Usefulness of numbers

Use this forum to discuss properties of number bases in general.

Usefulness of numbers

Ae596
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Jun 6 2017, 12:18 AM #1

PiotrGrochowski @ Jul 10 2014, 03:21 PM wrote:
icarus @ Oct 5 2012, 11:57 PM wrote:What's better than twelve! Double twelve? Let's check it out.
24 is minor improvement and 36 is major improvement.
Divisors of 24: 2, 3, 4, 6, 8
Divisors of 36: 2, 3, 4, 6, 9
9 is much more important than 8
Neighbors of 24: 23, 52
Neighbors of 36: 5*7, 37
I get what you're saying about the divisors - but why is 9 more important than 8? In my opinion, 8 is more important than 9 because more than one ternary division is less useful than two or every three binary divisions. Any thoughts?
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icarus
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Jun 6 2017, 12:42 PM #2

There's been discussion about a "theory" of which numbers are more "important" than other numbers. I can't recall right now where that discussion is in this forum, but I'll outline some thoughts behind this question.

First we have to define as best as possible what we mean by "important". Obviously "important" must have something to do with how people would use the number. From this, we immediately can see that the question might be subjective or "soft/social science," i.e., the province of statistical experimentation and short of that, speculation. In other words "importance" is probably not strictly mathematical. Both numbers are prime powers: 8 = 2^3 and 9 = 3^2. 8 is smaller than 9. The prime factor of 8 is the smallest prime and the exponent of 9 is smaller than that of 8. As m = {8,9} are prime powers of p {2,3} we observe no regular numbers outside of the divisors of m = powers of p and such powers greater than m. Because 9 involves the second prime it has significantly more frequent totatives and semi-coprime numbers than 8.

People have a tendency toward mediation/duplation and this is probably the reason for a preference toward evenness and small powers of 2. The fact that 9 is one less than the civilizational base 10 may artificially suppress its use but eightfoldness is a lot more commonly observed in society than ninefoldness. It is easier for people to arrive at 8 by applying twofoldness three times than to arrive at ninefoldness by the more difficult tripartition twice, thus it gets done more often. There is a lot of trouble in achieving ninefoldness though it is maybe conceptually simpler as there is less repetition.

Any notion of the "utility" of bases (implying _human_ utility) is an extrapolation; the further we extrapolate the more "wrong" we are, maybe to the point of absurdity because it might simply not be possible for a human mind en masse to achieve implementation of a given (large) base. This said, I derived the following list and wrote a program to "predict" which numbers are "important":

2
3, 4
5, 6, 8
7, 10, 12, 16
11, 14, 20, 24, 32,
etc.

This is a silly list. I can't see however we construct it that 11 is anywhere nearly as useful as 32 or 20 and even 14 in that list is questionable. Further, we don't see 9 or 25. The importance of 25 is artificially amplified in our native-decimal minds because of 25 being the decimal surrogate for the quarter, otherwise it is perhaps nearly as "weird" as 49 might be to us. 11 seems to be a "near miss" of 10 for the same reason, as 14 might be a "near miss" of 15, which is also not on the list.

The algorithm is the extrapolation of the first 6 terms and the latter 3 of the fourth row. Sevenfoldness does not seem to be too important; we'd rather have 9 perhaps, or 15. The triangle

2
3, 4
5, 6, 8
?, 10, 12, 16

derives from the "importance" of denominators, the first three and certain daughter terms of interest to dozenalists. Halves, thirds, and quarters being Neugebauer's "natural fractions". In this sort of study 8 appears and maybe 9 "deserves" to be in the 4th row. The list itself is my own extrapolation! It isn't as ludicrous as subsequent algorithmic extrapolations that are flawed in many ways because "importance" is, as said above, largely a subjective and cognitive-science determined quality.

