The Book of Nothing

by John D. Barrow

  Counting is one of those arts, like reading, into which we are thrust during our first days at school. Humanity learned the same lessons, but took thousands of years to do it. Yet whereas human languages exist by the thousand, their distinctiveness often enthusiastically promoted as a vibrant symbol of national identity and influence, counting has come to be a true human universal. After the plethora of our languages and scripts for writing them down, a present-day tourist from a neighbouring star would probably be pleasantly surprised by the complete uniformity of our systems of reckoning. The number system looks the same everywhere: ten numerals −1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 – and a simple system that allows you to represent any quantity you wish: a universal language of symbols. The words that describe them may differ from language to language but the symbols stay the same. Numbers are humanity’s greatest shared experience.

  The most obvious defining feature of our system of counting is its use of a base of ten. We count in tens. Ten ones make ten; ten tens make one hundred; and so on. This choice of base was made by many cultures and its source is clearly to be found close at hand with our ten fingers, the first counters. Sometimes one finds this base is mixed in with uses of 20 as a base (fingers plus toes) in more advanced cultures, whilst less advanced counting systems might make use of a base of two or five.6 The exceptions are so rare as to be worth mentioning. In America one finds an Indian counting system based on a base of eight. At first this seems very odd, until you realise that they were also finger counters – it is just that they counted the eight gaps between the fingers instead of the ten fingers.

  You don’t have to be a historian of mathematics to realise that there have been other systems of numbers in use at different times in the past. We can still detect traces of systems of counting that differ in some respects from the decimal pattern. We measure time in sets of 60, with 60 seconds in a minute, 60 minutes in an hour, and this convention is carried over to the measurement of angles, as on a protractor or a navigator’s compass. Else-where, there are relics of counting in twenties:7 ‘three-score years and ten’ is the expected human lifetime, whilst in French the number words for 80 and 90 are quatre-vingts and quatre-vingt-dix, that is four-twenties and four-twenties and ten. In the commercial world we often order by the gross or the dozen, witness to a system with a base of twelve somewhere in the past.

  The ten numerals 0, 1, …, 9 are used everywhere, but one other system for writing numbers is still in evidence around us. Roman numerals are often to be found on occasions where we want to emphasise something dynastic, like Henry VIII, or traditional, like the numbers on the clock face in the town square. Yet Roman numerals are rather different from those we use for arithmetic. There is no zero sign. And the information stored in the symbols is different as well. Write 111 and we interpret it as one hundred plus one ten plus one: one-hundred and eleven. Yet to Julius Caesar the marks 111 would mean one and one and one: three. These two missing ingredients, the zero sign and a positional significance when reading the value of a symbol, are features that lie at the heart of the development of efficient human counting systems.

  EGYPT – IN NEED OF NOTHING

  “Joseph gathered corn as the sand of the sea, very much, until he left numbering; for it was without number.”

  Genesis 41

  The oldest developed counting systems are those used in ancient Egypt and by the Sumerians in Southern Babylonia, in what is now Iraq, as early as 3000 BC. The earliest Egyptian hieroglyphic8 system used the repetition of a suite of symbols for one, ten, a hundred, a thousand, ten thousand, a hundred thousand and a million. The symbols are shown in Figure 1.1. The

  Figure 1.1 Egyptian hieroglyphic numerals.

  Egyptian symbols for the numerals one to nine are very simple and consist of the repetition of an appropriate number of marks of the vertical stroke, |, the symbol for one; so three is just |||. The symbols for the larger multiples of ten are more picturesque. Ten is denoted by an inverted u, a hundred by a coil, a thousand by a lotus flower, ten thousand by a bent finger, a hundred thousand by a frog or a tadpole with a tail, and a million by a man with his arms raised to the heavens. With the exception of the sign for one, they seem to have no obvious connection with the quantities they denote. Some connections are probably phonetic, deriving from the similar sounds for the things pictured and the original words used to describe the quantities. Only the bent finger marking ten thousand seems to hark back to a system of finger counting. We can only guess about the others. Perhaps the tadpoles were so numerous in the Nile when the frogs’ spawn hatched in the spring that they symbolised a huge number; maybe a million was just an awesomely large quantity, like the populations of stars in the heavens above.

