## The accidental story of two times five and base ten

Humans have used many different bases for number systems but the use of base 10 is overwhelmingly dominant. There are instances of the use of base 5, base 6, base 20 and even base 27. In spite of many attempts to replace it by base 10, base 60 has fended off all rationalist suggestions and remnants remain entrenched for our current mapping of time and space. For time periods, base 60 is used exclusively for hours, minutes and seconds but base 10 for subdivisions of the second. Similarly for spatial coordinates, degrees, minutes and seconds of arc are still used but subdivisions of the second use base 10. (Some of the other bases that appear in language are listed at the end of this post).

In terms of mathematics there is no great inherent advantage in the use of one particular number base or another. The utility of a particular choice is a trade off first between size and practicality. The size of the base determines how many unique number symbols are needed (binary needs 2, decimal needs 10 and hexagesimal 16). There are many proponents of the advantages of 2, 3, 8, 12 or 16 being used as our primary number base. Certainly base 12 is the most “fraction friendly”. But all our mathematics  could, in reality, be performed in any number base.

At first glance the reasons for the use of base 10 seems blindingly obvious and looking for origins seems trivial. Our use of base 10 comes simply – and inevitably – from two hands times five digits. In recent times other bases (binary – base 2- and hexadecimal – base 16 – for example) are used more extensively with computers, but base 10 (with some base 60) still predominates in human-human interactions (except when Sheldon is showing off). The use of base 10 predates the use of base 60 which has existed for at least 5,000 years.

It is ubiquitous now but (2 x 5) is not a consequence of design. It derives from a chain of at least three crucial, evolutionary accidents which gave us

1. four limbs, and
2. five digits on each limb, and finally
3. human bipedalism which reserved two limbs for locomotion and left our hands free.

The subsequent evolutionary accidents which led to increased brain size would still have been necessary for the discovery of counting and the invention of number systems. But if, instead of two, we had evolved three limbs free from the responsibilities of locomotion, with three digits on each limb, we might well have had base 9 at the foundations of counting and a nonary number system. The benefits of a place value system and the use of nonecimals would still apply.

It is more difficult to imagine what might have happened if limbs were not symmetrical or the number of digits on each limb were different. No human society has not been predominantly (c. 85%) right-handed. But left-handedness has never been a sufficient handicap to have been eliminated by evolution. Almost certainly right-handedness comes from the asymmetrical functions established in the left and right-brains. The distinction between the functions of the two sides of the brain goes back perhaps 500 million years and long before limbs and tetrapods. By the time limbs evolved, the brain functions giving our predilection for right-handedness must already have been established. So, it is possible to imagine evolution having led to, say, 6 digits on right fore-limbs and 5 digits on left fore-limbs.

I wonder what a natural base of 11 or 13 would have done to the development of counting and number systems?

#### Why four limbs?

All land vertebrates (mammals, birds, reptiles and amphibians) derive from tetrapods which have two sets of paired limbs. Even snakes evolved from four-limbed lizards.

Tetrapods evolved from a group of animals known as the Tetrapodomorpha which, in turn, evolved from ancient sarcopterygians around 390 million years ago in the middle Devonian period; their forms were transitional between lobe-finned fishes and the four-limbed tetrapods. The first tetrapods (from a traditional, apomorphy-based perspective) appeared by the late Devonian, 367.5 million years ago. Wikipedia

It would seem that – by trial and error – a land-based creature, fortuitously possessing two pairs of limbs, just happened to be the one which survived and become the ancestor of all tetrapods. The evolutionary advantage of having 4 limbs (two pairs)  – rather than one or three or five pairs – is not at all clear. Insects have evolved three pairs while arachnids have four pairs. Myriapoda are multi-segmented creatures which have a pair of limbs per segment. They can vary from having five segments (10 legs) to about 400 segments (800 legs). The genes that determine the number of limbs determine many other features also and why two pairs would be particularly advantageous is not understood.  It could well be that the two pairs of limbs were incidental and merely followed other survival characteristics. The best bet currently is that

“You could say that the reason we have four limbs is because we have a belly,”

All of us backboned animals — at least the ones who also have jaws — have four fins or limbs, one pair in front and one pair behind. These have been modified dramatically in the course of evolution, into a marvelous variety of fins, legs, arms, flippers, and wings. But how did our earliest ancestors settle into such a consistent arrangement of two pairs of appendages? — Because we have a belly.

According to our hypothesis, the influence of the developing gut suppresses limb initiation along the midgut region and the ventral body wall owing to an “endodermal predominance.” From an evolutionary perspective, the lack of gut regionalization in agnathans reflects the ancestral absence of these conditions, and the elaboration of the gut together with the concomitant changes to the LMD in the gnathostomes could have led to the origin of paired fins.

