0:00:10 > 0:00:15From measuring time to understanding our position in the universe,
0:00:15 > 0:00:19from mapping the Earth to navigating the seas,
0:00:19 > 0:00:24from man's earliest inventions to today's advanced technologies,
0:00:24 > 0:00:28mathematics has been the pivot on which human life depends.
0:00:34 > 0:00:37The first steps of man's mathematical journey
0:00:37 > 0:00:42were taken by the ancient cultures of Egypt, Mesopotamia and Greece -
0:00:42 > 0:00:49cultures which created the basic language of number and calculation.
0:00:49 > 0:00:51But when ancient Greece fell into decline,
0:00:51 > 0:00:54mathematical progress juddered to a halt.
0:00:58 > 0:01:00But that was in the West.
0:01:00 > 0:01:04In the East, mathematics would reach dynamic new heights.
0:01:08 > 0:01:11But in the West, much of this mathematical heritage
0:01:11 > 0:01:14has been conveniently forgotten or shaded from view.
0:01:14 > 0:01:18Due credit has not been given to the great mathematical breakthroughs
0:01:18 > 0:01:21that ultimately changed the world we live in.
0:01:21 > 0:01:24This is the untold story of the mathematics of the East
0:01:24 > 0:01:29that would transform the West and give birth to the modern world.
0:02:04 > 0:02:08The Great Wall of China stretches for thousands of miles.
0:02:08 > 0:02:12Nearly 2,000 years in the making, this vast, defensive wall
0:02:12 > 0:02:17was begun in 220BC to protect China's growing empire.
0:02:20 > 0:02:23The Great Wall of China is an amazing feat of engineering
0:02:23 > 0:02:26built over rough and high countryside.
0:02:26 > 0:02:28As soon as they started building,
0:02:28 > 0:02:31the ancient Chinese realised they had to make calculations
0:02:31 > 0:02:36about distances, angles of elevation and amounts of material.
0:02:36 > 0:02:38So perhaps it isn't surprising that this inspired
0:02:38 > 0:02:43some very clever mathematics to help build Imperial China.
0:02:43 > 0:02:46At the heart of ancient Chinese mathematics
0:02:46 > 0:02:49was an incredibly simple number system
0:02:49 > 0:02:53which laid the foundations for the way we count in the West today.
0:02:57 > 0:03:03When a mathematician wanted to do a sum, he would use small bamboo rods.
0:03:03 > 0:03:07These rods were arranged to represent the numbers one to nine.
0:03:14 > 0:03:16They were then placed in columns,
0:03:16 > 0:03:20each column representing units, tens,
0:03:20 > 0:03:23hundreds, thousands and so on.
0:03:25 > 0:03:28So the number 924 was represented by putting
0:03:28 > 0:03:33the symbol 4 in the units column, the symbol 2 in the tens column
0:03:33 > 0:03:36and the symbol 9 in the hundreds column.
0:03:43 > 0:03:46This is what we call a decimal place-value system,
0:03:46 > 0:03:49and it's very similar to the one we use today.
0:03:49 > 0:03:52We too use numbers from one to nine, and we use their position
0:03:52 > 0:03:57to indicate whether it's units, tens, hundreds or thousands.
0:03:57 > 0:04:00But the power of these rods is that it makes calculations very quick.
0:04:00 > 0:04:04In fact, the way the ancient Chinese did their calculations
0:04:04 > 0:04:06is very similar to the way we learn today in school.
0:04:12 > 0:04:14Not only were the ancient Chinese
0:04:14 > 0:04:17the first to use a decimal place-value system,
0:04:17 > 0:04:21but they did so over 1,000 years before we adopted it in the West.
0:04:21 > 0:04:25But they only used it when calculating with the rods.
0:04:25 > 0:04:28When writing the numbers down,
0:04:28 > 0:04:31the ancient Chinese didn't use the place-value system.
0:04:33 > 0:04:37Instead, they used a far more laborious method,
0:04:37 > 0:04:42in which special symbols stood for tens, hundreds, thousands and so on.
0:04:44 > 0:04:46So the number 924 would be written out
0:04:46 > 0:04:50as nine hundreds, two tens and four.
0:04:50 > 0:04:53Not quite so efficient.
0:04:54 > 0:04:56The problem was
0:04:56 > 0:04:59that the ancient Chinese didn't have a concept of zero.
0:04:59 > 0:05:02They didn't have a symbol for zero. It just didn't exist as a number.
0:05:02 > 0:05:04Using the counting rods,
0:05:04 > 0:05:08they would use a blank space where today we would write a zero.
0:05:08 > 0:05:12The problem came with trying to write down this number, which is why
0:05:12 > 0:05:15they had to create these new symbols for tens, hundreds and thousands.
0:05:15 > 0:05:20Without a zero, the written number was extremely limited.
0:05:23 > 0:05:26But the absence of zero didn't stop
0:05:26 > 0:05:29the ancient Chinese from making giant mathematical steps.
0:05:29 > 0:05:32In fact, there was a widespread fascination
0:05:32 > 0:05:34with number in ancient China.
0:05:34 > 0:05:39According to legend, the first sovereign of China,
0:05:39 > 0:05:41the Yellow Emperor, had one of his deities
0:05:41 > 0:05:44create mathematics in 2800BC,
0:05:44 > 0:05:49believing that number held cosmic significance. And to this day,
0:05:49 > 0:05:53the Chinese still believe in the mystical power of numbers.