I would assert, based only on anecdotal "evidence" and experience in 46 years of life and travels, that 8 is far more important to human societies than 9 is; that it would seem resolution of eightfoldness ought to be granted priority over that of ninefoldness.
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Oschkar
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Jun 6 2017, 05:43 PM #3

My preferred algorithm for this is to take the prime factors of a number, subtract 1 from them and add them up.

This results in a ranking that starts like this:

1
2
3, 4
6, 8
5, 9, 12, 16
10, 18, 24, 32
7, 15, 20, 27, 36, 48, 64
14, 30, 40, 54, 72, 96, 128
etc.

Unfortunately, this tends to give ternary powers a bit too much power for my taste; I can't see 27 being as useful as the rest of its row.
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icarus
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Jun 6 2017, 09:14 PM #4

oschkar this is a transform of OEIS A059975. This code generates the data:

Code: Select all

Array[Total[Flatten@ Apply[ConstantArray[#1, #2] &, FactorInteger@ #, 1] - 1] &, 12^2] 
Then we find the positions of all the values and arrange the data into rows:

Code: Select all

Values@ KeySort@
   PositionIndex@
    Array[Total[
       Flatten@ Apply[ConstantArray[#1, #2] &, FactorInteger@ #, 1] - 
        1] &, 12^3] // TableForm
This produces the table that starts with:

Code: Select all

   1                                                                           
   2                                                                           
   3    4                                                                        
   6    8                                                                        
   5    9   12   16                                                                  
  10   18   24   32                                                                  
   7   15   20   27   36   48   64                                                         
  14   30   40   54   72   96  128                                                         
  21   25   28   45   60   80   81  108  144  192  256                                             
  42   50   56   90  120  160  162  216  288  384  512                                             
  11   35   63   75   84  100  112  135  180  240  243  320  324  432  576  768 1024                           
  22   70  126  150  168  200  224  270  360  480  486  640  648  864 1152 1536 2048                           
  13   33   44   49  105  125  140  189  225  252  300  336  400  405  448  540  720  729  960  972 1280 1296 1728 2304 3072 4096
  26   66   88   98  210  250  280  378  450  504  600  672  800  810  896 1080 1440 1458 1920 1944 2560 2592 3456 4608 6144 8192
...
There appear to be OEIS A280954 terms in each row, i.e., {1,1,2,2,4,4,7,7,11,11,17,17,26,26,37,37,53,53,...}.

I would think this to be about right. Maybe 27 comes in too early, but if we're looking at an algorithm to try to make sense of it, this looks good.

I've generated 12^4 terms, but we need a power of two to complete rows. Of course the higher the rows, the more vertiginously ridiculous it gets to discern differences among the numbers. What's more important, 2742 or 2285? (answer: the former, but 3143 is more important than both of those.)
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Double sharp
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Jun 7 2017, 10:20 AM #5

I kind of imagine that if we somehow could define some function β(n) where we can plug in some natural number n and get a measure of "usefulness", it should be a multiplicative function. If n is a prime power p^k, I want to say that β(n) = β(p)^k, because "dividing into p parts k times" is exactly dividing into n. Similarly, if n = pq I want to say that β(n) = β(p) * β(q) for the same reason, and the same applies for any set of primes.

No doubt this makes things sound a lot more formal and certain than they really are, but it seems to be sensible. Part of the reason why large divisors in general are rarely encountered in real life is that even if your number is as nice as 72 = 2^3 * 3^2, you are unlikely to need quite that many repeated factors at once. I would almost want to say that you would want to divide into 11 or 13 parts more often than 72 because number of factors and size of factors both matter. With the right choices for β(p) for prime p, this could be achieved, although we would then be making up a function somewhat ad hoc and would be in danger of assuming our results.
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wendy.krieger
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Jun 7 2017, 10:54 AM #6

If i have divided an interval, the next division might be to restore it towards the base. The large primes, that are not in the base, would not serve much use.

