The Isopsephic Numerals

1.2.8?

The Isopsephic Numerals

Introduction

As we have seen throughout the last few articles, denominational systems became the dominant way to write numbers, replacing the more limited and older tally systems. We saw how they greatly improved peoples ability to grasp and manipulate larger numbers than before. We also saw however, that even in its most sophisticated form, exemplified by roman numerals, the system was little more than a glorified tally system, extending the range at best by a few orders of magnitude. The Attic system in particular proved to be only useful for a fairly limited range of numbers, less, it turns out, then even the Egyptian system.

But Number notation and writing in general became forever changed with the introduction of phonetic alphabets, in which each character represented ideally a single sound of the spoken language.

As we learned, language in the ancient world was a mess. Both the mesopotamian and Egyptian scripts were greatly limited by the fact that there most basic units of information were "nouns", and they did not lend themselves well to describing the far richer and more developed oral tradition of spoken language. The denomination systems reflect this early writing system in there very literal interpretation. But this also limited them quite a bit making them clunky and requiring the repetition of symbols.

Better systems are possible, but they were a later development. The newer number notations, which were what I call rank-and-file systems, were more abstract than the older denominational systems, owing to the more abstract meaning of the symbols themselves. Our story properly begins with the invention of the phoenician alphabet. From there we can see how this eventually led to a logically to a more sophisticated number system of the educated class.

Phonetic Symbolism

The older hieroglyphic style of writing, first used by the Babylonians and the Egyptians was a major step forward for civilization. It meant that society was no longer tethered to the limitations of human memory and the oral tradition. It provided a way to store vast records and information and preserve them for coming generations. This form of writing, which is fundamentally based on pictographs whose primary meaning is meant to be self-explanatory, also gave rise quite naturally to the idea of having different symbols for different size units in a specialized tally system. These different units or denominations is what all numerals were composed of. Hence my referral to them as denominational systems.

But this system of writing and numeration had severe drawbacks. In order to describe all of the different things, living and non-living, in their environment, the early scribes needed a symbol for each kind of thing. This ended up requiring the memorization of thousands of different signs and their meanings. To make matters worse, this was not enough to describe everything they wanted to express, in and of itself. So such symbols needed to be combined in various ways to form further ideas. Worse yet, it was hard to describe abstract concepts like actions. While we tend to think of writing as a way to record the spoken language, these early written languages were not quite languages in themselves. Rather they were more like memory aids, and were meant as a reminder of something, the specifics of which would be best understood by the original writer. Much of the writing's meaning was implied, symbolic, or simply left out. The oral tradition which was much older was miles ahead in development compare to this relatively new written form which had only appeared around the third millennium B.C.E in Sumer at earliest. As such the written form had some catching up to do, and was not able to capture the full richness of the spoken language.

One of the most basic ways that these early scripts got around the limitation of only having symbols representing objects, was by certain implications when objects were next to each other. For example if the symbol "mouth" was followed by the symbol for "bread" the implication was the action of eating. This, of course, relied on the readers intuition to come to the same conclusion. Due to this quirk, hieroglyphic writing was inherently ambiguous in nature. It was always necessary to interpret a text and it's true meaning or symbolic purpose would have only been best understood by those who first wrote it.

Eventually a partial solution was hit upon by the scribes. Since certain nouns had short names which might just be a single uttered sound, these could be used as symbols of the sound uttered, rather than the actual object. Then to describe any word in the spoken language, one simply broke it down into these smaller utterances that could be combined to suggest the word being written. One nicety of this idea is that one didn't have to come up with thousands of unintuitive symbols for each word in the spoken language. Instead one could essentially simulate spoken language by means of a sound-picture-puzzle. The downside to this was that one might not find the most useful set of sound-names pre-existing in the language. This game could only be played if there was conveniently the right sort of sounds to compose larger words out of. None the less we see the beginning of the idea of representing not an idea, but an actual sound in the spoken language as a better way to describe the spoken language.

Enter the Phoenicians. They took the basic idea of writing from the earlier Mesopotamian and Egyptian civilizations, but they abandoned the complicated specifics of their hieroglyphic systems. Instead they started from scratch borrowing the idea of simply using symbols that would represent utterances ... and nothing else. The signs themselves didn't matter too much, and could be abstract for two important reasons (1) they were representing an abstract concept anyway which didn't lend itself to any obvious visual representation and (2) there was only a very small number of symbols needed to be memorized.

The Phoenicians were able to describe their basic utterances with a set of just 22 symbols:

The specifics of what sound corresponded to what symbol, is not important for our purposes. The point is this system vastly simplified writing and made learning it almost trivial after learning the spoken language, and as we will see, it also eventually lead to important improvements in numerical notation.