0:05:56 > 0:06:01Odd numbers are seen as male, even numbers, female.
0:06:01 > 0:06:04The number four is to be avoided at all costs.
0:06:04 > 0:06:07The number eight brings good fortune.
0:06:08 > 0:06:11And the ancient Chinese were drawn to patterns in numbers,
0:06:11 > 0:06:15developing their own rather early version of sudoku.
0:06:17 > 0:06:20It was called the magic square.
0:06:24 > 0:06:28Legend has it that thousands of years ago, Emperor Yu was visited
0:06:28 > 0:06:32by a sacred turtle that came out of the depths of the Yellow River.
0:06:32 > 0:06:34On its back were numbers
0:06:34 > 0:06:37arranged into a magic square, a little like this.
0:06:46 > 0:06:47In this square,
0:06:47 > 0:06:51which was regarded as having great religious significance,
0:06:51 > 0:06:55all the numbers in each line - horizontal, vertical and diagonal -
0:06:55 > 0:06:59all add up to the same number - 15.
0:07:02 > 0:07:05Now, the magic square may be no more than a fun puzzle,
0:07:05 > 0:07:07but it shows the ancient Chinese fascination
0:07:07 > 0:07:10with mathematical patterns, and it wasn't too long
0:07:10 > 0:07:13before they were creating even bigger magic squares
0:07:13 > 0:07:18with even greater magical and mathematical powers.
0:07:25 > 0:07:28But mathematics also played
0:07:28 > 0:07:33a vital role in the running of the emperor's court.
0:07:33 > 0:07:35The calendar and the movement of the planets
0:07:35 > 0:07:38were of the utmost importance to the emperor,
0:07:38 > 0:07:43influencing all his decisions, even down to the way his day was planned,
0:07:43 > 0:07:47so astronomers became prized members of the imperial court,
0:07:47 > 0:07:50and astronomers were always mathematicians.
0:07:55 > 0:07:58Everything in the emperor's life was governed by the calendar,
0:07:58 > 0:08:02and he ran his affairs with mathematical precision.
0:08:02 > 0:08:05The emperor even got his mathematical advisors
0:08:05 > 0:08:08to come up with a system to help him sleep his way
0:08:08 > 0:08:11through the vast number of women he had in his harem.
0:08:11 > 0:08:14Never one to miss a trick, the mathematical advisors decided
0:08:14 > 0:08:19to base the harem on a mathematical idea called a geometric progression.
0:08:19 > 0:08:23Maths has never had such a fun purpose!
0:08:23 > 0:08:26Legend has it that in the space of 15 nights,
0:08:26 > 0:08:30the emperor had to sleep with 121 women...
0:08:36 > 0:08:38..the empress,
0:08:38 > 0:08:40three senior consorts,
0:08:40 > 0:08:43nine wives,
0:08:43 > 0:08:4527 concubines
0:08:45 > 0:08:47and 81 slaves.
0:08:49 > 0:08:52The mathematicians would soon have realised
0:08:52 > 0:08:56that this was a geometric progression - a series of numbers
0:08:56 > 0:08:58in which you get from one number to the next
0:08:58 > 0:09:03by multiplying the same number each time - in this case, three.
0:09:04 > 0:09:08Each group of women is three times as large as the previous group,
0:09:08 > 0:09:12so the mathematicians could quickly draw up a rota to ensure that,
0:09:12 > 0:09:14in the space of 15 nights,
0:09:14 > 0:09:18the emperor slept with every woman in the harem.
0:09:19 > 0:09:23The first night was reserved for the empress.
0:09:23 > 0:09:26The next was for the three senior consorts.
0:09:26 > 0:09:29The nine wives came next,
0:09:29 > 0:09:35and then the 27 concubines were chosen in rotation, nine each night.
0:09:35 > 0:09:38And then finally, over a period of nine nights,
0:09:38 > 0:09:42the 81 slaves were dealt with in groups of nine.
0:09:47 > 0:09:50Being the emperor certainly required stamina,
0:09:50 > 0:09:52a bit like being a mathematician,
0:09:52 > 0:09:54but the object is clear -
0:09:54 > 0:09:58to procure the best possible imperial succession.
0:09:58 > 0:10:00The rota ensured that the emperor
0:10:00 > 0:10:04slept with the ladies of highest rank closest to the full moon,
0:10:04 > 0:10:06when their yin, their female force,
0:10:06 > 0:10:11would be at its highest and be able to match his yang, or male force.
0:10:16 > 0:10:20The emperor's court wasn't alone in its dependence on mathematics.
0:10:20 > 0:10:23It was central to the running of the state.
0:10:23 > 0:10:28Ancient China was a vast and growing empire with a strict legal code,
0:10:28 > 0:10:30widespread taxation
0:10:30 > 0:10:33and a standardised system of weights, measures and money.
0:10:35 > 0:10:37The empire needed
0:10:37 > 0:10:41a highly trained civil service, competent in mathematics.
0:10:43 > 0:10:47And to educate these civil servants was a mathematical textbook,
0:10:47 > 0:10:51probably written in around 200BC - the Nine Chapters.