The critical evolutionary accident then is that the intrepid sea creature which first colonised the land, some 390 million years ago, and gave rise to all tetrapods was one with a developing belly and therefore just happened to have two pairs of appendages.

The tail, however, is an asymmetrical appendage which may also once have been a pair (one on top of the other) but is now generally a solitary appendage. But it is controlled by a different gene-set to those which specify limbs. In mammals it has disappeared for some and performs stability functions for others. In some primates it has functions close to that of a fifth limb. But in no case has a tail ever evolved digits.

#### Why five digits on each limb?

When our ancestor left the oceans and became the origin of all tetrapods, four limbs had appeared but the number of digits on each limb had not then been decided. It took another 50 million years before a split distinguished amphibians from mammals, birds and reptiles. The timeline is thought to be:

• 390 million years ago – tetrapod ancestor leaves the oceans
• 360 million years ago – tetrapods with 6,7 and 8 digits per limb
• 340 million years ago – amphibians go their separate way
• 320 million years ago – reptiles slither away on a path giving dinosaurs and birds
• 280 million years ago – the first mammals appear

SciAm

The condition of having no more than five fingers or toes …. probably evolved before the evolutionary divergence of amphibians (frogs, toads, salamanders and caecilians) and amniotes (birds, mammals, and reptiles in the loosest sense of the term). This event dates to approximately 340 million years ago in the Lower Carboniferous Period. Prior to this split, there is evidence of tetrapods from about 360 million years ago having limbs bearing arrays of six, seven and eight digits. Reduction from these polydactylous patterns to the more familiar arrangements of five or fewer digits accompanied the evolution of sophisticated wrist and ankle joints–both in terms of the number of bones present and the complex articulations among the constituent parts.

By the time we reach the mammals, five digits per limb has become the norm though many mammals then follow paths for the reduction of the number of effective digits in play. Moles and pandas evolve an extra sort-of adjunct digit from their wrists but do not (or cannot) create an additional digit.

…….. Is there really any good evidence that five, rather than, say, four or six, digits was biomechanically preferable for the common ancestor of modern tetrapods? The answer has to be “No,” in part because a whole range of tetrapods have reduced their numbers of digits further still. In addition, we lack any six-digit examples to investigate. This leads to the second part of the answer, which is to note that although digit numbers can be reduced, they very rarely increase. In a general sense this trait reflects the developmental-evolutionary rule that it is easier to lose something than it is to regain it. Even so, given the immensity of evolutionary time and the extraordinary variety of vertebrate bodies, the striking absence of truly six-digit limbs in today’s fauna highlights some sort of constraint. Moles’ paws and pandas’ thumbs are classic instances in which strangely re-modeled wrist bones serve as sixth digits and represent rather baroque solutions to the apparently straightforward task of growing an extra finger.

Five digits is apparently the result of evolutionary trial and error, but as with all things genetic, the selection process was probably selecting for something other than the number of digits.

Science Focus

All land vertebrates today are descended from a common ancestor that had four legs, with five toes on each foot. This arrangement is known as the pentadactyl limb. Some species have subsequently fused these fingers into hooves or lost them altogether, but every mammal, bird, reptile and amphibian traces its family tree back to a pentadactyl ancestor that lived around 340 million years ago. Before, there were animals with six, seven and even eight toes on each foot, but they all went extinct at the end of the Devonian period, 360 million years ago. These other creatures were more aquatic than the pentadactyl animals. Evidence in the fossil record suggests that their ribs weren’t strong enough to support their lungs out of water and their shoulder and hip joints didn’t allow them to walk effectively on land.

Five digits on our limbs are an evolutionary happenstance. There is nothing special that we can identify with being five. It could just as well have been six or seven or eight. That the number of digits on each limb are not unequal is also an evolutionary happenstance predating the tetrapods. It is more efficient genetically, when multiple limbs are needed, to duplicate the pattern (with some variations for mirror symmetry and for differences between paired sets). When each limb is to carry many digits it is more efficient to follow a base pattern and keep the necessary genetic variations to a minimum.

By 280 million years ago, four limbs with five digits on each limb had become the base pattern for all land-based creatures and the stage was set for base 20. And then came bipedalism.

#### Why bipedalism?

Bipedalism is not uncommon among land creatures and even birds. Some dinosaurs exhibited bipedalism. Along the human ancestral line, bipedalism first shows up around 7 million years ago (Sahelanthropus). It may then have disappeared for a while and then appeared again around 4 million years ago in a more resilient form (Australopithecus) which has continued through till us. What actually drove us from the trees to bipedalism is a matter of many theories and much conjecture. Whatever the reasons the large brain evolved only in bipedal hominins who had a straightened spine, and who had maintained two limbs for locomotion while freeing up the other two for many other activities. The advantages of being able to carry things and throw things and shape things are considered the drivers for this development. And these two free limbs became the counting limbs.

