wrote:
Kodegadulo wrote: AUTOMATE: Write a program that does the above for you.

Think of it as  Zen answer. 😉
Well here's the kicker for me, Kodo. If you are going to get the computer to do it for you, really, what difference does it make what base you use?  The computer is going to translate it into binary anyways.  

For me, anyways, the utility of alternate bases is that it makes everyday mental calculations easier.  If I *don't* have to pull out my cell phone, and type the problem into the calculator--if I can just choose a handy base and do the problem easily in my head--it just accellerates the whole process of life.   And in meetings, it gives you a substantial social leverage---if you can multiply a number by 7/5 in your head to convert from man-weeks to calender weeks--while everybody else in the meeting is patting their pockets and rumaging around in their purses to find their cell phones--people think you are a goddamn genius.  

And this isn't turning yourself into a robot. It only enhances your humanity if you can connect your creativity with calculation.
Heh, well, I was being a little tongue in cheek. I'm completely with you on the mental arithmetic story when it comes to simple pure bases. I might even agree if we were talking about simple cases of a superbase encoded in a single subbase, for example base 60d (decimal-encoded sexagesimal), or even base 120d (decimal-encoded centovigesimal, which would need 3 subdigits instead of two per superdigit). In those cases, you're still essentially doing the calculations in a single pure base, you're just needing to modulo by the superbase and carry at every superdigit.

But with alternating subbases you wind up having to shift gears with each subdigit, alternating your context from one base to another, and dealing with the combinatorial explosion of cases of criss-crossing subbase-times-subbase.  Heh try three subbases sometime. 😉

My take is, why buy into all that complexity? It may be easier to convert the problem into a single subbase, do all the calculations in that, then just convert the superdigits from single-subbase to equivalent alternating subbases for display purposes.  Look at that as a variation on the idea I described in my last post. I'm pretty sure that's how I've always tackled arithmetic in alternating bases: convert it into a simpler problem, and convert back at the end, even if I was doing that in my head.
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SenaryThe12th
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3:52 PM - Oct 03, 2018 #16

Kodegadulo wrote:My take is, why buy into all that complexity?
Not an unreasonable question.  
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Double sharp
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4:06 PM - Oct 04, 2018 #17

SenaryThe12th wrote:
Double sharp wrote: Yup, I agree with that. I don't think criss-cross multiplication is really an improvement for twelfty, though, much less a stroke of genius ..  it really wasn't that difficult to figure out 
Well that's just because you are smart too.  For us mere mortals, its a game-changer, bringing even very large twistaff bases into the realm of practical usability.
Well, thank you for the compliment, but I think everyone here deserves it for thinking out of the decimal box! ^_^ I think we all sometimes forget the time when things like this weren't yet obvious, and I cheerfully admit that I just did so. ^_^
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