4000 YEARS OF NUMBERS
Professor Robin Wilson
In earlier lectures we looked at 4000 years of geometry and 4000 years of algebra. Today we turn our attention to 4000 years of number.
Number is the basis of our counting systems, so let's start with some number names from various cultures and look at their similarities and differences.
English Gothic Latin Greek French
one ains unus heis un
two twai duo duo deux
three threis tres treis trois
four fidwor quattuor tettares quatre
five fimf quinque pente cinq
six saihs sex hex six
seven sibun septeni hepta sept
eight ahtau octo okto huit
nine niun novem ennea neuf
ten taihun decem deka dix
eleven ainlif undecim hendeka onze
twelve twalif duodecim dodeka douze
twenty twaitigjus viginti eikosi vingt
Note, in particular, the similarities between six, sex and hex, and quattuorand quatre (which has a different root from four). Also, 11 is sometimes 1+10 and sometimes a new word. French also has the remnants of a base-20 system (as in quatre-vingt for 80).
The Egyptians and Mesopotamians
As before, we start with the Egyptians and their counting system. Like most counting systems it was based on ten, but used different symbols for one, ten, a hundred, a thousand, and so on: a vertical rod for 1, a heel bone for 10, a coiled rope for 100, a lotus flower for 1000, . . . (recall that the Romans similarly used different symbols for 1, 10, 100 and 1000). Each number is then represented with the appropriate multiple of each symbol, written from right to left. To add the representations of 367 and 756 (say), we add them together by collecting the symbols together and replacing a group of ten by the next symbol (ten rods = one heel bone, ten coiled ropes = one lotus flower, etc.).
Multiplication is more interesting, and was done mainly by successivedoubling and halving - though multiplication by 10 was also simple: just replace each symbol by the next. To calculate 80 × 14 (from the Rhind papyrus Problem 69) we first write 80, and then replace each rod by a heel bone to get 800. We next return to 80 and double it twice to give 160 and 320. Adding the rows for 10 and 4, we get the answer, 1120.
Egyptian fractions were very different from ours. Apart from 2/3, they were all unit fractions or reciprocals 1/n. For example, where we'd write2/11 they'd write 1/6 1/66, and where we'd write 2/13 they'd write 1/81/52 1/104.
Their ability to calculate with these unit fractions can be seen from Problem 31 of the Rhind papyrus. A quantity, its 2/3, its ½, and its 1/7, added together become 33. What is the quantity? In our algebraic notation, we must solve the equation x + 2/3x + ½x + 1/7x = 33. Their answer (which we'd write as 1428/97) was 14 ¼ 1/56 1/97 1/194 1/3881/679 1/776 - an impressive feat of calculation.
How did they do it? They used extensive tables of numbers, breaking each fraction down to a succession of fractions of the form 2/n, and then combining these repeatedly. For this purpose, the Rhind papyrus starts with a table of fractions of the form 2/n, for all odd numbers n from 5 up to 101.
In contrast, the Mesopotamians used a place-value sexagesimal system (based on 60) that used only two symbols. In their place-value system, the actual number depends on the context - the same symbol can represents 41, or 41 × 60 (= 2460), or (40 × 60) + 1 (= 2401), and it is only from the context that we can know which is intended. This idea of context is quite familiar to us: 6-50 might represent time (ten to seven), or the cost of a bus trip to Cambridge (£6.50), or the cost of a flight to Singapore (£650).
To see how the sexagesimal system works, we look at a table of numbers from Larsa, but which ones? As we move down the table, the second and fourth columns don't change, but how about the third column? We have the numbers 49, 50, 51, ... , increasing to 59, and then (not 1 but) 60. In the first column we interpret the first number (not as 41 but) as four 60s plus 1, which is 2401. Then we have forty-one 60s plus 40, which is 2500. The last number in the column is (not 1 or 60 but) 3600. So the table, when properly interpreted, gives a list of perfect squares, from 492up to 602.
