From 0 to Infinity in 26 Centuries

by Chris Waring

  Court eunuch Jia Xian (AD 1010–70) is credited with being the first individual to investigate what later became known in Europe as Pascal’s triangle (see here). Chinese mathematician Yang Hui (AD 1238–98) published Xian’s findings in 1261, four hundred years before Pascal’s discoveries would be revealed. Xian was also interested in what we call magic squares, which had long fascinated Chinese mathematicians. A magic square is a square of numbers in which all the rows, columns and diagonals add up to a particular number. For example:

  In the example above, each row, column and diagonal has a sum of 15. This particular magic square is known as the Lo Shu square because of its connection to a legend in which the River Lo floods and a magic turtle carries the magic square on its shell to aid the afflicted people.

  The Chinese Abacus

  At some point in c. AD 1000 the Chinese began to use the suanpan (Chinese abacus) in favour of the counting rods, although the suanpan had been around for some time and may have influenced the abacus in the West.

  This particular abacus has counters suspended on rods, which are layered on two decks. On the lower deck there are five beads per rod, each of which is worth one; on the top deck there are two beads, each worth five. Each rod represents a decimal column (unit, tens, hundreds, etc.) and pushing the beads towards the separator in the middle signifies the number. For example, 123,456 would look something like this:

  The extra bead on each deck can be used for performing calculations.

  INDIAN MATHEMATICS

  In 1920 archaeological excavations in north-western India unearthed the Indus Valley civilization, which existed from c. 3500 BC to 2000 BC. These Bronze Age settlements, contemporary to the first urban areas in Egypt and Mesopotamia, indicated the Ancient Indians had a good understanding of basic mathematical concepts, and possessed a standardized system of weights and measures.

  Ancient Indian religious texts also contain evidence of mathematical knowledge; in Hinduism, mathematics, astronomy and astrology were considered to be in the same field, and they each had important religious implications. It was a religious requirement that all altars should occupy the same amount of floor space, even if they weren’t the same shape or used different configurations of bricks – all of which required a good knowledge of geometry. Texts from 700 BC show the Ancient Indians possessed knowledge of Pythagoras’ theorem, irrational numbers and methods for calculating them.

  Astronomical discoveries

  Brahmagupta (AD 598–668), an astronomer, was the first person to treat zero as a number. The Hindu numeral system, predecessor to the Hindu-Arabic numeral system that we use today (see here), developed over time and was fully established by the end of the first millennium AD. Up until Brahmagupta’s treatment of zero as a number, it had been used merely as a place-holder within various number systems in order to show a gap. Brahmagupta, however, thought of 0 simply as a whole number or integer that lies between 1 and -1. He wrote down rules for its use in arithmetic, alongside rules for using negative numbers.

  Useful Functions

  Aryabhata (AD 475–550) was an astronomer who is credited with being the first person to introduce trigonometry, which we use to work out lengths and angles in triangles, and the concept of the sine, cosine and tangent functions.

  Brahmagupta recognized that an equation could have a negative solution and, as a result, that any positive integer would have a positive and negative square root. For example, the square roots of 36 are 6 and -6, because, as Brahmagupta himself stated, a negative multiplied by a negative gives a positive.

  Brahmagupta is also famous for developing Brahmagupta’s formula, which tells us the area of a cyclic quadrilateral – a four-sided shape, the corners of which lie on a circle:

  Modern Indian Mathematics

  Srinvasa Ramanujan (1887–1920) was an Indian mathematical genius. After dropping out of university, he became an accounting clerk at a government office, from where he sent papers to various British mathematicians for consideration. The English mathematician Godfrey Hardy (1877–1947) recognized Ramanujan’s genius and arranged for him to have a research post at the University of Madras.

  In 1914 Ramanujan joined Hardy at Cambridge University and remained in England for five years, in which time he became one of the youngest ever members of the Royal Society, had work published and finally gained a degree. However, Ramanujan was often ill.