Order Matters

One key, seemingly unnecessary, feature of the phoenician letters turned out to be the key to the development of a totally new way of writing numbers: the letters had a conventional order to them. What's curious about this is, that it is not strictly necessary to have a conventional order of your letters in order to have a functioning phonetic language, and yet the order that the Phoenicians imposed on there letters had a profound influence on all later alphabets, including our own. The order of the letters, regardless of the variations in the figures themselves, almost invariably remained the same. And so it is no coincidence, that the first letter of greek is alpha, the first letter of hebrew is aleph, and the first letter of english is 'a'. Slight variations did occur simply because not all spoken languages have the same set of sounds. Some share the same sounds, and in those instances it was convenient to simply borrow the exact letter from some earlier alphabet, but if one had sounds not represented one could always invent another letter. Some languages had less sounds and so would discard the ones they didn't need. In this way this proto-alphabet would be added to and subtracted from, but the strict order of the original 22 letters remained intact.

This seemingly trivial feature of order proved to be very useful to mathematicians. Why had such an order been imposed? Most likely this order was intended as a memory aid for learning the alphabet, in the same way that school children today recite their ABC's to memorize the letters. The human mind remembers things best as a network of related ideas, and so it is sensible that the letters should be organized into some sort of relation with each other. Other more complicated relations could have been devised besides succession, but this structure is perhaps the simplest, seeing as each letter need only be associated with it's nearest two neighbors. This successive property lends itself much to the relationship of numbers which are themselves successive in nature.

So through simple habit and repetition, the letters would subtly take on numeric connotations as well. We can see this with our own language. You may notice an almost unconscious association between "A" and "1", "B" and "2", "C" and "3", and so on. Furthermore, sometimes letters are even used to number Chapters and sub-headings in writing. In the same way each letter would have assumed it's ordinal value based on its position in the alphabet. This suggests that the alphabet would be used to simply count all the numbers from 1 to 22, or to however many letters were in that particular alphabet. For reasons I will elaborate on, this is not what happened.

The Greek Alphabet

Let us now turn to the development of the greek alphabet, from which the Isopsephic Numerals come from. Originally the greeks borrowed directly from the phoenician alphabet.

One key difference between the phoenician abjad and greek alphabet, that we touched upon briefly in the article on Attic Numerals, is that the phoenicians lacked vowel sounds. The greeks needed vowel sounds, but had no use for some of the guttural sounds of the phoenicians, so they simply replaced the gutteral sounds with their vowels, thus forming what is believed to be the first true alphabet containing both consonants and vowels. When this was accomplished the original greek letters in their order was:

alpha, beta, gamma, delta, epsilon, digamma,

zeta, eta, theta, iota, kappa, lambda, mu, nu,

xi, omicron, pi, san, koppa, rho, sigma, tau

That is 22 letters in all. Of these digamma, san, and koppa were quickly abandoned:

alpha, beta, gamma, delta, epsilon,

zeta, eta, theta, iota, kappa, lambda, mu, nu,

xi, omicron, pi, rho, sigma, tau

Leaving just 19 letters. The greeks also had aspirated consonants that had no equivalents in phoenician, so they simply invented three new letters, phi,chi, and psi:

alpha, beta, gamma, delta, epsilon,

zeta, eta, theta, iota, kappa, lambda, mu, nu,

xi, omicron, pi, rho, sigma, tau, phi, chi, psi

Bringing the total back up to 22. Next omega was invented to distinguish the long oh sound from the short oh sound represented by omicron:

alpha, beta, gamma, delta, epsilon,

zeta, eta, theta, iota, kappa, lambda, mu, nu,

xi, omicron, pi, rho, sigma, tau, phi, chi, psi, omega

Thus bringing the total to 23. Lastly the semitic vov was incorporated as the letter upsilon in greek:

alpha, beta, gamma, delta, epsilon,

zeta, eta, theta, iota, kappa, lambda, mu, nu,

xi, omicron, pi, rho, sigma, tau, upsilon, phi, chi, psi, omega

Bringing us to the classic 24 letters. We now summarize this in the following Table:

The Spoken Numbers

Now to fully discuss and understand the new greek numerals we need to go on a bit of a detour. Although we have mostly focused on the written form of numbers, numbers had a spoken form as well, and much like the oral tradition was superior to the new written tradition prior to phonetic writing, the spoken numbers were in fact a vastly superior system to the actually written ones. For this reason, and due to the ambiguity in many systems of numeration, the actual names of numbers was often written below the numerals themselves as a form of disambiguation. Recall the first I I I I I I I I I numerals of the attic system:

I

II

III

IIII

Π

ΠI

ΠII

ΠIII

ΠIIII

We learned that the greeks called I "ενα" which is pronounced ena. This also corresponds with the Attic numeral I. Furthermore Π was called "πεντε" which is pronounced pente. The Attic numerals, of course, didn't necessarily require spoken forms. One could record, and manipulate these symbols without recourse to speaking. This presents something of a problem however if you want to tell someone how much something is, or you want to make an offer. One could of course write it on a piece of paper, but this might not be readily available. One could make hand gestures, but this is best suited only for small quantities. It is really most convenient in this instance to have a spoken form. Now we might suppose that we could simply name them according to their written form as follows:

ενα

ενα ενα

ενα ενα ενα

ενα ενα ενα ενα

πεντε

πεντε ενα

πεντε ενα ενα

πεντε ενα ενα ενα

πεντε ενα ενα ενα ενα

There are a couple of reasons why this doesn't work quite as well in spoken form as it does in written form. The most basic reason is that human visual memory is superior to auditory memory. Also our visual number-sense is superior to both our auditory number-sense and tactile number-sense. Hearing more than III repetitions of something is hard for us to recognize without counting, whereas visually we can handle up to IIII. If the listener was not paying close attention it would be easy to confuse πεντε ενα ενα ενα with πεντε ενα ενα ενα ενα. Furthermore, as we saw earlier the Attic numerals could easily involve dozens of symbols. In spoken form this would mean the person trying to understand the number would have to carefully listen and tally up mentally at least a dozen separate words. This is not terribly practical! But it was also not necessary because the greek language already had a special name for all the numbers from I to Δ, most likely due to finger counting, and associating a special name with each finger and then eventually to each number. The greeks had the following names:

Like the greek letters themselves, these number names have special significance for us as we will be using them quite extensively later on when we want to create some names for some very very large numbers ... but that is still a very long way off. For now however, memorize these names, they will come in very handy later.

In any case, in this spoken system it was not necessary to repeat any word. Instead each of the first Δ numbers had its own name. In fact the first ΔΔ numbers had their own unique name. This suggests a trace of a vigesimal system at work. However for the next Δ numbers the names are simply formed by combining the name deka with one of the other number names. Interestingly deka comes last for ΔΙ and ΔΙΙ, but after that comes first. Is this a trace of a duodecimal system? In any case here are the next Δ names:

The numbers from ΔΔ to ΔΔΠΙΙΙΙ now are formed in the predictable way. We can think of numbers being broken into files. That is to say we can take the portion of the numeral that is composed of Ι's and Π's and collectively they add up to the "rank" of the "first file". The "second file" is then composed of Δ's and

's. Notice that there is exactly ennea possible ranks for each file, as well as the possible of a file being "empty". The greek language had special names for the various possible ranks of the second file as follows:

????

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Greek Numeration

The Greek numeration system was a somewhat awkward and difficult notational system. It made use of the entire Greek alphabet, each symbol representing a different value. The table below displays each of the symbols used, and their default value_[1]:

The order of the letters in greek was the same as the order of ascending value. Thus the first 9 symbols were used for the ones position, the next 9 for the tens position, and the last 9 for the hundreds position. Thus 27 symbols were used to notate every number from 1 to 999. This is actually not a very efficient method of numeration, although it has certain advantages over roman numerals as the expressions don't get prohibitively long.

To write a number you would simply write symbols whose sum was the number in question. For example to write "564" you would write:

φξδ

The symbols were always written in descending order from left to right, same as with our decimals. In order to denote the thousands place, a ones place symbol would have an apostrophe added to the left. For example "3285" would be:

'γσπε

In this manner, one could provide a written form for every counting number up to 9999. 10,000 was known as the myriad, which means uncountable in greek_[2]. This name is particularly poignant for our story since Archimedes was going to use the "uncountable" as the base unit of his system. To notate a myriad, a capital m was used:

M

One could continue by adding any number from 1 to 9999 to the myriad. For example:

M φξδ

would be 1564. To specify a multiple of a myriad any number could be raised next to M. Some examples:

M^δ 'γσπε = 43,285

M^ξδ φξδ = 640,564

M^χα 'δσξε = 6,014,265

Using this system one could get up to the "myriad myriad" without too much trouble. This would be written as:

M^M = 100,000,000

This is probably as far as the system was ever carried, although the logic can certainly be used to extend to the "myriad myriad myriad", the "myriad myriad myriad myriad", etc. simply by stacking Ms. Clearly this would get cumbersome pretty quickly. Could any stack of Ms exceed the number of grains of sand over all the earth? Not only was Archimedes able to show that it would only take a stack of 16 Ms to fill the entire universe with sand, but he also devised an improved system for expressing numbers of this size and much much greater.

Works Cited

[1] http://www.math.wichita.edu/history/topics/num-sys.html#greek

[2] http://physics.weber.edu/carroll/archimedes/sand.htm