For example, if i divide the circle to four, the next step in twelfty, is three and then two, it is only then that i will guess the fives. The minute hand on the wall is pointing to 87, which is in the third quarter, third hour, past the half point, two fifths.

This is a lot of repeated divisions of the primes 2, 3, 5.

The division into 42, takes fewer steps, because this number has three factors, rather than five.

The books on pre-metric europe seem to follow this rule. Any odd number greater than three, is the binary fraction of some larger number (15, 25), or the result of units of diverse source (a dutch rod = 13 rhenish feet of 11 dutch inches), or the additions of augments of waste (bakers dozen = net 12 good, printers ream = 20 quires of 24 sheets, plus an extra 16 for spoilage, net 480. This is in metrics 500).

Powers of 3 would tend to accumulate, since it is the integer for which x^(1/x) is maximal. Powers of 3 dominate the canonical sum (sum of factor-primes). For example, 18 is the largest number with a canonical sum of 8, and 324 for a sum of 16. But for 24, the canoncal sum is not 5832, but 6561.

Of course, what works well, is always a matter of preference here. The needs of metrology are different to those of pure maths. The process here is to tip the scales in favour of two. One must remember that the 'racetrack' here is really not infinite, but something in the order of a millionred, something like 12^8 to 12^9, with some extension on either side.
Twelfty is 120 dec, as 12 decades. V is teen, the '10' digit, E is elef, the '11' digit. A place is occupied by two staves (digits).
Digits group into 2's and 4's, and . , are comma points, : is the radix.
Numbers writen with a single point, in twelfty, like 5.3, means 5 dozen and 3. It is common to push 63 into 5.3 and viki verka.
Exponents (in dec): E = 10^x, Dx=12^x, H=120^x, regardless of base the numbers are in.
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icarus
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Jun 7 2017, 07:44 PM #7

Note: I split this topic from Acrobatic Tetravigesimal; the first post of this is a response to this post.

Double Sharp: I agree with you. It seems that things that relate to primes and their powers should be multiplicative as primes themselves are generally multiplicative concepts. Maybe beta(n) is inverse multiplicative (sorry, that's clumsy: don't have the term right now and am in a hurry). Or maybe beta is a "golf" style function where "higher" is "worse". It's an interesting and complicated consideration.

Wendy: I do not follow the second paragraph, specifically "is three then two". I am thinking about what you wrote though. I am not sure what twelfty has to do with it but perhaps it is just an example. I also don't follow "racetrack".
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wendy.krieger
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Jun 8 2017, 07:14 AM #8

If you have a base structured for division, then you would think in those units. So for example, you would divide pretty much as the currency flows in 12 or 120, (i am more familiar with the latter). So you divide a range into littler ranges, and keep dividing. So while 9 and 7 look pretty much the same, you are more likely to end up with 9th in base 12 or 120, rather than seven.

If i were going to divide something in base twelfty, i would start with quarters, then thirds of quarters, and then halves of the thirds. It is only at this point would i start to divide by five. But this is how the money runs: 30p, 10p, 5p, 1p.

The 'racetrack' is the range of most metrological quantities. It's around 120^4.5, or so. So while you can make up these fancy primorial bases and bases with large primes, in the end you have to make a string of prime factors, and try and fit that into the racetrack a few times over.

Of course, if you plan to go to eighths and sixtyfourths, there is less guff included in base 120 (ie 225), than in 60 (3375), and twelfty is still not much behind 12 (27) in this regard. This is why it's best to increase the '2' content of a base. Twelfty has logrithmically, 3/7 or 43% 2's while 60 makes only 33%. If you go over 50% in a three-divisor base, other things kick in, like that there are only 1/3 of the numbers whose reciprocals are the same size, like 10.80 and 11.30 in twelfty.
Twelfty is 120 dec, as 12 decades. V is teen, the '10' digit, E is elef, the '11' digit. A place is occupied by two staves (digits).
Digits group into 2's and 4's, and . , are comma points, : is the radix.
Numbers writen with a single point, in twelfty, like 5.3, means 5 dozen and 3. It is common to push 63 into 5.3 and viki verka.
Exponents (in dec): E = 10^x, Dx=12^x, H=120^x, regardless of base the numbers are in.
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icarus
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Jun 8 2017, 12:47 PM #9

In this thread we are discussing what are the numbers that are most relevant to everyday human situations and a potential logical extrapolation of these.