  During one bout of illness, Hardy visited him and mentioned that the number of his taxi, 1729, was ‘rather dull’. Ramanujan replied instantly that 1,729 was the lowest number that could be written as the sum of two cubes in two different ways, and as such, was actually quite interesting:

  13 + 123 = 1 + 1728 = 1729

  93 + 103 = 729 + 1000 = 1729

  There are lower numbers that can be written as the sum of two cubes, but 1,729 is the lowest number that can be written like this in two ways, and Ramanujan’s instant recognition of this was nothing short of miraculous.

  In his short life Ramanujan came up with nearly 4,000 theorems, equations and identities that still inspire mathematical research to this day.

  If you find half the perimeter of the quadrilateral (let’s call it ‘s’) then the area of the shape can be found using Brahmagupta’s formula:

  √(s-a)(s-b)(s-c)(s-d)

  Although the Indians were clearly excellent mathematicians, when the British began to take control of the country in the 1700s they assumed the backward pagan Hindus had nothing of worth to contribute beyond vast natural resources and cheap labour. It has only been in the last hundred years that we have come to appreciate the mathematical heritage of the sub-continent.

  ISLAMIC MATHEMATICS

  Mohammed, the founder of Islam, was born in AD 570. In the two centuries following Mohammed’s birth the Islamic Empire came to dominate all of the Middle East, Central Asia, North Africa and what would become Spain and Portugal. This Islamic Golden Age saw much important mathematical progress emerge from the countries in the empire, while Europe remained still in its Dark Ages.

  The Islamic religion itself is particularly open to the idea of science, which contrasted strongly with the ideas prevalent in medieval Europe, where it was often considered heretical to question or investigate the workings of a world made by God.

  The Islamic Empire too was committed to gathering the knowledge of the ancient world. Texts in Classical Greek and Latin, Ancient Egyptian, Mesopotamian, Indian, Chinese and Persian were all translated by Islamic scholars, broadening their availability to the empire’s scientists and mathematicians.

  AL-KHWARIZMI (c. 790–c. 850)

  Mathematician Al-Khwarizmi hailed from an area situated in present-day Uzbekistan, and he is credited with providing several significant contributions to mathematics. Although some of his original works have survived, he is familiar to us through editions of his work translated into Latin for use later in Europe.

  The new number system

  One of Al-Khwarizmi’s significant legacies was what is now known as the Hindu-Arabic numeral system, which we still use to this day. Derived from his Book of Addition and Subtraction According to the Hindu Calculation, Al-Khwarizmi’s system of numbers, developed over time in India from c. 300 BC and passed through into Persia, revolutionized arithmetic.

  Up to this point, no culture had a system of numerals with which it was really possible to use in arithmetic. Numbers would always be converted into letters or symbols (either mentally or using counters, abacuses or other such tools), the calculation performed and the result reconverted back into numerals. Lots of symbols were often needed to show a number, many of which were difficult to decipher at a glance.

  The Hindu-Arabic system contains just ten symbols – 0 1 2 3 4 5 6 7 8 9 – that could be used to write any number. It is important to note that these symbols were exactly that – they were not associated with the value they represented through stripes or dots. The zero (from the Arabic zifer, meaning ‘empty’) meant that the symbols could have a diffe
rent value depending on where they were positioned in the number – which freed people of the difficulty the Mesopotamians had faced. Today, the concept of place-value is taken for granted. But the idea that the 8 in 80 is worth eight tens, and yet could be used, with the help of those friendly zeros, to also mean 800 or 8 million was revolutionary at the time. In fact, some European scholars were deeply suspicious of this heathen method of calculating, despite its advantages.

  In the Book of Addition and Subtraction According to the Hindu Calculation Al-Khwarizmi describes how to do arithmetic using these new numbers. His translators referred to him by the Latinized name Algorism. Over time Al-Khwarizmi’s methods of calculation became known as algorithms, a word still in use today and which refers to a set of instructions to perform a calculation – which is exactly what Al-Khwarizmi provided.