0:10:54 > 0:10:58The book is a compilation of 246 problems
0:10:58 > 0:11:02in practical areas such as trade, payment of wages and taxes.
0:11:05 > 0:11:08And at the heart of these problems lies
0:11:08 > 0:11:13one of the central themes of mathematics, how to solve equations.
0:11:16 > 0:11:19Equations are a little bit like cryptic crosswords.
0:11:19 > 0:11:21You're given a certain amount of information
0:11:21 > 0:11:24about some unknown numbers, and from that information
0:11:24 > 0:11:27you've got to deduce what the unknown numbers are.
0:11:27 > 0:11:29For example, with my weights and scales,
0:11:29 > 0:11:31I've found out that one plum...
0:11:32 > 0:11:35..together with three peaches
0:11:35 > 0:11:39weighs a total of 15g.
0:11:41 > 0:11:42But...
0:11:43 > 0:11:45..two plums
0:11:45 > 0:11:48together with one peach
0:11:48 > 0:11:50weighs a total of 10g.
0:11:50 > 0:11:55From this information, I can deduce what a single plum weighs
0:11:55 > 0:11:59and a single peach weighs, and this is how I do it.
0:12:00 > 0:12:02If I take the first set of scales,
0:12:02 > 0:12:05one plum and three peaches weighing 15g,
0:12:05 > 0:12:11and double it, I get two plums and six peaches weighing 30g.
0:12:14 > 0:12:18If I take this and subtract from it the second set of scales -
0:12:18 > 0:12:20that's two plums and a peach weighing 10g -
0:12:20 > 0:12:23I'm left with an interesting result -
0:12:23 > 0:12:25no plums.
0:12:25 > 0:12:28Having eliminated the plums,
0:12:28 > 0:12:31I've discovered that five peaches weighs 20g,
0:12:31 > 0:12:34so a single peach weighs 4g,
0:12:34 > 0:12:39and from this I can deduce that the plum weighs 3g.
0:12:39 > 0:12:42The ancient Chinese went on to apply similar methods
0:12:42 > 0:12:45to larger and larger numbers of unknowns,
0:12:45 > 0:12:50using it to solve increasingly complicated equations.
0:12:50 > 0:12:52What's extraordinary is
0:12:52 > 0:12:55that this particular system of solving equations
0:12:55 > 0:12:59didn't appear in the West until the beginning of the 19th century.
0:12:59 > 0:13:03In 1809, while analysing a rock called Pallas in the asteroid belt,
0:13:03 > 0:13:05Carl Friedrich Gauss,
0:13:05 > 0:13:08who would become known as the prince of mathematics,
0:13:08 > 0:13:09rediscovered this method
0:13:09 > 0:13:13which had been formulated in ancient China centuries earlier.
0:13:13 > 0:13:17Once again, ancient China streets ahead of Europe.
0:13:21 > 0:13:23But the Chinese were to go on to solve
0:13:23 > 0:13:27even more complicated equations involving far larger numbers.
0:13:27 > 0:13:31In what's become known as the Chinese remainder theorem,
0:13:31 > 0:13:35the Chinese came up with a new kind of problem.
0:13:35 > 0:13:38In this, we know the number that's left
0:13:38 > 0:13:42when the equation's unknown number is divided by a given number -
0:13:42 > 0:13:44say, three, five or seven.
0:13:46 > 0:13:50Of course, this is a fairly abstract mathematical problem,
0:13:50 > 0:13:54but the ancient Chinese still couched it in practical terms.
0:13:56 > 0:13:59So a woman in the market has a tray of eggs,
0:13:59 > 0:14:02but she doesn't know how many eggs she's got.
0:14:02 > 0:14:05What she does know is that if she arranges them in threes,
0:14:05 > 0:14:08she has one egg left over.
0:14:08 > 0:14:13If she arranges them in fives, she gets two eggs left over.
0:14:13 > 0:14:15But if she arranged them in rows of seven,
0:14:15 > 0:14:18she found she had three eggs left over.
0:14:18 > 0:14:22The ancient Chinese found a systematic way to calculate
0:14:22 > 0:14:26that the smallest number of eggs she could have had in the tray is 52.
0:14:26 > 0:14:29But the more amazing thing is that you can capture
0:14:29 > 0:14:31such a large number, like 52,
0:14:31 > 0:14:34by using these small numbers like three, five and seven.
0:14:34 > 0:14:36This way of looking at numbers
0:14:36 > 0:14:40would become a dominant theme over the last two centuries.
0:14:49 > 0:14:53By the 6th century AD, the Chinese remainder theorem was being used
0:14:53 > 0:14:57in ancient Chinese astronomy to measure planetary movement.
0:14:57 > 0:15:00But today it still has practical uses.
0:15:00 > 0:15:05Internet cryptography encodes numbers using mathematics
0:15:05 > 0:15:09that has its origins in the Chinese remainder theorem.
0:15:17 > 0:15:19By the 13th century,
0:15:19 > 0:15:22mathematics was long established on the curriculum,
0:15:22 > 0:15:26with over 30 mathematics schools scattered across the country.
0:15:26 > 0:15:30The golden age of Chinese maths had arrived.
0:15:32 > 0:15:36And its most important mathematician was called Qin Jiushao.
0:15:38 > 0:15:43Legend has it that Qin Jiushao was something of a scoundrel.
