HOW TEN LOSES TO TIME
IN FINGER COUNTING

INTRODUCTION

Worldwide, contemporary human civilization is still affected enormously by its choice of ten as the base of its most frequently used 'decimal' numeral system in everyday life. People who are so strongly affected psychosocially that one could call them "ten-fingered fetishists" even arrange their greatest celebrations around the number ten, whether it concerns their personal days of birth and death, historic events decades or hundreds of years ago or less often occurring millennium festivals. I hope i^* will not need to prove that there is neither a logical nor a physical connection whatsoever between universal time and the empirical fact that human beings have almost always two hands with five fingers each, ten together. But on top of that it turns out that there is not even a fundamental connection between these ten fingers and the number we can count up to by using them in finger counting or 'dactylonomy'! Quite ironically, time does play a decisive role here in a different way when we consider the several arithmetic stages civilization has gone thru^**:

• The first stage was that of addition only, a stage without any awareness of multiplication as an operation in itself.
• The second stage saw the introduction of a systematic repetition of addition (iterative addition) in what became to be called "multiplication"
• And in the third stage humankind grew familiar with the systematic repetition of multiplication (iterative multiplication) in the shape of raising a number (the base) to another number (the exponent) in exponentiation.

I will discuss how the maximum numbers which can be reached in each of these arithmetic stages vary considerably, and show that in the dactylonomic maximum of our present-day third stage there is nothing left anymore which favors the decimal-denary system. (Decimal-denary because a complete numeral system is not supposed to deal with 'fractions' exclusively, nor with integers exclusively.)

IN THE ERA OF MERE ADDITION

The average 'base-tenner' never thought about it that with five fingers a human body also has four spaces between these fingers, and that some fellow humans once used to base their numeral system on the number of these spaces; or that a human body does not only normally have ten fingers, but also ten toes, that is, twenty digits altogether, and that elsewhere this number was a reason to adopt a base-twenty system instead. Not only do human bodies normally have all these things, their fingers, in turn, normally have fourteen clearly distinguishable parts with their own bones or 'phalanxes' as well: three each on the digits which anyone will call "fingers" and two on the thumb. Even if we confine ourselves to addition we have always been able to count to twelve or fourteen, or up to twenty-four or twenty-eight. It can be argued that the twelve phalanxes are the (or a) source of the ancient base-12 numeral system. However, this 'phalanx counting' differs very much from finger counting in its limited public visibility and, more importantly, in that you cannot activate a phalanx in the way you can activate a finger by stretching it. I will therefore discuss finger counting proper, and only mention phalanx counting occasionally for comparison.

Whether four- or eight manual interdigital spaces, ten fingers, sixteen interdigital spaces on hands and feet or twenty digits (or whether twelve or fourteen phalanxes, whether on one or two hands, or on two hands and two feet), it is generally taken for granted that with these parts of the body one cannot count any further than the number of these parts That conclusion, however, is plainly false, or only true with an additional assumption of primitivity.

No doubt, primeval finger counting consisted of something like holding up one hand and using the forefinger of the other to point successively to each of the digits (thumb or 'fingers' in a narrow sense) on the hand held up until the number of (thumb and) fingers thus pointed to was the same as the number of things one had in mind. If five was not enough, you would just use the fingers of both your hands to point to and to be pointed to, until ten, the end. (When counting the phalanxes of your fingers other than your thumbs you can use your thumb to count their number on the same hand.) Those were the days of addition and the beginning of number naming, when an iterative operation such as multiplication was still too abstract to be comprehended. People literally did not 'know better' then. And without multiplication in their tool box, the idea of assigning different values to their fingers, values different from the elementary one to ten (or one to twenty-four in phalanx counting), did not emerge, simply because there was no framework within which to put them. All you could do after reaching ten was to lay a little twig, stick or suchlike thing in front of you and return to your first finger, a thumb or little finger, to start a new round of ten again. The primeval numeral system simply amounted to a base-one or 'unary' system, a system which works and suffices for the smallest of integers.