  Transforming Mathematics

  Al-Khwarizmi also wrote The Compendious Book on Calculation by Transformations and Dividing, which set out to show how to solve different types of quadratic equations (equations in which the unknown numbers are squared). ‘Transformations’ in Arabic is Al-Jabr, from which we derive (via Latin) the English term algebra. While Al-Khwarizmi himself did not replace unknown numbers with letters, he did pave the way for this to happen.

  OMAR KHAYYÁM (1048–1131)

  Persian scholar Omar Khayyám is best known for his The Rubaiyat of Omar Khayyám, a selection of poems that were later translated into English in the nineteenth century by the poet Edward Fitzgerald. Multi-talented, Khayyám spent a great proportion of his life as a court astronomer to a sultan, while also working as a scientist and mathematician.

  Khayyám’s mathematical works were far-reaching. He expanded on Al-Khwarizmi’s earlier work in algebra, and he was one of the first mathematicians to use the replacement of unknown numbers with letters to make solving equations easier. He also devised techniques for solving cubic equations, where the unknown term has been cubed. Khayyám’s insight enabled him to be one of the first people to connect geometry and algebra, which had until that point been separate disciplines.

  Endless possibilities

  Khayyám also investigated something now called the binomial theorem. This has many applications in mathematics, many of which involve rather tricky algebra. One side product of binomial theorem is something called Pascal’s triangle, named after the seventeenth-century French mathematician Blaise Pascal, who borrowed the triangle from Khayyám, who in turn borrowed it from the Chinese (see here). Unlike the binomial theorem, Pascal’s triangle is simple to understand: the number in each cell of a triangle is made by adding together the two numbers above it.

  Pascal’s triangle is useful because each horizontal row shows us the binomial coefficients that the binomial theorem spits out. These can tell us how many combinations of two different things it is possible to have.

  For example, imagine you have planted a row of four flower bulbs. It says on the packet that the flowers can be blue or pink, with an equal chance of having either.

  There is one way for you to grow four blues:

  BBBB

  There are four ways for you to grow three blues and one pink:

  BBBP

  BBPB

  BPBB

  PBBB

  There are six ways for you to end up with two of each:

  BBPP

  BPPB

  PBBP

  PPBB

  BPBP

  PBPB

  There are four ways for you to have three pinks and one blue:

  PPPB

  PPBP

  PBPP

  BPPP

  And one way for you to have four pinks:

  PPPP

  If you look across the fourth row of the triangle, it says 1, 4, 6, 4, 1, which corresponds to the number of ways worked out in the example above. Because there is an equal chance of a flower being either pink or blue, you can also see that you’re most likely to end up with two of each colour because there are 6 out of 16 total ways this could happen.

  A new geometry

  Khayyám also wrote a book that tackled Euclid’s fifth postulate, which had long rankled a contingent of mathematicians. The fifth postulate Euclid wrote concerned parallel lines, and it is therefore normally referred to as the parallel postulate.

  Imagine two lines (PQ and RS) with a third (XY) crossing them. Inside PQ and RS we now have four angles, two on each side of the XY: a, b, c and d:

  The parallel postulate suggests that if you add the pairs of angles on the same side of XY together (e.g. a+b and c+d) then PQ and RS will cross on the side of the line where the sum of the angles is less than 180°. If the angles on each side add up to 180° then PQ and RS are parallel and therefore will never cross.

  Mathematicians, however, have argued over the ages that this postulate is not quite as obvious as Euclid made out. Khayyám was the first to come up with a counter-example, arguing that Euclid’s parallel postulate does not always work if the surface you are drawing on is curved. Thus Khayyám instigated the ideas of elliptical and hyperbolic geometry, a direct challenge to the simple Euclidean geometry that had gone before. This kind of thinking would eventually help Albert Einstein to come up with his ideas of space-time and gravity.