0:15:43 > 0:15:47He was a fantastically corrupt imperial administrator
0:15:47 > 0:15:50who crisscrossed China, lurching from one post to another.
0:15:50 > 0:15:54Repeatedly sacked for embezzling government money,
0:15:54 > 0:15:57he poisoned anyone who got in his way.
0:15:59 > 0:16:02Qin Jiushao was reputedly described as
0:16:02 > 0:16:04as violent as a tiger or a wolf
0:16:04 > 0:16:07and as poisonous as a scorpion or a viper
0:16:07 > 0:16:10so, not surprisingly, he made a fierce warrior.
0:16:10 > 0:16:13For ten years, he fought against the invading Mongols,
0:16:13 > 0:16:17but for much of that time he was complaining that his military life
0:16:17 > 0:16:19took him away from his true passion.
0:16:19 > 0:16:22No, not corruption, but mathematics.
0:16:34 > 0:16:36Qin started trying to solve equations
0:16:36 > 0:16:39that grew out of trying to measure the world around us.
0:16:39 > 0:16:41Quadratic equations involve numbers
0:16:41 > 0:16:46that are squared, or to the power of two - say, five times five.
0:16:47 > 0:16:49The ancient Mesopotamians
0:16:49 > 0:16:52had already realised that these equations
0:16:52 > 0:16:55were perfect for measuring flat, two-dimensional shapes,
0:16:55 > 0:16:57like Tiananmen Square.
0:17:00 > 0:17:02But Qin was interested
0:17:02 > 0:17:06in more complicated equations - cubic equations.
0:17:08 > 0:17:10These involve numbers which are cubed,
0:17:10 > 0:17:15or to the power of three - say, five times five times five,
0:17:15 > 0:17:19and they were perfect for capturing three-dimensional shapes,
0:17:19 > 0:17:21like Chairman Mao's mausoleum.
0:17:23 > 0:17:26Qin found a way of solving cubic equations,
0:17:26 > 0:17:28and this is how it worked.
0:17:32 > 0:17:34Say Qin wants to know
0:17:34 > 0:17:37the exact dimensions of Chairman Mao's mausoleum.
0:17:39 > 0:17:42He knows the volume of the building
0:17:42 > 0:17:45and the relationships between the dimensions.
0:17:47 > 0:17:49In order to get his answer,
0:17:49 > 0:17:53Qin uses what he knows to produce a cubic equation.
0:17:53 > 0:17:57He then makes an educated guess at the dimensions.
0:17:57 > 0:18:01Although he's captured a good proportion of the mausoleum,
0:18:01 > 0:18:03there are still bits left over.
0:18:05 > 0:18:09Qin takes these bits and creates a new cubic equation.
0:18:09 > 0:18:11He can now refine his first guess
0:18:11 > 0:18:15by trying to find a solution to this new cubic equation, and so on.
0:18:18 > 0:18:21Each time he does this, the pieces he's left with
0:18:21 > 0:18:26get smaller and smaller and his guesses get better and better.
0:18:28 > 0:18:31What's striking is that Qin's method for solving equations
0:18:31 > 0:18:34wasn't discovered in the West until the 17th century,
0:18:34 > 0:18:39when Isaac Newton came up with a very similar approximation method.
0:18:39 > 0:18:41The power of this technique is
0:18:41 > 0:18:46that it can be applied to even more complicated equations.
0:18:46 > 0:18:49Qin even used his techniques to solve an equation
0:18:49 > 0:18:51involving numbers up to the power of ten.
0:18:51 > 0:18:56This was extraordinary stuff - highly complex mathematics.
0:18:58 > 0:19:00Qin may have been years ahead of his time,
0:19:00 > 0:19:03but there was a problem with his technique.
0:19:03 > 0:19:05It only gave him an approximate solution.
0:19:05 > 0:19:09That might be good enough for an engineer - not for a mathematician.
0:19:09 > 0:19:13Mathematics is an exact science. We like things to be precise,
0:19:13 > 0:19:16and Qin just couldn't come up with a formula
0:19:16 > 0:19:19to give him an exact solution to these complicated equations.
0:19:27 > 0:19:30China had made great mathematical leaps,
0:19:30 > 0:19:34but the next great mathematical breakthroughs were to happen
0:19:34 > 0:19:37in a country lying to the southwest of China -
0:19:37 > 0:19:40a country that had a rich mathematical tradition
0:19:40 > 0:19:43that would change the face of maths for ever.
0:20:13 > 0:20:18India's first great mathematical gift lay in the world of number.
0:20:18 > 0:20:22Like the Chinese, the Indians had discovered the mathematical benefits
0:20:22 > 0:20:24of the decimal place-value system
0:20:24 > 0:20:28and were using it by the middle of the 3rd century AD.
0:20:30 > 0:20:34It's been suggested that the Indians learned the system
0:20:34 > 0:20:38from Chinese merchants travelling in India with their counting rods,
0:20:38 > 0:20:42or they may well just have stumbled across it themselves.
0:20:42 > 0:20:46It's all such a long time ago that it's shrouded in mystery.
0:20:48 > 0:20:51We may never know how the Indians came up with their number system,
0:20:51 > 0:20:54but we do know that they refined and perfected it,
0:20:54 > 0:20:58creating the ancestors for the nine numerals used across the world now.