  The symbols were written differently if they were to be read from right to left or left to right in an inscription.

  Hieroglyphs were generally written down from right to left so that our number 3,225,578 would appear as shown in Figure 1.2.

  One of the oldest examples of these numerals appears on the handle of a club belonging to King Narmer, who lived in the period 3000–2900 BC, celebrating the fact that the loot seized in one of his military campaigns amounted to 400,000 bulls, 1,422,000 goats and 120,000 human prisoners. The symbols for these quantities beneath pictures of a bull, a goat and a seated figure can be seen on the bottom right of Figure 1.3. The order in which the symbols are written is not important because there are different symbols for one, ten and a hundred. The hieroglyph

  would signal exactly the same quantity if written forwards or backwards. The symbols can be laid out in any way at all without changing the value of the number they are representing. However, Egyptian stonemasons were given strict rules of style for writing numbers: signs were to appear from right to left in descending order of size on a line underneath the symbol for the object that was being counted (as in Figure 1.3). However, there was a tendency to group similar symbols together over two or three lines to help the reader quickly read off the total, as shown in Figure 1.4.

  Figure 1.2 The hieroglyph for our number three million, two hundred and twenty-five thousand, five hundred and seventy-eight.

  Thus we see that the relative positions of the Egyptian counting symbols carry no numerical information and so there is no need for a symbol for zero. When the number symbols can sit in any location without altering the total quantity they are representing, there is no possibility of an empty ‘slot’ and no meaning to a signal of its presence. The need for a zero arises when you have nothing to count – but in that case you write no symbols at all. The Egyptian system is an early example of a decimal system (the collective unit is 10) with symbols for numbers which carry no positional information. In such a system there is no place for a zero symbol.

  Figure 1.3 Hieroglyphics inscribed on the handle of King Narmer’s war club,9 3000–2900 BC.

  Figure 1.4 The grouping of number signs to help the reader.

  BABYLON – THE WRITING IS ON THE WALL

  “In the same hour came forth fingers of a man’s hand, and wrote over against the candlestick upon the plaster of the wall of the king’s palace … And this is the writing that was written, Mene, Mene, Tekel, Upharsin. This is the interpretation. Mene; God hath numbered thy kingdom and finished it. Tekel; Thou art weighed in the balances, and art found wanting. Peres; thy kingdom is divided.’

  Daniel 510

  The earliest Sumerian system, also in use around 3000 BC, was more complex than that employed by the Egyptians and seems to have developed independently. It was later adopted by the Babylonians and so the two civilisations are usually regarded as different parts of a single cultural development. The motivation for their systems of writing and counting was at first administrative and economic. They kept detailed records and accounts of exchanges, stores and wages. Often, a detailed list of items will be found on one side of a tablet, with the total inscribed on the reverse.

  The counting system of early Sumer was not solely decimal. It made good use of t
he base ten to label quantities but it also introduced 60 as a second base number.11

  It is from this ancient system that we inherited our pattern of time-keeping with 60 seconds to the minute and 60 minutes to the hour. Expressing 10 hours 10 minutes and 10 seconds in seconds shows us how to unfold a base-60 counting system. We have a total of (10 × 60 × 60) + (10 × 60) + 10 = 36,000 + 600 + 10 = 36,610 seconds.

  The Sumerians had number words for the quantities 1, 60, 60 × 60, 60 × 60 × 60, … and so on. They also had words for the numbers 2, 3, 4, 5, 6, 7, 8, 9 and 10, together with the multiples of ten below 60. A distinct word was used for 20 (unrelated to the words for 2 and 10) but ‘thirty’ was a compound word meaning ‘three tens’, ‘forty’ meant ‘two twenties’, and ‘fifty’ meant ‘forty and ten’. So there was a weaving of base 10 and base 20 elements to ease the jump up from one to sixty.