    1          1    
  1_|_1      1_|_1  
1___|___1  1___|___1

AFTER THE ADVENT OF MULTIPLICATION

But reality was pushing. Counting beyond ten was possible by remembering the number of ten-finger rounds, but remembering more and more rounds could not go on indefinitely. Two rounds was no problem, because if you even could not remember those, you still had twenty digits, fingers and toes, on your own body. The villain of the scene, you may imagine, was the then still unnamed first number ('twenty-one') after the two rounds or twenty digits. This number and its successors were part of the third round, and whether the numbers of the second round had already names or not, people had to come up with something more systematic, if only to be able to name all these numbers, since you cannot go on wasting and/or remembering numerical morphemes forever. (The very reason why a consistent truly base-60 numeral system is impossible in practice.) And yet, the rounds of ten-fingered counting introduced repetition in arithmetic: the repetition of adding up to ten, a process in which the first ten was still ten, but the second ten did not refer to just ten anymore, but to two times ten, the third to three times ten, the fourth to four times ten, and so on ad infinitum. Multiplication was born, if not out of necessity, for the sake of convenience.

Among those who continued to count with their number of left- and right-hand fingers, and not the spaces between them, and with full disregard of their toes, the ultra-primitive unary system was replaced with a more 'civilized' prototype of the base-ten system when the official counting was not done anymore with fingers in the air, but with the symbols for numbers carved in bark, clay or stone, written on paper or another physical medium, including light pixels on a dark background. With this concretion of counting in the shape of symbols and, later, whole formulaic expressions, ten-finger counting never got a chance anymore to prove its incredible numerical power, a power reaching far beyond the number ten. Up to today the simple-minded are sure that you cannot count further on your fingers than ten; that counting further than ten would require little sticks or suchlike physical means to keep track of the numerical record.

With our modern notion of numeral systems and the base or 'radix' of a numeral system, we know that the symbols for numbers, that is figures, in the denary system are assigned different values dependent on the place of the figure in the full number notation. Why couldn't we assign place values to fingers in a similar way? In primitive ten-finger counting every finger is implicitly assigned the value 1, but with multiplication having entered our toolbox we could assign different values to different fingers, albeit without making explicit use of exponentiation (which is not yet in the toolbox of the time). Thus, we could consider a 'base-5' system with multiples of 5, or a 'base-10' system with multiples of 10, as in the arithmetric sequences {5,10,15,20,25,30,35,40,45,50} and {10,20,30,40,50,60,70,80,90,100} of which 5 and 10 are the common differences. However, this would not make continuous counting possible, if only because 1 to 4 are not covered, or, worse, 1 to 9. Nonetheless, we have two hands and can split things up:

• the light hand for unit values (among which all single values needed to fill up the gaps between the multiples)
• the heavy hand for multiplication values (the multiples of whatever number that serves as a base of multiplication)

If there are fewer than five unit values, we can, of course, start with one or more of the lower multiplication values on the light hand. (Note that in this context i do not apply the terms left and right to our hands, altho it has the added advantage of doing away with any skewed connotation these terms may have in some circles. The reason is that in the communication between people facing one another there must be agreement, not on what is to be called "right" or "left" —with my right being your left, and vice versa— but on what both the sender and the receiver of the numerical message will have to consider the light and the heavy hand.)