Still not sure about racetrack: what are metrological quantities? The bases/fundamental values of systems of measure? Which systems? Current, operational ones like SI and USC?

You're asserting the following, as I understand.

1. Your discussion of money; is this an extant system or a sketch of how base-120 money would be set up? With as many divisors and regular numbers that 120 has {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120}, {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120}, respectively, there are very many avenues for division. I think this has been studied, by Dan from recollection, elsewhere in the forum. Not sure we would have to conform to preserving tenfoldness, etc. unless there are other constraints on division. If we want to explore this in greater depth we can start a new thread.

2. "Reciprocal" here means the following. Let d divide base b. Then "reciprocal" d' is such that b = d * d'. I'll call it "complementary" to avoid the usual definition of reciprocal (admittedly a term I used in the very same way a few years ago having read it in material regarding research into Babylonian sexagesimal arithmetic practices).

3. "Guff" is unwanted complementary divisors d' that are significantly larger and more complex than d. You are suggesting that we have fewer of these if we inflate the base b by doubling it such that we bring sqrt(B) higher. This is fine for some but most people would be hard pressed to deal with a base twice as large as the last, beyond a certain point. Consider children that must be educated in the use of a larger base with twice as many numerals, etc. Not sure where the limit is (it's been discussed) but a number in the hundreds is very likely beyond the capacity of the preponderance of humanity.

4. Your examples of 225 and 3375 in bases 120 and 60 relate to 1/64. Thus, 64 * 225 = 120^2 and 64 * 3375 = 60^3.

I do not follow the percentages in the last part of the last paragraph.

Assuming a two-part digit representation of base 120 where the higher part represents decades and the lower part units, 1,0;8,0 = 1280 and 1,1;3,0 = 1350. These are very large numbers that seem far from any priority usage. If we're considering which numbers have the greatest appeal toward human purpose, these are "far off the highway" and deep in the boonies. Creating a large base just to bring these into the fold is clearly going much too far. However as a tool to intuitively sense factorization of numbers in this range, it's interesting.

At any rate, introducing the notion of a base the size of 120 to settle problems common folk have in everyday life seems like overkill despite its interesting qualities. And I say this as a friend to considering a panoply of bases as tools toward bettering our grasp on arithmetic.
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icarus
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Jun 8 2017, 01:13 PM #10

Granting that the chart I produced is not likely exactly what we are thinking of but understanding that it could be "close", and also granting that using "sub-bases" or two places for each digit of the higher bases in this post muddies things a bit, here are some studies of what the "important numbers" might look like. Tell me if you want to see them in another base I didn't convert them to:

Here is the chart in base 12:

Code: Select all

    1
    2
    3    4
    6    8
    5    9   10   14
    a   16   20   28
    7   13   18   23   30   40   54
   12   26   34   46   60   80   a8
   19   21   24   39   50   68   69   90  100  140  194
   36   42   48   76   a0  114  116  160  200  280  368
    b   2b   53   63   70   84   94   b3  130  180  183  228  230  300  400  540  714
   1a   5a   a6  106  120  148  168  1a6  260  340  346  454  460  600  800  a80 1228
Chart in decimal-coded sexagesimal:

Code: Select all

   1
   2
   3    4
   6    8
   5    9   12   16
  10   18   24   32
  07   15   20   27   36   48  104
  14   30   40   54  112  136  208
  21   25   28   45  100  120  121  148  224  312  416
  42   50   56  130  200  240  242  336  448  624  832
  11   35  103  115  124  140  152  215  300  400  403  520  524  712  936 1248 1704
  22  110  206  230  248  320  344  430  600  800  806 1040 1048 1424 1912 2536 3408
Decimal-coded base 120, with the even-numbered places in dozenal (i.e., "twelfty"):