0:20:58 > 0:21:01Many rank the Indian system of counting
0:21:01 > 0:21:05as one of the greatest intellectual innovations of all time,
0:21:05 > 0:21:09developing into the closest thing we could call a universal language.
0:21:27 > 0:21:29But there was one number missing,
0:21:29 > 0:21:33and it was the Indians who would introduce it to the world.
0:21:39 > 0:21:44The earliest known recording of this number dates from the 9th century,
0:21:44 > 0:21:48though it was probably in practical use for centuries before.
0:21:49 > 0:21:53This strange new numeral is engraved on the wall
0:21:53 > 0:21:57of small temple in the fort of Gwalior in central India.
0:22:01 > 0:22:05So here we are in one of the holy sites of the mathematical world,
0:22:05 > 0:22:08and what I'm looking for is in this inscription on the wall.
0:22:09 > 0:22:12Up here are some numbers, and...
0:22:12 > 0:22:14here's the new number.
0:22:14 > 0:22:16It's zero.
0:22:21 > 0:22:25It's astonishing to think that before the Indians invented it,
0:22:25 > 0:22:28there was no number zero.
0:22:28 > 0:22:31To the ancient Greeks, it simply hadn't existed.
0:22:31 > 0:22:35To the Egyptians, the Mesopotamians and, as we've seen, the Chinese,
0:22:35 > 0:22:39zero had been in use but as a placeholder, an empty space
0:22:39 > 0:22:42to show a zero inside a number.
0:22:45 > 0:22:48The Indians transformed zero from a mere placeholder
0:22:48 > 0:22:51into a number that made sense in its own right -
0:22:51 > 0:22:54a number for calculation, for investigation.
0:22:54 > 0:22:58This brilliant conceptual leap would revolutionise mathematics.
0:23:02 > 0:23:06Now, with just ten digits - zero to nine - it was suddenly possible
0:23:06 > 0:23:09to capture astronomically large numbers
0:23:09 > 0:23:12in an incredibly efficient way.
0:23:15 > 0:23:18But why did the Indians make this imaginative leap?
0:23:18 > 0:23:20Well, we'll never know for sure,
0:23:20 > 0:23:24but it's possible that the idea and symbol that the Indians use for zero
0:23:24 > 0:23:27came from calculations they did with stones in the sand.
0:23:27 > 0:23:31When stones were removed from the calculation,
0:23:31 > 0:23:33a small, round hole was left in its place,
0:23:33 > 0:23:37representing the movement from something to nothing.
0:23:39 > 0:23:44But perhaps there is also a cultural reason for the invention of zero.
0:23:44 > 0:23:47HORNS BLOW AND DRUMS BANG
0:23:47 > 0:23:50METALLIC BEATING
0:23:53 > 0:23:57For the ancient Indians, the concepts of nothingness and eternity
0:23:57 > 0:24:00lay at the very heart of their belief system.
0:24:04 > 0:24:07BELL CLANGS AND SILENCE FALLS
0:24:09 > 0:24:13In the religions of India, the universe was born from nothingness,
0:24:13 > 0:24:17and nothingness is the ultimate goal of humanity.
0:24:17 > 0:24:18So it's perhaps not surprising
0:24:18 > 0:24:22that a culture that so enthusiastically embraced the void
0:24:22 > 0:24:25should be happy with the notion of zero.
0:24:25 > 0:24:30The Indians even used the word for the philosophical idea of the void,
0:24:30 > 0:24:33shunya, to represent the new mathematical term "zero".
0:24:47 > 0:24:52In the 7th century, the brilliant Indian mathematician Brahmagupta
0:24:52 > 0:24:55proved some of the essential properties of zero.
0:25:01 > 0:25:04Brahmagupta's rules about calculating with zero
0:25:04 > 0:25:08are taught in schools all over the world to this day.
0:25:09 > 0:25:12One plus zero equals one.
0:25:13 > 0:25:16One minus zero equals one.
0:25:16 > 0:25:19One times zero is equal to zero.
0:25:24 > 0:25:28But Brahmagupta came a cropper when he tried to do one divided by zero.
0:25:28 > 0:25:31After all, what number times zero equals one?
0:25:31 > 0:25:35It would require a new mathematical concept, that of infinity,
0:25:35 > 0:25:38to make sense of dividing by zero,
0:25:38 > 0:25:41and the breakthrough was made by a 12th-century Indian mathematician
0:25:41 > 0:25:45called Bhaskara II, and it works like this.
0:25:45 > 0:25:51If I take a fruit and I divide it into halves, I get two pieces,
0:25:51 > 0:25:54so one divided by a half is two.
0:25:54 > 0:25:57If I divide it into thirds, I get three pieces.
0:25:57 > 0:26:00So when I divide it into smaller and smaller fractions,
0:26:00 > 0:26:04I get more and more pieces, so ultimately,
0:26:04 > 0:26:06when I divide by a piece
0:26:06 > 0:26:10which is of zero size, I'll have infinitely many pieces.
0:26:10 > 0:26:14So for Bhaskara, one divided by zero is infinity.
0:26:22 > 0:26:26But the Indians would go further in their calculations with zero.
0:26:27 > 0:26:31For example, if you take three from three and get zero,
0:26:31 > 0:26:35what happens when you take four from three?