What is most appealing about a multiplication base of 5, is that with five unit values the values of the light-hand fingers will be 1, 2, 3, 4 and 5, after which the values of the heavy-hand fingers are multiples of 5: 10, 15, 20, 25 and 30. The whole light hand will have unit values then, and the whole heavy hand multiples only. (I will call the unit values before the base "single (unit) values" or "singles" for short.) All numbers between the numbers assigned as place values can be shown in dactylonomic equations: 6=5+1, 7=5+2, 8=5+3, 9=5+4, 11=10+1, 12=10+2, 13=10+3, 14=10+4, 16=15+1, et cetera. Above 30 we can make use of equations such as 35=30+5, 40=30+10, 45=30+15, 50=30+20, 55=30+25, 60=30+20+10, 65=30+20+15, 70=30+25+15, 75=30+25+20, 80=30+25+15+10, 85=30+25+15+10+5, 90=30+25+20+15, 95=30+25+20+15+5, 100 = 30+25+20+15+10, 105 = 30+25+20+15+10+5, 110 = 30+25+20+15+10+5+4+1 and 115 = 30+25+20+15+10+5+4+3+2+1. All addends are arranged in order, descending here, but ascending would also be possible, depending on where the light and heavy hands are located. What is absolutely impossible, however, is to use a place value which is assigned only once more than once. For example, after 80=30+25+15+10 you might be inclined to make 85=30+25+15+15, but that would transgress against this rule. The number 115 is the dactylonomic maximum on this scheme, because, firstly, it makes use of all place values, without making use of any place value more than once, and, secondly, because all values between 1 and 115 can also be expressed with the same place values. Just like we expressed the numbers 6 to 9, so we can express the numbers 31 to 34 as 31=30+1, 32=30+2, 33=30+3, 34=30+4, and 111 to 114 as 111 = 30+25+20+15+10+5+4+2, 112 = 30+25+20+15+10+5+4+3, 113 = 30+25+20+15+10+5+4+3+1 and 114 = 30+25+20+15+10+5+4+3+2. Whatever you may think of this (not yet perfect) scheme, it is already a great leap forward from the ancient ten.

    3           20     
  2_|_4      15_||_25  
1___|___5  10___||___30

At first sight it looks like we could never use a multiplication base of 10, because we would need nine fingers first with the unit values 1 to 9; and that followed by 10, which would not even leave one finger for the multiples of 10. Fortunately, this train of thought is mistaken, because 3=1+2, so we do not need a 3-valued finger, so long as we have a 4-valued finger; we just activate the 1-valued pinkie, if that is the one, and the 2-valued 'fourth' finger (counting from the thumb) by stretching them. (The terms ring finger and annular finger are not biologically or otherwise acceptable, while it may also be the second finger, rather than the fourth.) A 5-, 6-, 7-, or 9-valued finger is likewise not necessary, because their values can also be expressed by means of two lower-valued fingers or three lower-valued ones in the case of the number 7=4+2+1. (Strictly speaking, the 8-valued finger is not absolutely necessary either, as long as we have a 5-, 6- or 7-valued finger, but the choice of 8 yields the highest maximum.) So, if we assigned the single values 1, 2, 4, 8 and 10 to the fingers of the light hand, and the multiples 20, 30, 40, 50 and 60 to the fingers of the heavy hand, we could count on our ten fingers until (1+2+4+8+10) + (20+30+40+50+60) = 25+200=225. This can be shown schematically as:

    4            40     
  2_|_8       30_||_50  
1___|___10  20___||___60

In the previous scheme we moved from multiples of 5 to multiples of 10, but it left us with an ugly, if not dirty, light hand in which the transition of the unit value 8 to the unit value 10 is purely ad hoc. So, let us return to the beautiful light hand of the earlier scheme, and completely forget about 10 as a multiplication base. What should that base be, then, in order to produce a higher maximum? It follows from the light-hand scheme with place values 1, 2, 3, 4 and 5 that any base will do that does not produce intervals larger than 1+2+3+4+5=15. By stretching all five fingers of the light hand we can express any number that is 15 larger than a place value on the heavy hand. The number 16 cannot be formed anymore in this way (since we can stretch each finger only once). Hence, with this light hand, the value 16 should be assigned to the next finger on the heavy hand, after which the other fingers automatically receive the values 32, 48, 64 and 80, the multiplication base being 16. On this scheme the maximum number is 15+(16+32+48+64+80) = 15+240=255 for the two hands together. As far as the heavy hand is concerned this leaves neither any number between 0 and 256 that cannot be represented at all nor any number that can be represented in more than one way.