Code: Select all

    1                                                
    2                                                
    3    4                                             
    6    8                                             
    5    9   12   16                                       
   10   18   24   32                                       
    7   15   20   27   36   48   64                              
   14   30   40   54   72   96  108                              
   21   25   28   45   60   80   81   108   124   172   216                  
   42   50   56   90  100  140  142   196   248   324   432                  
   11   35   63   75   84  100  112   115   160   200   203   280   284   372   496   648   864
   22   70  106  130  148  180  204   230   300   400   406   540   548   724   972  1296  1708
Chart in base-14-coded base 210:

Code: Select all

   1
   2
   3    4
   6    8
   5    9   12   16
  10   18   24   32
  07   15   20   27   36   48  104
  14   30   40   54  112  136  208
  21   25   28   45  100  120  121  148  224  312  416
  42   50   56  130  200  240  242  336  448  624  832
  11   35  103  115  124  140  152  215  300  400  403  520  524  712  936 1248 1704
  22  110  206  230  248  320  344  430  600  800  806 1040 1048 1424 1912 2536 3408
Note that putting on alternate base "glasses" can resolve some but not all the numbers in the triangle. Aside from 5 and "a", the terms in the first 6 rows are regular to dozen. Aside from 7, "14", the first 8 rows have terms regular to bases that have {2,3,5} as distinct prime divisors, etc.

* note: I am leery of the fact that the algorithm I used for base 120 might have introduced errors in the alternating even places despite my checking for them.
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wendy.krieger
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Jun 8 2017, 01:31 PM #11

You can spot the occasional errors in the twelfty, by looking at the sixty-table.

Look for the patterns o4X and o5X in base 60, where o is an odd digit. These produce things like 148 gives 108, in place of 0a8. Likewise 152 gives 112 for e2. The fix would be to look at the routine for values > 99, and apply a test there.

The table for base 210 is actually base 60 repeated.

Also, the thing to look at with regulars, is that they can be fractions. Thus 10.80 can be 1280, but it can also be 10 2/3. Its reciprocal is 11.30 (1350), can be read as 11 1/4. Fractions involving nineths produce large significant integers, like 1.13.40 = 16000, but this should be though of like the decimal 625, which is like a sixteenth.
Twelfty is 120 dec, as 12 decades. V is teen, the '10' digit, E is elef, the '11' digit. A place is occupied by two staves (digits).
Digits group into 2's and 4's, and . , are comma points, : is the radix.
Numbers writen with a single point, in twelfty, like 5.3, means 5 dozen and 3. It is common to push 63 into 5.3 and viki verka.
Exponents (in dec): E = 10^x, Dx=12^x, H=120^x, regardless of base the numbers are in.
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dgoodmaniii
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Jul 6 2017, 03:08 PM #12

icarus @ Jun 6 2017, 09:14 PM wrote: oschkar this is a transform of OEIS A059975. This code generates the data:

Code: Select all

Array[Total[Flatten@ Apply[ConstantArray[#1, #2] &, FactorInteger@ #, 1] - 1] &, 12^2] 
This is really interesting, and would be a great Bulletin article. I shouldn't have been absent from the forum so long; I've been missing stuff like this!