  Whereas the Egyptians carved their signs in stone with hammers and chisels or painted them on to papyri with reeds, Sumerian records were kept by making marks in tablets of wet clay. Stone was not common in Sumer and other media like papyrus or wood would rapidly perish or rot, but clay was readily available. The inscriptions were made by impressing the wet clay with two types of reed or ivory stylus, shaped like pencils of differing widths. The round blunt end allowed notches or circular shapes to be impressed whilst the sharp end allowed lines to be drawn. The sharp end was used for writing whilst the blunt end was used for representing numbers. The original symbols are shown in Figure 1.5 and are called curvilinear signs. The number symbols12 usually appeared over an image of the thing being enumerated and reveal a new feature, not present in Egypt. The symbol for 600 combines the large notch, representing 60, with the small circle, representing 10. Likewise, the symbol for 36,000 combines the large circle, for 3,600, with the small circle, for 10. This economical scheme creates a multiplicative notation. There are fewer symbols to learn and the symbols for large numbers have an internal logic that enables larger numbers to be generated from smaller ones without inventing new symbols. However, notice that you have to do a little bit of mental arithmetic every time you want to read a large number! The system is additive and there is again no significance to the positions of the symbols when they are inscribed on the clay tablets. As in Egypt, similar symbols were grouped together for stylistic reasons and for ease of reckoning. The early style was to gather marks into pairs. For example, the decimal number 4980 is broken down as

  and this would be written as shown in Figure 1.6 since tablets were read from right to left and from top to bottom.

  Figure 1.5 The impressed shapes representing Sumerian numerals on clay tablets.

  A tedious feature of this system is the huge number of marks that have to be made in order to represent large numbers that are not exact multiples of 60. To overcome this problem, scribes developed a shorthand subtraction notation, introducing a ‘wing’ sign that played the role of our minus sign so that they could write a number like 59 as 60 minus 1 by means of the three symbols (Figure 1.7) instead of the fourteen marks that would otherwise have been required.13

  By 2600 BC a significant change had occurred in the way that the Sumerian number characters were written. The reason: new technology – in the form of a change of writing implement. A wedge-shaped stylus was introduced which could produce sharper lines and wedge-shapes of different sizes. These became known as ‘cuneiform’14 signs and only two marks are used, a vertical wedge denoting ‘one’ and a chevron representing ‘ten’ (Figure 1.8). Again, the fusion of symbols can be used to build up large numbers from smaller ones. If the symbols for 60 and 10 were in contact they signified a multiplication of values (600) whereas if they were separated they signified an addition (70). However, some care was needed to make sure that juxtapositions of signs like these did not become confused. The Sumerian combinations of symbols avoided this problem because the individual marks were much more distinct.

  Figure 1.6 The number 4980 in early Sumerian representation, before 2700 BC.

  Another problem was the distinction of the signs for 1 and for 60. Their shapes are identical wedges and at first they were distinguished simply by making the 60 wedge bigger. Later, it was done by separating the wedge shape for 60 from those for numbers less than nine. The writing of the number 63 is shown in Figure 1.9.

  Figure 1.7 The number 59 written as 60 minus 1.

  Figure 1.8 The cuneiform impressions made by the two ends of the scribe’s stylus, denoting the numbers 1 and 10.

  Many other systems of counting can be found around the ancient world which use the same general principles as these. The Aztecs (AD 1200) had an additive base-20 system with symbols for 1, 20, 400 = 20 × 20, and 8000 = 20 × 20 × 20. The Greeks (500 BC) used a base-10 system with different signs for 1, 10, 100, 1000, 10,000 but supplemented them with a further sign for 5 which they then added to the other signs to generate new symbols for 50, 500, 5000, and so on (see Figure 1.10).