0:26:35 > 0:26:37It looks like you have nothing,
0:26:37 > 0:26:39but the Indians recognised that this
0:26:39 > 0:26:43was a new sort of nothing - negative numbers.
0:26:43 > 0:26:47The Indians called them "debts", because they solved equations like,
0:26:47 > 0:26:51"If I have three batches of material and take four away,
0:26:51 > 0:26:53"how many have I left?"
0:26:56 > 0:26:58This may seem odd and impractical,
0:26:58 > 0:27:01but that was the beauty of Indian mathematics.
0:27:01 > 0:27:04Their ability to come up with negative numbers and zero
0:27:04 > 0:27:08was because they thought of numbers as abstract entities.
0:27:08 > 0:27:11They weren't just for counting and measuring pieces of cloth.
0:27:11 > 0:27:15They had a life of their own, floating free of the real world.
0:27:15 > 0:27:19This led to an explosion of mathematical ideas.
0:27:30 > 0:27:34The Indians' abstract approach to mathematics soon revealed
0:27:34 > 0:27:38a new side to the problem of how to solve quadratic equations.
0:27:38 > 0:27:42That is equations including numbers to the power of two.
0:27:43 > 0:27:47Brahmagupta's understanding of negative numbers allowed him to see
0:27:47 > 0:27:50that quadratic equations always have two solutions,
0:27:50 > 0:27:52one of which could be negative.
0:27:55 > 0:27:57Brahmagupta went even further,
0:27:57 > 0:28:00solving quadratic equations with two unknowns,
0:28:00 > 0:28:04a question which wouldn't be considered in the West until 1657,
0:28:04 > 0:28:05when French mathematician Fermat
0:28:05 > 0:28:08challenged his colleagues with the same problem.
0:28:08 > 0:28:11Little did he know that they'd been beaten to a solution
0:28:11 > 0:28:14by Brahmagupta 1,000 years earlier.
0:28:20 > 0:28:24Brahmagupta was beginning to find abstract ways of solving equations,
0:28:24 > 0:28:27but astonishingly, he was also developing
0:28:27 > 0:28:31a new mathematical language to express that abstraction.
0:28:32 > 0:28:36Brahmagupta was experimenting with ways of writing his equations down,
0:28:36 > 0:28:40using the initials of the names of different colours
0:28:40 > 0:28:42to represent unknowns in his equations.
0:28:44 > 0:28:47A new mathematical language was coming to life,
0:28:47 > 0:28:49which would ultimately lead to the x's and y's
0:28:49 > 0:28:52which fill today's mathematical journals.
0:29:07 > 0:29:10But it wasn't just new notation that was being developed.
0:29:13 > 0:29:15Indian mathematicians were responsible for making
0:29:15 > 0:29:19fundamental new discoveries in the theory of trigonometry.
0:29:22 > 0:29:26The power of trigonometry is that it acts like a dictionary,
0:29:26 > 0:29:29translating geometry into numbers and back.
0:29:29 > 0:29:33Although first developed by the ancient Greeks,
0:29:33 > 0:29:35it was in the hands of the Indian mathematicians
0:29:35 > 0:29:37that the subject truly flourished.
0:29:37 > 0:29:42At its heart lies the study of right-angled triangles.
0:29:44 > 0:29:48In trigonometry, you can use this angle here
0:29:48 > 0:29:52to find the ratios of the opposite side to the longest side.
0:29:52 > 0:29:55There's a function called the sine function
0:29:55 > 0:29:58which, when you input the angle, outputs the ratio.
0:29:58 > 0:30:01So for example in this triangle, the angle is about 30 degrees,
0:30:01 > 0:30:05so the output of the sine function is a ratio of one to two,
0:30:05 > 0:30:10telling me that this side is half the length of the longest side.
0:30:12 > 0:30:16The sine function enables you to calculate distances
0:30:16 > 0:30:21when you're not able to make an accurate measurement.
0:30:21 > 0:30:25To this day, it's used in architecture and engineering.
0:30:25 > 0:30:28The Indians used it to survey the land around them,
0:30:28 > 0:30:32navigate the seas and, ultimately, chart the depths of space itself.
0:30:34 > 0:30:37It was central to the work of observatories,
0:30:37 > 0:30:39like this one in Delhi,
0:30:39 > 0:30:42where astronomers would study the stars.
0:30:42 > 0:30:45The Indian astronomers could use trigonometry
0:30:45 > 0:30:48to work out the relative distance between Earth and the moon
0:30:48 > 0:30:49and Earth and the sun.
0:30:49 > 0:30:53You can only make the calculation when the moon is half full,
0:30:53 > 0:30:56because that's when it's directly opposite the sun,
0:30:56 > 0:31:01so that the sun, moon and Earth create a right-angled triangle.
0:31:02 > 0:31:04Now, the Indians could measure
0:31:04 > 0:31:07that the angle between the sun and the observatory
0:31:07 > 0:31:09was one-seventh of a degree.
0:31:10 > 0:31:14The sine function of one-seventh of a degree
0:31:14 > 0:31:18gives me the ratio of 400:1.
0:31:18 > 0:31:23This means the sun is 400 times further from Earth than the moon is.
0:31:23 > 0:31:25So using trigonometry,
0:31:25 > 0:31:28the Indian mathematicians could explore the solar system
0:31:28 > 0:31:31without ever having to leave the surface of the Earth.