    3           48     
  2_|_4      32_||_64  
1___|___5  16___||___80

The light hand is a different matter, because of the superfluity of first 3 and then 5, which contribute 8 to the maximum, but prevent 96 and 112 from contributing much more — one would say. This reasoning is erroneous, however: by reducing the number of unit values to three (1, 2 and 4) their sum will only be 1+2+4=7 and the distance between multiples on the heavy hand can be no more than 7, so that the multiplication base cannot be larger than 8. The result on this purist scheme is therefore (1+2+4+8+16) + (24+32+40+48+56) = 31+200=231, 24 less than on the last scheme above. Yet, this is merely the first variant of it, as we may also consider 8 a unit value, so that the distance between multiples may be as large as 1+2+4+8=15, in which case the maximum will be (1+2+4+8+16) + (32+48+64+80+96) = 31+320=351, 96 more than in the last scheme. And yet, with our five fingers on the light hand, we may even consider 16 a unit value so that the largest distance allowed between multiples on this scheme will be 1+2+4+8+16=31. This means that the first finger on the heavy hand should be given the value 32. (The numbers between 16 and 32 will then have the values 17=16+1, 18+16+2, 19=16+2+1, 20=16+4, 21=16+4+1, 22=16+4+2, 23=16+4+2+1, 24=16+8, 25=16+8+1, 26=16+8+2, 27=16+8+2+1, 28=16+8+4, 29=16+8+4+1, 30=16+8+4+2, 31=16+8+4+2+1.) From now on the last four fingers can be assigned multiples of 32: 64, 96, 128 and 160. With a multiplication base of 32 we will reach the largest maximum for a scheme of this type. Our ten-finger counting will now get us as far as (1+2+4+8+16) + (32+64+96+128+160) = 31+480=511, 256 more than the last scheme shown above!

    4           96      
  2_|_8       64||128   
1___|___16  32__||___160

AFTER THE ADVENT OF EXPONENTIATION

In this rather heuristic way we have ascertained that it is possible to count from 1 up to 511 on our ten fingers without leaving any integer gap in between; even without any number having more than one representation in finger language. This is already an enormously far cry from the days in very ancient times when people could not count further than ten on their fingers. In that remote past, we may assume, people never thought about some kind of place-value system, while —or, rather, precisely because— multiplication was not known at the time. Yet, after iterative addition, that is, multiplication, has lifted us up from 10 (ten) to 511 (five hundred and eleven), nowadays we also have easy access to iterative multiplication, that is, exponentiation. Will the maximum number to which we can count on our ten fingers now move even further to six, seven, eight hundred or more? And, if so, is there such a number, then, that can be proved to be the last one, the maximum of dactylonomic maximums?

Radix-ten addicts who are most familiar with ten and the denary morphemes hundred, thousand and million which follow in its wake may be delighted now. For what is more self-evident with a ubiquitous radix of ten and a modern operation of exponentiation at your disposal than that your ten fingers should be assigned the place values 1 (10ˆ0), 10 (10ˆ1), 100 (10ˆ2), 1000 (10ˆ3), 10 000 (10ˆ4), 100 000 (10ˆ5), 1 000 000 (10ˆ6), 10 000 000 (10ˆ7), 100 000 000 (10ˆ8) and 1 000 000 000 (10ˆ9)? Unfortunately, it is far too soon for them to start cheering; for the same reason that in the era of multiplication, when exponentiation was still inconceivable, we could not assign the place values 10 (1×10), 20 (2×10), 30 (3×10) up to 100 (10×10), even not the place values 1, 10 (1×10), 20 (2×10) up to 90 (9×10), to our ten fingers. And whereas the gaps of inexpressible integers left before 10 and between 10 and 20, 20 and 30, et cetera are relatively small in a scheme of multiples of ten, these gaps grow ever bigger in a scheme of powers of ten. For example, you could express the numbers 10, 100 and 110=100+10, or 1, 10 and 11=1+10, but not any of the numbers before 10 or between 1 and 10; and with the numbers 100 000 000 and 1 000 000 000 you could express numbers such as 100 000 010 or 100 000 001 and 111 111 110 or 111 111 111, but not the great majority of other numbers in between. (Don't forget that you can only use a finger once; that you cannot use your 1-valued finger twice in order to express the number 2, or your 100-valued finger twice in order to express the number 200.)