There are essentially two types of mathematical tablet - table texts, listing tables of numbers for use in calculations, and problem texts in which problems are posed and solved. Several table texts present multiplication tables: here are the 9-times table and the 5-times table.
Of great importance to the Mesopotamians were reciprocals of numbers, since by using them one could both multiply and divide. A table text of reciprocals gives the reciprocals of all the numbers that can be made up from 2, 3 and 5, since these are the only ones with finite expressions; for example, the reciprocal of 9 is 0;06 40, since 9 × 0;06 40 is 1 (or, equivalently, 9 × 62/3 is 60).
We now move forward a millennium to Greek arithmetic. To the Pythagoreans of about 550 BC, arithmetic apparently meant either 'arithmos' (ordinary arithmetic, calculating with whole numbers) or what we now call number theory - though we have little hard evidence about their mathematics. It seems that they represented numbers geometrically - for example, by adding consecutive integers or odd numbers they could obtain the triangular numbers and square numbers (15 = 1 + 2 + 3 + 4 + 5 and 16 = 1 + 3 + 5 + 7).
The Pythagoreans were apparently interested in commensurable andincommensurable numbers, which came to play an important role later in Euclid's Elements. We say that 12 and 8 are commensurable because each can be 'measured' an exact number of times by a ruler of length 4, and 5pi and 3pi are commensurable because each can be measured by a ruler of length pi In general, two numbers are commensurable if their ratio can be written as a fraction (a ratio of whole numbers) - so 5pidivided by 3pi is just 5/3, which is a fraction. However, as they discovered, the diagonal and side of a square are not commensurable.
The proof of this last fact is typical of the Greek approach. They would have couched everything in geometrical terms, but I'll use modern algebraic notation. The proof is by contradiction: by Pythagoras's theorem the ratio of the diagonal and side of a square is √2, and we must prove that this number cannot be written as a fraction - as a/b, where aand b are whole numbers.
To obtain our contradiction we assume that √2 can be written as a fraction, and we may assume that it is written in its lowest terms, so a and b have no common factor. By squaring, we can rewrite this as a2 = 2b2, so a2 must be an even number. But if a2 is even, then a must also be even (because otherwise, a is odd, so a2 is odd). Since a is even, we can write a = 2k, for some integer k. So 2b2 = 4k2, which gives b2 = 2k2, so b2 is even, and b is even. This gives us our contradiction: a andb are both even, and so divisible by 2, contradicting the fact that a and bhad no common factor. This contradiction arises from our original assumption (that √2 can be written as a fraction), so this assumption is wrong: √2 cannot be written as a fraction, and so the diagonal and side of a square are incommensurable.
The Greeks adopted a decimal counting system in which separate Greek letters were used for 1, 2, 3, ... , 9; then new letters for 10, 20, ... , 90; and then nine further letters taking them to 900. Here's a later Greek multiplication table - for example, 7 times 9 is 63.
Before we leave the Greeks, we mention Archimedes's Sand-reckoner, in which he ridiculed the idea that the number of grains of sand in the universe is infinite by constructing some very large numbers. He started by counting as far as he could - up to a myriad (10,000) - and used this to go to the next stage - a myriad myriad (= 100,000,000). Next he went to (100,000,000)2, (100,000,000)3, and so on up to P = (100,000,000)100,000,000. He then formed powers of P, eventually stopping when he reached a number whose size was about 1 followed by 80,000,000,000,000,000 zeros!
The Chinese, Indians and Mayans
Let's now move to China, India and Central America. But first let's tackle the problem of zero.
Our cultures so far all needed to count objects around them - 5 cows, 12 people - but if there were no things there, they didn't feel the need to count them: they felt no need to introduce a symbol for 0, and even less did they need to introduce negative numbers: -20 cows would have been meaningless.