0:31:39 > 0:31:42The ancient Greeks had been the first to explore the sine function,
0:31:42 > 0:31:46listing precise values for some angles,
0:31:46 > 0:31:50but they couldn't calculate the sines of every angle.
0:31:50 > 0:31:55The Indians were to go much further, setting themselves a mammoth task.
0:31:55 > 0:31:57The search was on to find a way
0:31:57 > 0:32:01to calculate the sine function of any angle you might be given.
0:32:17 > 0:32:21The breakthrough in the search for the sine function of every angle
0:32:21 > 0:32:24would be made here in Kerala in south India.
0:32:24 > 0:32:27In the 15th century, this part of the country
0:32:27 > 0:32:31became home to one of the most brilliant schools of mathematicians
0:32:31 > 0:32:33to have ever worked.
0:32:35 > 0:32:38Their leader was called Madhava, and he was to make
0:32:38 > 0:32:42some extraordinary mathematical discoveries.
0:32:45 > 0:32:49The key to Madhava's success was the concept of the infinite.
0:32:49 > 0:32:52Madhava discovered that you could add up infinitely many things
0:32:52 > 0:32:54with dramatic effects.
0:32:54 > 0:32:57Previous cultures had been nervous of these infinite sums,
0:32:57 > 0:33:00but Madhava was happy to play with them.
0:33:00 > 0:33:02For example, here's how one can be made up
0:33:02 > 0:33:05by adding infinitely many fractions.
0:33:06 > 0:33:11I'm heading from zero to one on my boat,
0:33:11 > 0:33:15but I can split my journey up into infinitely many fractions.
0:33:15 > 0:33:18So I can get to a half,
0:33:18 > 0:33:21then I can sail on a quarter,
0:33:21 > 0:33:24then an eighth, then a sixteenth, and so on.
0:33:24 > 0:33:29The smaller the fractions I move, the nearer to one I get,
0:33:29 > 0:33:33but I'll only get there once I've added up infinitely many fractions.
0:33:36 > 0:33:38Physically and philosophically,
0:33:38 > 0:33:41it seems rather a challenge to add up infinitely many things,
0:33:41 > 0:33:45but the power of mathematics is to make sense of the impossible.
0:33:45 > 0:33:47By producing a language
0:33:47 > 0:33:49to articulate and manipulate the infinite,
0:33:49 > 0:33:52you can prove that after infinitely many steps
0:33:52 > 0:33:54you'll reach your destination.
0:33:57 > 0:34:01Such infinite sums are called infinite series, and Madhava
0:34:01 > 0:34:04was doing a lot of research into the connections
0:34:04 > 0:34:07between these series and trigonometry.
0:34:08 > 0:34:12First, he realised that he could use the same principle
0:34:12 > 0:34:14of adding up infinitely many fractions to capture
0:34:14 > 0:34:19one of the most important numbers in mathematics - pi.
0:34:20 > 0:34:25Pi is the ratio of the circle's circumference to its diameter.
0:34:25 > 0:34:29It's a number that appears in all sorts of mathematics,
0:34:29 > 0:34:32but is especially useful for engineers,
0:34:32 > 0:34:36because any measurements involving curves soon require pi.
0:34:38 > 0:34:42So for centuries, mathematicians searched for a precise value for pi.
0:34:48 > 0:34:52It was in 6th-century India that the mathematician Aryabhata
0:34:52 > 0:34:57gave a very accurate approximation for pi - namely 3.1416.
0:34:57 > 0:34:58He went on to use this
0:34:58 > 0:35:02to make a measurement of the circumference of the Earth,
0:35:02 > 0:35:05and he got it as 24,835 miles,
0:35:05 > 0:35:09which, amazingly, is only 70 miles away from its true value.
0:35:09 > 0:35:12But it was in Kerala in the 15th century
0:35:12 > 0:35:15that Madhava realised he could use infinity
0:35:15 > 0:35:17to get an exact formula for pi.
0:35:21 > 0:35:24By successively adding and subtracting different fractions,
0:35:24 > 0:35:28Madhava could hone in on an exact formula for pi.
0:35:30 > 0:35:34First, he moved four steps up the number line.
0:35:34 > 0:35:36That took him way past pi.
0:35:38 > 0:35:41So next he took four-thirds of a step,
0:35:41 > 0:35:44or one-and-one-third steps, back.
0:35:44 > 0:35:46Now he'd come too far the other way.
0:35:47 > 0:35:51So he headed forward four-fifths of a step.
0:35:51 > 0:35:56Each time, he alternated between four divided by the next odd number.
0:36:03 > 0:36:06He zigzagged up and down the number line,
0:36:06 > 0:36:08getting closer and closer to pi.
0:36:08 > 0:36:12He discovered that if you went through all the odd numbers,
0:36:12 > 0:36:15infinitely many of them, you would hit pi exactly.
0:36:19 > 0:36:22I was taught at university that this formula for pi
0:36:22 > 0:36:26was discovered by the 17th-century German mathematician Leibniz,
0:36:26 > 0:36:29but amazingly, it was actually discovered here in Kerala
0:36:29 > 0:36:31two centuries earlier by Madhava.
0:36:31 > 0:36:34He went on to use the same sort of mathematics
0:36:34 > 0:36:36to get infinite-series expressions
0:36:36 > 0:36:38for the sine formula in trigonometry.