Perhaps, we should, again, distinguish between unit values on the light hand, and now powers instead of multiples on the heavy hand. It should be immediately clear then, that a scheme such as the ones above with a maximum of 115 and of 255, in which the unit values were 1, 2, 3, 4 and 5 will not be acceptable with powers of 16 and 10 on the heavy hand. With base 16 it means that 32 (2×16) will be replaced with 256 (16^2), but the range that can be covered is still 1+2+3+4+5=15, which leaves a gap of unrepresented integers from 32 to 255 inclusive. With base 10 it means that 20 will be replaced with 100, but also here this leaves a gap, albeit a smaller one, from 10+15+1=26 up to 99. We should, therefore, stick to the light-hand place values of the last scheme with enabled us to count up to 511; at least insofar as they are not larger than the base of the powers which follow. (In this stage we may look at the set {1,2,4,8,16} as lower powers of 2, altho they were selected because they turned out to be the minimum number of smallest numbers needed to fill up the gaps on the heavy hand.) After 16 and 10, let us consider 5 as base of the powers. The scheme will then start with 1, 2, 4, 5, 25, ... Since 1+2+4=7 and 5+7=12, it will not be possible to represent 13 to 24, and we will have to abandon this scheme too. A scheme with base 4 for the powers will start with 1, 2, 4, 16, ..., and since 1+2=3 and 4+3=7, it will not be able to represent the numbers 8 to 15, and many other integers to follow later. A scheme with base 3 for the powers will fare no better or, more appropriately, it will fare less badly: starting with 1, 2, 3, 9, ... it will leave the numbers 7 and 8 unaccounted for even before we have reached 10. It is not a watertight mathematical proof which i have given here, yet enough reason not to consider mixed schemes anymore with powers of 2 on the (beginning of the) light hand and powers of 3 or larger on the heavy hand.

From the moment we recognized multiplication as a presently existing operation suitable for our purpose we opened the door for the introduction of finger place values. However, we cannot but notice that rash applications of the concept of dactylonomic place values may lead to deplorable drawbacks rather than an advancement in the search for the largest number in ten-finger counting; that is, the largest number that does not leave any integer gap on the way! It is high time i explicitly listed all the conditions that apply in any search for the largest number that can be reached in finger counting:

• You have ten fingers, and no other things, at your disposal. (The fact that five of them are on your left, and five on your right hand, may but does not have to be taken into account.)
• To each finger only one positive integer may be assigned as place value. (This does not preclude two or more fingers from having the same place value.)
• There are only two finger positions: stretched or folded (like on or off). In a stretched position a finger or set of fingers is activated, in a folded position it or they are not. (Other) gestures do not 'count'.
• Every finger may be activated only once. (If so, its place value will count; if not, it will not count.)
• The number shown is the sum of the place values of the fingers stretched or otherwise activated. (A place value can only occur once in the addition, unless it is assigned to a number of different fingers not smaller than the number of times the same place value is added.)
• A number may only be claimed to be the largest that can be shown, if all integers between 1 and that number, including 1 and the number itself, can be expressed by means of activating or not activating one or more of your ten fingers. (It remains possible that a number can be expressed in more than one way.)