Recall that the Egyptians used separate symbols for 1, 10, 100, . . . , repeated as many times as necessary, but with no symbol for 0. The Greeks used separate symbols for 1 to 9, 10 to 90 and 100 to 900, but in their geometrical style numbers were drawn as lines, so 0 would have had zero length and wouldn't have appeared.
The situation was rather different for the Mesopotamians with their base-60 place-value system. In the early Mesopotamian tablets of around 1800 BC, gaps were sometimes left to indicate no entry in that position, but in the later Babylonian period (around 600 BC) a special place-holder symbol was used to represent 0 - just as we use 0 to distinguish 305 from 35 or from the two numbers 3 and 5. However, they didn't employ a final 0, so context was still needed to distinguish between 4 and 4 × 60.
The other use of zero is as a number to calculate with. Here, the difference between 2 and 2 (which is 0) has the same mathematical status as the difference between 3 and 2 (which is the number 1). As we'll see, this use of 0 as a number did not emerge until much later, in India.
Let's now look at the mathematics of China and India. Around 250 BC in India, King Ashoka's edicts were written on pillars around the kingdom, and numerical information appeared there. Written in a decimal place system, it seems to have been the origin of what we now call the Hindu-Arabic numerals, with separate columns for units, tens, hundreds, and so on.
The Chinese used a similar scheme for their counting boards, with separate compartments for units, tens, hundreds, . . . - here are 6736 and 2101 (as we now write them). There are only nine different symbols (1 to 9), although each has two forms (horizontal and vertical) so that the calculator could distinguish more easily between adjacent compartments. In this context it would have been natural to introduce a zero symbol for an empty box - though the Chinese didn't do so.
The Indians did, however. Whether they were familiar with Chinese counting boards is unknown, though the Chinese visited India and their boards were transportable (like lap-tops), so it is quite possible. In any case, the Indian number system developed as a place-value system based on 10, using only the numbers 1 to 9 (unlike the Egyptian and Greek systems), and eventually (possibly around 400 AD) including also the number 0.
In the 7th century the Indian mathematician Brahmagupta gave rules for calculating with zero (or cipher) and positive or negative numbers:
The sum of cipher and negative is negative; of positive and nought, positive; of two ciphers, cipher. Negative taken from cipher becomes positive, and positive from cipher is negative; cipher taken from cipher is nought. The product of cipher and positive, or of cipher and negative, is nought; of two ciphers is cipher.
He then becomes confused: Cipher divided by cipher is nought. Positive or negative divided by cipher is a fraction with that as denominator. It would be many centuries before mathematicians really understood the problems caused on dividing by 0.
For the Mayans of Central America, the idea of zero was firmly established. There has been a Mayan culture for many thousands of years, but the heyday of their activities was focussed around the period 300 - 900 AD.
We have primary sources, in the form of stone columns called stelae and a handful of codices - but regrettably most of the latter were destroyed by the Spanish conquerors who arrived here in the 1500s. The codices were drawn on tree bark and folded, and were intended to guide Mayan priests in ritual ceremonies involving hunting, planting and rainmaking.
The dots and lines represent Mayan numerals, with a dot representing 1 and a line representing 5, and the symbol in the middle has 13 below 12. What does this mean? It means thirteen 1s combined with twelve 20s, which is 253, since the Mayans essentially used a number system based on 20. Let's see how this works.
Here are the dot-and-line representations of the numbers from 1 to 19, and you can also see that there's a special symbol, looking rather like an eye, for zero. One jolly feature of the Mayan numerals is that each number had an alternative form, the head-form shown here. They can vary a bit, but 0 always has clasped hands across the lower part of the face, while 3 has a banded headdress and 4 has a bulging eye with square irid, snag tooth, and curling fangs from the back of the mouth. These head-forms appear on various columns, which can sometimes be interpreted as giving the particular date when it was constructed.