  The Middle Ages in Europe

  Despite Europe’s plunge into the Dark Ages – so-called because it was thought that following the fall of the Roman Empire the continent had reverted back to a barbaric state of tribal warfare and religious fundamentalism – there remained a coterie of individuals intent on pushing the boundaries of mathematics even during these difficult times.

  BEDE (672–735)

  The Venerable Bede is known more perhaps for his contribution as a historian than for the role he played in the development of mathematics. Bede was a monk living in north-eastern England and his translation of a number of scholarly works into the English of the time helped to spread an enormous amount of knowledge.

  Bede’s contribution to mathematics began when he attempted to develop a way to calculate accurately when Easter would fall. At the time it was thought to fall on the first Sunday after the first full moon following the spring equinox. Missing Easter mass following the calculation of an incorrect date would have resulted in excommunication, and therefore damnation, so Bede’s was no trivial task.

  Dating in the Dark Ages

  In order to calculate the date of Easter, it was necessary for Bede to rationalize the date of the spring equinox with the lunar calendar. This was a difficult task in itself because the date of the equinox varied because the Julian calendar in use at the time was unreliable. Because the date of the equinox varied each year, and full moons come at alternate 29- or 30-day intervals, it meant that there was a 19-year cycle of possible dates for Easter. The procedure for calculating the date of Easter has been known as computus (meaning ‘computation’) ever since.

  Once Bede had completed the computus, he decided to sort out dating the rest of history as well. Prior to Bede’s endeavours, historians had been dating things in reference to the lifetime of the current emperor or king, for example: ‘the Vikings first attacked in the third year of Aethelred’s reign.’ This method, of course, relied on the reader knowing when Aethelred was around in the first place. Bede decided that it would be far more sensible to date everything occurring either before or after the birth of Jesus Christ. Although not originally Bede’s idea – that responsibility lay with Dionysius Exiguus, a south-eastern European monk active during the sixth century – such was his influence that we have been using AD (Anno Domini, Year of the Lord) and BC (Before Christ) ever since.

  Finger Talk

  Bede also wrote a book called On Counting and Speaking With the Fingers, which allowed the reader to use hand signals for numbers into the millions – a super-sized version of the systems we saw Stone Age cultures using. Again, such was his influence, people were still referencing Bede’s book 1,000 years later.

  ALCUIN OF YORK (730–804)

  A gifted p
oet, scholar, teacher and mathematician, Alcuin of York began his academic life under the instruction of Archbishop Ecgbert of York, who in turn had been tutored by Bede. Alcuin’s main mathematical work was a textbook for students titled Propositiones ad acuendos juvenes (Problems to Sharpen the Young). The book contains many word-based logic puzzles, a few of which have become quite famous, including the following two river-crossing problems.

  Heavy load

  The first problem relates to a man trying to cross a river with a wolf, a goat and a cabbage. The man’s boat is very small and he can only fit one thing in the boat with him at a time. However, if he leaves the goat and the wolf together, the wolf will eat the goat. If he leaves the goat and the cabbage together, the goat will eat the cabbage. How does he get them all safely across the river?

  Answer: this is a good medieval example of lateral thinking. Clearly, on his first run the man can only take the goat across the river. On his second trip he brings the wolf across but takes the goat back with him; he then leaves the goat there and takes the cabbage across, and then makes a final trip for the well-travelled goat.

  Family matters

  In the second problem, a couple, who are of equal weight, have two children, each of whom weighs half the weight of one of the adults. All four people need to cross a river, but their boat will only hold the weight of one adult. How can they cross in safety?

  Answer: the children cross the river in the boat. One child stays on the far bank while the other child returns. Dad crosses to the far bank and the child returns with the boat to be with the mum and the other child. The two children cross again and one remains on the far side with Dad. The other child returns to be with Mum. Mum crosses to the far side, and the child with the dad returns to collect the other child to reunite the family.