In my last scheme from the era of multiplication, in which the light hand of units covers precisely, like the heavy hand of multiples, one hand of fingers, the numbers 1, 2, 4, 8 and 16 were not selected but left over, because the other numbers in the same range were the sum of two or three other, smaller numbers. Nevertheless, it will strike a person familiar with exponentiation immediately that 1, 2, 4, 8 and 16 are successive integer powers of 2: 2^0, 2^1, 2^2, 2^3 and 2^4. When looking at each new subrange of integers added to each old range, the procedure starts with 1 (2^0), after which 2 (2^1) follows, with the old range containing one integer (1) and the new range two integers, among which one added integer (2); after 2 the power 4 (2^2) follows, which adds two integers (3 and 4) to the two previous ones; after 4 the power 8 (2^3) follows, which adds four integers (5, 6, 7 and 8) to the four previous ones; and after 8 the power 16 (2^4) follows, which adds eight integers (9, 10, 11, 12, 13, 14, 15 and 16) to the eight previous ones, exactly the same number as the number of integers which the old range contains already. This pattern is of tremendous importance, because it demonstrates that every newly added integer between the last, new power and the previous, old power can be expressed as the sum of that previous power and one or more smaller powers.
If 2ⁿ is the last power and 2^m the second-last (m=n-1), then for any integer x between these two (2^m<x<2ⁿ), and for either a_i=0 or a_i=1, it holds that

(Altho purely arithmetically also 2ⁿ=2^m+2^m, we cannot claim this in our finger place value scheme, because by using the same value twice we are bound to miss out on a higher place value which could be useful later on.)

When selecting the powers of 2 as place values in finger counting each added range of integers is of the same length as the range added to. This unique feature was the underlying reason why we assigned the first five powers of two to the five fingers on the light hand, and now it should be the very same reason for assigning the following five powers of two to the five fingers on the heavy hand. In other words, to get the largest maximum the total distribution of place values must be (1+2+4+8+16) + (32+64+128+256+512) = 31+992=1023. (Note that the numbers 1 to 512 are the powers 2^0 to 2^9, but that the maximum number we can express this way is 2^10-1=1024-1=1023.) If my pretty well-founded conjecture above (that all schemes which mix powers of 2 with other powers such as 3 leave gaps of integers which cannot be expressed) is accepted as true, 1023 is not just some binary maximum, but the maximum for any type of ten-finger counting.

   4          128      
 2_|_8      64_|_256   
1__|__16  32___|____512

Something definitely remarkable occurred on the path thru time we have trodden. We started off with the idea that we could not count any further on our ten fingers than the number ten, while taking mere addition for granted. We even shared the common belief that there is an intimate, if not exclusive, relationship between the human body counting ten fingers and the historical hegemony of the base-ten numeral system. Nevertheless we have now arrived at the conclusion that we cannot just count up to ten on our ten fingers, but up to one thousand and twenty-three instead. And that there is a most intimate relationship between this 1023, not at all with the decimal-denary numeral system, but with the binary numeral system which is part of a micromacrobinary supersystem. Those who have six fingers on each hand, that is, twelve together (which is called "bilateral polydactyly") should even be able to count up to 4095, because 2^12=4096.

POSTSCRIPT

When i wrote the draft for this article i came to the conclusion that the largest number one could reach in finger counting was 1023. I considered it my own discovery, because i had never heard or read about someone arguing the same. However, between the draft and the final version i decided to have a closer look at what others might have written about this subject, and had published on the internet. That is when i hit upon the Wikipedia article Finger-counting (at https://en.wikipedia.org/[]wiki/Finger-counting) with a special section titled "Non-decimal finger-counting". That section has a two-sentence paragraph about binary finger counting, called "finger binary", and a four-sentence paragraph about senary finger counting (which counts up to only 35, but which also explains how to express sixths and thirty-sixths). In a special article on the subject, finger binary is defined as system for counting and displaying binary numbers on [one's] fingers. This illustrates very well how my own article above takes an opposite approach: altho i have in no uncertain terms displayed an aversion to any exclusive or disproportionate attention paid to the decimal-denary system, i did not in this article start out from any particular other numeral system and wanted to know what the maximum was regardless of the numeral system used. Displaying binary numbers is not at all the purpose of the present article. Hence, i decided to publish it, in spite of my not being the discoverer of the binary number 1111111111 in its capacity as dactylonomic maximum for the ten-fingered.