The Mayans had a fixation on the calendar, and they had two basic types. The first was a ritual calendar of 260 days, known as the tzolkin, used for forecasting and consisting of thirteen months of 20 days; each day had a number and a name, such as 1 Imix, which was followed not by 2 Imix and 3 Imix but by 2 Ik and 3 Akbal, etc. But they also had another calendar, with eighteen months of 20 days, plus an extra five inauspicious days to make up the usual 365 days. These two calendars operated independently, and were also combined to give a long-count, orcalendar round, in which the number of days was the least common multiple of 260 and 365, which is 18,980 days, or 52 calendar years; for example, this day of 4 Ahau and 8 Cumku won't come around again for 52 years. These periods of 52 years were then packaged up into even longer time-periods, and you'll be alarmed to learn that they predicted that the world will come to an end at the end of the current period, in the year 2012.
Their units were based on the following scheme: 1 kin = 1 day; 20 kins = 1 uinal = 20 days; 18 uinals = 1 tun = 360 days; 20 tuns = 1 katun = 7200 days; 20 katuns = 1 baktun = 144,000 days, etc. They had no problem with calculating with such large numbers. The largest number found on a codex is (starting at the bottom) one 1, fifteen 20s, thirteen 360s, fourteen 7200s, and so on, giving a grand total of 12,489,781 days, or over 34,000 years.
The Hindu-Arabic numerals
In Baghdad the caliphs actively promoted mathematics and astronomy - in particular, Caliph Harun al-Rashid established the 'House of Wisdom', a scientific academy with an extensive library and observatory.
One of the earliest scholars at the House of Wisdom was al-Khwarizmi (c.780-850), who is remembered primarily for two books on arithmetic and algebra. Neither was a work of great originality, but both had great influence in the West when translated into Latin in the twelfth century - in particular, his Arithmetic helped to spread the Hindu decimal place-value system (the so-called Hindu-Arabic numerals) throughout Christian Europe. Indeed, his name, transmuted into 'algorism' was frequently used in Europe to mean arithmetic, and we now use the word 'algorithm', named after him, to refer to a step-by-step procedure for solving a problem.
It was during the 8th and 9th centuries, the Islamic world spread along the northern coast of Africa and up through southern Spain and Italy, and with it the Hindu-Arabic numerals.
Europe had been going through a lengthy period known as the Dark Ages. The legacy of the ancient world was almost forgotten, schooling became infrequent, and the general level of culture had remained low. Mathematical activity was generally sparse, but included writings on the calendar and on finger reckoning by the Venerable Bede (around 700 AD), and an influential book of Problems for the quickening of the mind by Alcuin of York, educational adviser to Charlemagne, about fifty years later.
Revival of interest in mathematics began with Gerbert of Aurillac (938-1003), who trained in Catalonia and is believed to be the first to introduce the Hindu-Arabic numerals to Christian Europe, using an abacus that he'd designed for the purpose; he was crowned Pope Sylvester II in 999.
The development of the Hindu-Arabic numerals led to the use of finger counting. Hindu-Arabic methods of calculation were also used by Leonardo Fibonacci (Leonardo of Pisa) in his Liber abaci (Book of calculation) of 1202. This celebrated book contained many problems in arithmetic and algebra, including the celebrated problem of the rabbits that leads to the 'Fibonacci sequence'.
Johann Gutenberg's invention of the printing press (around 1440) revolutionised mathematics, enabling classical mathematical works to be widely available for the first time. Previously, scholarly works, such as the texts of Euclid and Archimedes had been available only in manuscript form, but the printed versions made these works much more widely available.
At first the new books were printed in Latin or Greek for the scholar, but increasingly, vernacular works began to appear at a price accessible to all. These included introductory texts in arithmetic, algebra and geometry, as well as vernacular commercial arithmetics, containing computational rules and tables to help with financial transactions designed to prepare young men for a commercial career.
The first major arithmetic text to be published in England was by Cuthbert Tonstall, whose 1522 De arte supputandi was the best of its time. The first arithmetic book in English, in 1537, was An introduction for to lerne to recken with the pen, or with the counters according to the trewe cast of Algorithme, in hole numbers or in broken, newly corrected . . .