It seems unlikely that a large brain could have developed in a creature which did not have some limbs freed from the tasks of locomotion. Locomotion itself and the preference for symmetry would have eliminated a three-limbed creature with just one free limb.

Two limbs for counting, rather than 3 of 4 or 4 of 4, is also happenstance. But it may be less accidental than the 4 limbs to begin with and the 5 digits on each limb. An accidental four limbs reduced inevitably to two counting limbs. Together with an accidental five digits they gave us base 10.

Other bases

1. Oksapmin, base-27 body part counting

The Oksapmin people of New Guinea have a base-27 counting system. The words for numbers are the words for the 27 body parts they use for counting, starting at the thumb of one hand, going up to the nose, then down the other side of the body to the pinky of the other hand …… . ‘One’ is tip^na (thumb), 6 is dopa (wrist), 12 is nata (ear), 16 is tan-nata (ear on the other side), all the way to 27, or tan-h^th^ta (pinky on the other side).

2. Tzotzil, base-20 body part counting

Tzotzil, a Mayan language spoken in Mexico, has a vigesimal, or base-20, counting system. ….. For numbers above 20, you refer to the digits of the next full man (vinik). ..

3. Yoruba, base-20 with subtraction

Yoruba, a Niger-Congo language spoken in West Africa, also has a base-20 system, but it is complicated by the fact that for each 10 numbers you advance, you add for the digits 1-4 and subtract for the digits 5-9. Fourteen (??rinlá) is 10+4 while 17 (eétàdílógún) is 20-3. So, combining base-20 and subtraction means 77 is …. (20×4)-3.

4. Traditional Welsh, base-20 with a pivot at 15

Though modern Welsh uses base-10 numbers, the traditional system was base-20, with the added twist of using 15 as a reference point. Once you advance by 15 (pymtheg) you add units to that number. So 16 is un ar bymtheg (one on 15), 36 is un ar bymtheg ar hugain (one on 15 on 20), and so on.

#### 5. Alamblak, numbers built from 1, 2, 5, and 20

In Alamblak, a language of Papua New Guinea, there are only words for 1, 2, 5, and 20, and all other numbers are built out of those. So 14 is (5×2)+2+2, or tir hosfi hosfihosf, and 59 is (20×2)+(5x(2+1))+(2+2) or yima hosfi tir hosfirpati hosfihosf.

#### 6. Ndom, base-6

Ndom, another language of Papua New Guinea, has a base-6, or senary number system. It has basic words for 6, 18, and 36 (mer, tondor, nif) and other numbers are built with reference to those. The number 25 is tondor abo mer abo sas (18+6+1), and 90 is nif thef abo tondor ((36×2)+18).

#### 7. Huli, base-15

The Papua New Guinea language Huli uses a base-15, or pentadecimal system. Numbers which are multiples of 15 are simple words. Where the English word for 225 is quite long, the Huli word is ngui ngui, or 15 15. However 80 in Huli is ngui dau, ngui waragane-gonaga duria ((15×5)+the 5th member of the 6th 15).

#### 8. Bukiyip, base-3 and base-4 together

In Bukiyip, another Papua New Guinea language also known as Mountain Arapesh, there are two counting systems, and which one you use depends on what you are counting. Coconuts, days, and fish are counted in base-3. Betel nuts, bananas, and shields are counted in base-4. The word anauwip means 6 in the base-3 system and 24 in the base-4 system!

#### 9. Supyire, numbers built from 1, 5, 10, 20, 80, and 400

Supyire, a Niger-Congo language spoken in Mali has basic number words for 1, 5, 10, 20, 80 and 400, and builds the rest of the numbers from those. The word for 600 is kàmpwòò ná ?kwuu shuuní ná bééshùùnnì, or 400+(80×2)+(20×2)

#### 10. Danish, forms some multiples of ten with fractions

Danish counting looks pretty familiar until you get to 50, and then things get weird with fractions. The number 50 is halvtreds, a shortening of halv tred sinds tyve (“half third times 20” or 2½x20). The number 70 is 3½x20, and 90 is 4½x20.

#### 11. French, mix of base-10 and base-20

French uses base-10 counting until 70, at which point it transitions to a mixture with base-20. The number 70 is soixante-dix (60+10), 80 is quatre-vingts (4×20), and 90 is quatre-vingts-dix ((4×20)+10).

#### 12. Nimbia, base-12

Even though, as the dozenalists claim, 12 is the best base mathematically, there are relatively few base-12 systems found in the world’s languages. In Nimbia, a dialect of the Gwandara language of Nigeria, multiples of 12 are the basic number words around which everything else is built. The number 29 is gume bi ni biyar ((12×2)+5), and 95 is gume bo’o ni kwada ((12×7)+11).

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