  All these systems of writing numbers are cumbersome and laborious to use if you want to do calculations that involve multiplication or division. The notation does not do any work for you, it is just like a shorthand for writing down the number words in full. The next step in sophistication, a step that was to culminate in the need to invent the zero symbol, was to introduce a positional or place value system in which the locations of symbols determined their values. This allows fewer symbols to be used because the same symbol can have different meanings in different locations or when used in different contexts.

  Figure 1.9 Two ways of writing the number 63: (a) using a larger version of the 60 symbol to separate 63 as 60 and 3, or (b) by leaving a space between the symbol for 60 and those for 3.

  Figure 1.10 Greek numerals, which first appeared around 500 BC, used combinations of symbols to generate higher numbers. As an example, we have written the number 6668.

  A positional system appeared first in Babylonia around 2000 BC. It simply extended the cuneiform notation and the old additive base-60 system to include positional information. It was used by mathematicians and astronomers rather than for everyday accounting because the old system allowed the reader to see the relative sizes of numbers more easily. Many scribes must therefore have practised with both systems. However, it was used in the recording of royal decrees and so must have been understood by a broad cross-section of the Babylonian public. Thus, a number like 10,292 would be conceived in our notation as [2; 51; 32] = (2 × 60 × 60) + (51 × 60) + 32, and written in cuneiform as shown in Figure 1.11. This is just like our representation of a number like 123 as (1 × 10 × 10) + (2 × 10) + 3. Our notation just reads off the number that multiplies the number of contributions by each power of 10. We still retain the Babylonian system for time measures. Seven hours and five minutes and six seconds is just (7 × 60 × 60) + (5 × 60) + 6 = 25,506 seconds.

  Figure 1.11 The number 10,292 in cuneiform.

  The earliest positional decimal system like our own did not appear until about 200 BC when the Chinese introduced the place value system into their base-10 system of signs. Their rod number symbols, together with an example of their positional notation in action, are shown in Figure 1.12.

  THE NO-ENTRY PROBLEM AND THE BABYLONIAN ZERO

  “There aren’t enough small numbers to meet the many demands made of them.”

  Richard K. Guy15

  These advances were not without their problems. The Babylonian system was really a hybrid of positional and additive systems because the marking of the number of each power of 60 was still denoted in an additive fashion. This could produce ambiguity if sufficient space was not left between one order of 60 and the next. For instance, the symbols for 610 = [10; 10] = (10 × 60) + 10 could easily be misread for 10 + 10 (see Figure 1.13). This was generally dealt with by separating the different orders of 60 clearly. Eventually, a separation marker was introduced to make the divisions unambiguous. It consisted of two wedge marks, one on top of the other, as shown in Figure 1.14.

  Any difficulties
of interpretation would be compounded further if there was no entry at all in one of the orders. The spacing would then be more tricky to interpret. Imagine that our system had no 0 symbol and relied on careful spacing to distinguish 72 (seventy-two) from 7 2 (seven hundred and two). With different writing styles to contend with there would be many problems which are exacerbated if one has to distinguish 7 2 (seven thousand and two) as well as 7 2 and 72. The more spaces that you need to leave, the harder it becomes to judge.16 This is why positional notation systems eventually need to invent a zero symbol to mark an empty slot in their positional representation of a number. The more sophisticated their commercial systems the greater is the pressure to do so. For nearly 1500 years the Babylonians worked without a symbol for ‘no entry’ in their register of different powers of ten or sixty; they merely left a space. Their success required a good feeling for the magnitudes of the astronomical and mathematical problems they were dealing with, so that large discrepancies from expected answers could be readily detected.

  Figure 1.12 (a) Chinese rod numerals. They are pictures of bamboo or bone calculating rods. When these symbols were used in the tens or thousands position they were rotated, and written as in (b), so our number 6666 would have been expressed as shown in (c).