The best known English texts were by Robert Record, and ran to many editions. The ground of artes of 1543 was an arithmetic book explaining the various rules so simply that 'everie child can do it'. As with all his books, it was written in the form of a Socratic dialogue between a scholar and his master.
It also explains how to carry out multiplication. To multiply 8 by 7, for example, we write them on the left, and opposite we subtract each from 10 to give 2 and 3. Now 8 - 3 (or 7 - 2) is 5 and 3 ´ 2 = 6, so we get 56. The cross eventually shrank and became the multiplication sign we use today.
These improvements in notation went hand in hand with developments in calculation. Decimal fractions had taken many centuries to become established throughout Europe. In the late 15th century the Flemish mathematician Simon Stevin wrote a popular book De thiende [The tenth] that explained decimal fractions, advocated their widespread use for everyday calculation, and proposed a decimal system of weights and measures. This work and its translations into other languages really seemed to do the trick at last.
Can we represent numbers geometrically? Descartes had employed various geometrical devices to draw a number of curves, and to multiply two numbers together, and found geometrical constructions for the square root of a positive quantity and for the positive root of the quadratic equation x2 = ax + b2. Influenced by him, other 17th-century mathematicians attempted to represent algebraic ideas geometrically.
It was also around this time that calculating devices were introduced. In Germany, Schickhard designed a calculating machine, and later ones were constructed by Pascal and Leibniz for the purpose of carrying out simple arithmetical calculations.
There were also developments with the calendar. Before the Romans many different calendars were in use. As early as 4000 BC the Egyptians used a 365-day solar calendar of twelve 30-day months and five extra days. The early Roman year had just 304 days, but this was later extended to 355 days by adding two new months Januarius and Februarius.
In 45 BC Julius Caesar introduced his 'Julian calendar' with 365¼ days, the fraction arising from an extra 'leap day' every four years. The beginning of the year was moved to January and the lengths of the months alternated between 30 and 31 days (apart from a 29-day February in leap years); then Augustus Caesar stole a day from February to add to August and altered September to December accordingly.
Later writers determined the length of the solar year with increasing accuracy. In particular, the Islamic scholars Omar Khayyam and Ulugh Beg independently measured it as 365 days, 5 hours and 49 minutes - just a few seconds out.
The Julian year was thus 11 minutes too long, and by 1582 the calendar had drifted by ten days with respect to the seasons. In that year Pope Gregory XIII issued an Edict of Reform, removing the extra days. He corrected the over-length year by omitting three leap days every 400 years, so that 2000 was a leap year, but 1700, 1800 and 1900 were not. The Gregorian calendar was quickly adopted by the Catholic World and other countries eventually followed suit: Protestant Germany and Denmark in 1700, Britain and the American colonies in 1752, Russia in 1917, and China in 1949.
Meanwhile, the line from which time is measured (0° longitude) was located at the Royal Observatory in Greenwich in 1884, giving rise to an international date line near Tonga. In 1972, atomic time replaced earth time as the official standard, and the year was officially measured as 290,091,200,500,000,000 oscillations of atomic caesium.
Different types of number
Let's recall how our usual number system is built up. As the 19th-century German mathematician Leopold Kronecker remarked:
'God created the natural numbers, and all the rest is the work of man.'
So, starting with the natural numbers, 1, 2, 3, ... , we then obtain all theintegers - positive, negative and zero. This was not a trivial process, taking thousands of years, and negative numbers were treated with the same ridicule that the imaginary numbers would later face - after all, what is meant by 'minus 2 sheep'? These days we have no difficulty understanding negative temperatures in our weather forecasts, and it seems hard to see why negative numbers caused so much disbelief.