[quote=""Double Sharp""]
I kind of imagine that if we somehow could define some function β(n) where we can plug in some natural number n and get a measure of "usefulness", it should be a multiplicative function. If n is a prime power p^k, I want to say that β(n) = β(p)^k, because "dividing into p parts k times" is exactly dividing into n. Similarly, if n = pq I want to say that β(n) = β(p) * β(q) for the same reason, and the same applies for any set of primes.
[/quote]

I agree that this probably makes it more mechanical than it really is, but it seems to give a pretty good "rough guide," considered anecdotally/completely according to my own use and predilections. I wonder if we can flesh this out to a more comlete theoretical justification? E.g., why should it be multpilicative, and why should primes be given pride of place?
All numbers in my posts are dozenal unless stated otherwise.
For ten, I use :A or X; for elv, I use :B or E. For the digital/fractional/radix point, I use the Humphrey point, ";".
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Kodegadulo
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Jul 6 2017, 03:32 PM #13

Hey, Don, welcome back! Good to see you returning. Don't be a stranger. :)

As for the gist of this thread, I think icarus got it right when he stressed the this concept of "usefulness" is going to be more of a "social science" property, rather than a mathematical one. If it can be defined at all, i think the criteria will be highly subjective, and vulnerable to cultural and ideological biases. "Proving" any hypotheses about it requires actual psychological experimentation and evidence, not computation and syllogisms.

Perhaps a place to start is with the notion of subitizing numbers. Factors of 2, 3, 4 are most "useful" because humans can directly perceive them. But right away we see that, just because "some" is "good", it doesn't necessarily follow that "more" is "better". Factors of five, seven, eleven rapidly fall off in "importance", because they are increasingly difficult to visualize. But we need actual scientific data to confirm that.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
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dgoodmaniii
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Jul 7 2017, 06:30 PM #14

Kodegadulo @ Jul 6 2017, 03:32 PM wrote: Perhaps a place to start is with the notion of subitizing numbers. Factors of 2, 3, 4 are most "useful" because humans can directly perceive them. But right away we see that, just because "some" is "good", it doesn't necessarily follow that "more" is "better". Factors of five, seven, eleven rapidly fall off in "importance", because they are increasingly difficult to visualize. But we need actual scientific data to confirm that.
Hmm; I seem to remember reading some paper or other that confirmed that, but I can't put my finger on it right now. I'll have to dig around a while.
All numbers in my posts are dozenal unless stated otherwise.
For ten, I use :A or X; for elv, I use :B or E. For the digital/fractional/radix point, I use the Humphrey point, ";".
TGM for the win!
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Double sharp
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Dozens Demigod
Joined: Sep 19 2015, 11:02 AM

Nov 8 2017, 08:58 AM #15

An interesting possibility for usefulness and roundness is given in Hardy and Wright, section 22.10, where the functions ω(n) and Ω(n) are defined.

ω(n) is defined as the number of distinct prime factors of n, and Ω(n) as its total number of prime factors. Hence ω(60) = 3 and Ω(60) = 4.

If we declare n to be the product of the primes p_1 through p_r with multiplicities a_1 through a_r, then we have ω(n) = r, Ω(n) = Σ a_i, and d(n) = Π (1 + a_i). Since 1 + a_i is at least 2 and is never more than 2^(a_i), we have sandwiched d(n) between 2^ω(n) and 2^Ω(n).

To make this useful, though, we need to compare ω and Ω values to those of numbers around n. And perhaps we would like to weight various functions, since ω is very unkind to prime powers and Ω is too kind to them.

In the end, usefulness and roundness are subjective measures that are dictated by a number of considerations (not least the base in use; I think we all agree that some of the usefulness of 5 comes from decimal trumping it up, even though it has some value in and of itself). So a good handle on them seems to be obtainable only by examining a lot of possibilities for functions.

It is true that ω and Ω by themselves seem to be quite excellent at finding very convenient numbers to use as groupings in the grand-base range.

Regarding the subitisation idea, it should be remarked that 5 seems to go into the subitising range somewhere in the school years. And furthermore, a trick for reading scales in bases with 5s and 7s as factors is to look for the nearest subdivision, not the immediately preceding ones. For this reason, I would think that factors of 5 and even 7 shouldn't be dismissed out of hand, and that there is a physiological reason to want 5 around if we can get it to maximise our subitising abilities.
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