The next step is to divide one integer by another, and we get fractions, orrational numbers: these are fractions of the form a/b, where a is an integer and we can take b to be a natural number (since we mustn't divide by 0). But even here we have to be careful: the fractions 1/2, 2/4and 3/6 are all different, yet they represent the same rational number.
But many numbers cannot be written as fractions - for example, pi, √2,3√7, and the number e that we'll meet next time. These are irrational numbers, which when combined with the rational numbers form the real numbers. But how do we define the real numbers? We'll come to that later.
Let's now go back to our various types of numbers and look at them from a different point of view. If we're restricted to the natural numbers, then we can solve certain equations: for example, the equation x + 3 = 7 has the solution x = 4. But to solve the equation x + 7 = 3 we need to expand our number system to the negative integers, and the solution is x= -4.
The next stage is to bring in fractions: using them we can now solve an equation such as 7x = 5: the solution is x = 5/7. We can then solve alllinear equations of the form ax = b, where a and b are integers or rationals.
Once we introduce real numbers, we can go beyond linear equations and find solutions for equations such as x2 = 2 or x3 = 7, or for the equationx4 - 10x2 + 1 = 0, which has √2 + √3 as a solution. But even now we cannot solve all quadratic equations - to solve the equation x2 = -1 we need to introduce another type of number, the square root of -1.
But is there such a thing as √-1? After all, if you square either 1 or -1 you get 1, so what can you square to get -1?
They first arose in the 16th century, when Cardano asked how one can divide 10 into two parts whose product is 40. If the parts are taken to bex and 10 - x, then x(10 - x) = 40, and Cardano obtained the solutions 5 + √-15 and 5 - √-15. He could see no meaning to these, but observed 'Nevertheless we will operate, putting aside the mental tortures involved', and found that everything works out correctly:
(5 + √-15) × (5 - √-15) = 52 - (√-15)2 = 40.
Shortly after, the situation was clarified by Rafael Bombelli, an expert in draining swampy marshes, who developed means for calculating with such 'imaginary' quantities.
But even three centuries later, there was much anguish about them. The Astronomer Royal, George Airy, said that he had not the smallest confidence in any result which is essentially obtained by the use of imaginary symbols, while Augustus De Morgan, Professor of Mathematics at University College, London, said We have shown the symbol √-1 to be void of meaning, or rather self-contradictory and absurd.
Suppose that we do agree to allow this mysterious object called '√-1'. We can then form many more 'numbers' such as 3 - 4√-1. Ignoring for the moment what this actually means, we can then carry out calculations with such objects. We shall usually follow Leonhard Euler who in 1777 introduced the letter i to mean √-1, so that i2 = -1.
Addition is easy: (2 + 3i) + (4 + 5i) = 6 + 8i,
and so is multiplication:
(2 + 3i) × (4 + 5i) = (8 - 15) + (12 + 10)i = -7 + 22i.
In fact, we can carry out all the standard operations of arithmetic on these new objects.
What happens when we look at higher-degree equations, such as this one:
x6 - 12x5 + 60x4 - 160x3 + 239x2 - 188x + 60 = 0?
Can this be solved with just real and complex numbers, or do we need to introduce yet another type of number?
To get an idea of the answer, let's try to take the square root of i. Do we need to introduce further numbers, or are our existing complex numbers enough? If the latter, then we can write:
x2 = (a + bi)2 = i, so (a2 - b2) + 2abi = i.
So a2 - b2 = 0 and 2ab = 1, giving a = b = √1/2 or a = b = -√1/2, so that the solutions are ± √1/2 (1 + i). So in this case, complex numbersare enough.
In fact, complex numbers are always enough for any polynomial equation; for example,
x6 - 12x5 + 60x4 - 160x3 + 239x2 - 188x + 60 = 0
has solutions 1, 3, 2 (twice) and 2 ± i. This is a special case of what became known as the Fundamental Theorem of Algebra: every polynomial p(x) can be factorized completely into linear factors with complex coefficients;
Even as late as the 1830s there was still a great deal of suspicion about complex numbers, and about so-called 'imaginary' numbers that don't seem to exist. Sir William Rowan Hamilton, the Astronomer Royal of Ireland diffused much of this suspicion by saying that the complex numbers a + bi should be defined as pairs (a, b) of real numbers, which we combine by using the following rules:
(a, b) + (c, d) = (a + c, b + d) and (a, b) × (c, d) = (ac - bd, ad+ bc),
corresponding to the equations
(a + bi) + (c + di) = (a + c) = (b + d)i and (a + bi) × (c + di) = (ac- bd) + (ad + bc)i.
The pair (a, 0) corresponds to the real number a, the pair (0, 1) corresponds to the number i, and we have the equation (0, 1) × (0, 1) = (-1, 0), corresponding to the equation i × i = -1.
Hamilton then tried to extend his ideas to three terms of the form a + bi +cj, where i2 = j2 = -1. Certainly, addition works well:
(a + bi + cj) + (d + ei + fj) = (a + d) + (b + e)i + (c + f)j.
But he couldn't make multiplication work:
(a + bi + cj) × (d + ei + fj) = (ad - be - cf) + (ae + bd)i + (af + cd)j + (bf + ce)ij.
This gives four terms, rather than three. How can we get rid of the last term? We can't let ij = 0, because then 0 = (ij)2 = i2j2 = (-1)( -1) = 1.
Hamilton tried everything, such as writing ij = 1 or-1, but nothing seemed to work: in a letter to one of his sons he later wrote: Every morning, on my coming down to breakfast, your little brother William Edwin and yourself used to ask me, 'Well Papa, can you multiply triples?' Whereto I was obliged to reply, with a shake of the head: 'No, I can only add and subtract them'.
Hamilton struggled with the problem for fifteen years, until one day he took a walk along the canal: As I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations exactly as I have used them ever since. I pulled out on the spot a pocket book and made an entry - it is fair to say that this was because I felt a problem to have been at that moment solved - an intellectual want relieved which had haunted me for at least fifteen years since.
What he had come up with was his quaternions: objects of the form
a + bi + cj + dk, where i2 = j2 = k2 = -1.
In order to make multiplication work, he had to abandon the commutative law, in which we can multiply numbers either way round (3 × 4 = 4 × 3). The rules that made these quaternions work are ij = k, but ji = -k; jk =i, but kj = -i; ki = j, but ik = -j - or, more concisely, i2 = j2 = k2 = ijk = -1.
Hamilton was so excited that he carved these basic rules on the bridge. There is now a plaque to commemorate their discovery, and over the years the Irish Post Office has issued several stamps featuring Hamilton and his discovery. Quaternions have proved to be of enormous importance, both for their theoretical properties and also in their applications to physics and engineering.
Can we go further? It turns out that we can take just one further step, tooctonians. These have the form α + βi + γj + δk + εl + ζm + ηn + θo, where i2 = j2 = . . . .= o2 = -1, and the multiplication of these letters is defined in a complicated way. But where the complex numbers were both commutative (XY = YX) and associative ((XY)Z = X(YZ)), and the quaternions were associative but not commutative, the octonians are neither. Even so, they have recently caused quite a lot of interest to mathematicians and physicists.
We end by looking at some more basic questions. As mathematics developed in so many ways, its practitioners found it increasingly important to define all terms precisely, and the 1860s witnessed an examination of the very fundamentals of the subject: mathematicians were even asking such basic questions as What is a number?
In 1817, Bolzano had produced a pamphlet that gave a purely analytic proof of the theorem that, between any two values that give a result of opposite sign, there lies at least one real root of the equation. This result, now called the intermediate value theorem, tells us that if we have a continuous graph that at one place is below the x-axis and at another is above it, then at some point in between it must cross the x-axis. Although this is obvious intuitively, it had never been rigorously proved before. Using it, we can prove that there is a number whose square is 2, by applying it to the graph of y = x2 - 2 over the range 0 to 2 (since 02 - 2 < 0 and 22 - 2 > 0).
But the big question that arose in the 19th century is: what exactly is a real number? They're all the points that lie on a line, but how do we actually define them? We might try to introduce the irrational numbers, such as pi, but they won't help us unless we can then say what an irrational number is.
It's not difficult to prove that every rational number can be written as a finite or recurring decimal fraction; for example, 1/8 = 0.125, 1/3 = 0.333... and 1/7 = 0.14285714285714... We can also go the other way and prove that every finite or recurring decimal can be written as a fraction. So we might try to define the irrational numbers like √2 or pi as infinite decimals that don't recur.
At one level this answers the difficulty - we can now say that a number is a decimal: finite, recurring or infinite as the case may be. But we then run into trouble when we try to do arithmetic, because if we define √2 to be the infinite decimal 1.414213..., then how do we prove that √2 × √2 = 2 (try multiplying two infinite decimals together!).
Much time was spent in the second half of the 19th century sorting out such difficulties. The German mathematician Georg Cantor and others introduced set theory, starting from a few basic undefined terms such as a set and an element of a set. By developing mathematics as a hierarchical structure based on sets and elements, just as Euclid had built up geometry from the undefined notions of point and line, he hoped to remove all difficulties.
I'd like to end with a result of Cantor that was truly revolutionary. This was the idea that some infinities are larger than others. This idea had its origins in the musings of Galileo over 300 years earlier, who had noticed that there are far fewer perfect squares 1, 4, 9, 16, ... than natural numbers 1, 2, 3, ... , and yet we can match them up exactly:
1 ↔ 1, 2 ↔ 4, 3 ↔ 9, 4 ↔ 16, etc.
We can also match the natural numbers with some larger sets, such as the integers. We simply list them in the order:
0, 1, -1, 2, -2, 3, -3, 4, etc.
Notice that every integer occurs somewhere in the list - none is omitted.
Cantor's first amazing discovery was that we can also list all the rational numbers in order, so that the set of all rational numbers has 'the same size' as the set of all natural numbers, even though it seems so much bigger. His second amazing discovery was that the set of all real numbers cannot be made to match up with the natural numbers, and so must be a 'bigger' infinite set. So some infinite sets are genuinely bigger than others: not all infinities are the same. Indeed, to take his idea further, there are infinitely many infinities, all of different sizes.
In fact, we can construct an 'arithmetic of infinities'. The size of the set of all integers (or of all rationals) is called aleph-0 (aleph is a Hebrew letter), and we can write 'infinite equations' such as aleph-0 + 5 = aleph-0, and aleph-0 + aleph-0 = aleph-0. The size of the set of all real numbers is called aleph-1 (which is larger than aleph-0). A question now arises, is there an infinite set whose size lies between these two infinite numbers? This problem became known as Cantor's continuum problem.
The answer was unexpected. In 1902 Bertrand Russell upset the set theory apple-cart by producing a famous logical paradox that seemed to have no answer. One version of this paradox is: the barber in a village shaves everyone who doesn't shave themselves, but obviously doesn't shave those who do shave themselves; who shaves the barber? (Whatever answer you give must be false, as you'll see after a moment's thought.) The paradox was eventually overcome by Zermelo and Fraenkel who produced a system of logic that could deal with it - this logical system became universally accepted and used. Then, in the 1930s, Kurt Gödel produced another bombshell: in any axiomatic system that one can construct, there are true results that cannot be proved, and there are statements that are undecidable - they cannot be proved true or false.
In 1963 Paul Cohen stunned the mathematical world by proving that Cantor's continuum problem cannot be decided within the Zermelo-Fraenkel logical system - it is undecidable.
We've come along way from the natural numbers and from the Egyptian and Mesopotamian counting systems. I hope you'll agree that we've seen some exciting mathematics along the way.
©Professor Robin Wilson, Gresham College, 7 November 2007