0:36:38 > 0:36:42And the wonderful thing is that you can use these formulas now
0:36:42 > 0:36:46to calculate the sine of any angle to any degree of accuracy.
0:36:56 > 0:37:00It seems incredible that the Indians made these discoveries
0:37:00 > 0:37:03centuries before Western mathematicians.
0:37:06 > 0:37:10And it says a lot about our attitude in the West to non-Western cultures
0:37:10 > 0:37:14that we nearly always claim their discoveries as our own.
0:37:14 > 0:37:18What is clear is the West has been very slow to give due credit
0:37:18 > 0:37:22to the major breakthroughs made in non-Western mathematics.
0:37:22 > 0:37:25Madhava wasn't the only mathematician to suffer this way.
0:37:25 > 0:37:28As the West came into contact more and more with the East
0:37:28 > 0:37:30during the 18th and 19th centuries,
0:37:30 > 0:37:33there was a widespread dismissal and denigration
0:37:33 > 0:37:35of the cultures they were colonising.
0:37:35 > 0:37:38The natives, it was assumed, couldn't have anything
0:37:38 > 0:37:40of intellectual worth to offer the West.
0:37:40 > 0:37:43It's only now, at the beginning of the 21st century,
0:37:43 > 0:37:45that history is being rewritten.
0:37:45 > 0:37:49But Eastern mathematics was to have a major impact in Europe,
0:37:49 > 0:37:53thanks to the development of one of the major powers
0:37:53 > 0:37:54of the medieval world.
0:38:17 > 0:38:20In the 7th century, a new empire began to spread
0:38:20 > 0:38:23across the Middle East.
0:38:23 > 0:38:25The teachings of the Prophet Mohammed
0:38:25 > 0:38:28inspired a vast and powerful Islamic empire
0:38:28 > 0:38:30which soon stretched from India in the east
0:38:30 > 0:38:35to here in Morocco in the west.
0:38:41 > 0:38:46And at the heart of this empire lay a vibrant intellectual culture.
0:38:51 > 0:38:56A great library and centre of learning was established in Baghdad.
0:38:56 > 0:38:59Called the House of Wisdom, its teaching spread
0:38:59 > 0:39:01throughout the Islamic empire,
0:39:01 > 0:39:05reaching schools like this one here in Fez.
0:39:05 > 0:39:08Subjects studied included astronomy, medicine,
0:39:08 > 0:39:10chemistry, zoology
0:39:10 > 0:39:11and mathematics.
0:39:13 > 0:39:18The Muslim scholars collected and translated many ancient texts,
0:39:18 > 0:39:20effectively saving them for posterity.
0:39:20 > 0:39:23In fact, without their intervention, we may never have known
0:39:23 > 0:39:27about the ancient cultures of Egypt, Babylon, Greece and India.
0:39:27 > 0:39:30But the scholars at the House of Wisdom weren't content
0:39:30 > 0:39:33simply with translating other people's mathematics.
0:39:33 > 0:39:36They wanted to create a mathematics of their own,
0:39:36 > 0:39:37to push the subject forward.
0:39:42 > 0:39:46Such intellectual curiosity was actively encouraged
0:39:46 > 0:39:49in the early centuries of the Islamic empire.
0:39:51 > 0:39:54The Koran asserted the importance of knowledge.
0:39:54 > 0:39:58Learning was nothing less than a requirement of God.
0:40:01 > 0:40:05In fact, the needs of Islam demanded mathematical skill.
0:40:05 > 0:40:07The devout needed to calculate the time of prayer
0:40:07 > 0:40:10and the direction of Mecca to pray towards,
0:40:10 > 0:40:13and the prohibition of depicting the human form
0:40:13 > 0:40:15meant that they had to use
0:40:15 > 0:40:18much more geometric patterns to cover their buildings.
0:40:18 > 0:40:22The Muslim artists discovered all the different sorts of symmetry
0:40:22 > 0:40:26that you can depict on a two-dimensional wall.
0:40:34 > 0:40:37The director of the House of Wisdom in Baghdad
0:40:37 > 0:40:40was a Persian scholar called Muhammad Al-Khwarizmi.
0:40:43 > 0:40:48Al-Khwarizmi was an exceptional mathematician who was responsible
0:40:48 > 0:40:52for introducing two key mathematical concepts to the West.
0:40:52 > 0:40:55Al-Khwarizmi recognised the incredible potential
0:40:55 > 0:40:57that the Hindu numerals had
0:40:57 > 0:41:00to revolutionise mathematics and science.
0:41:00 > 0:41:03His work explaining the power of these numbers
0:41:03 > 0:41:06to speed up calculations and do things effectively
0:41:06 > 0:41:09was so influential that it wasn't long before they were adopted
0:41:09 > 0:41:13as the numbers of choice amongst the mathematicians of the Islamic world.
0:41:13 > 0:41:16In fact, these numbers have now become known
0:41:16 > 0:41:18as the Hindu-Arabic numerals.
0:41:18 > 0:41:21These numbers - one to nine and zero -
0:41:21 > 0:41:25are the ones we use today all over the world.
0:41:29 > 0:41:34But Al-Khwarizmi was to create a whole new mathematical language.
0:41:36 > 0:41:38It was called algebra
0:41:38 > 0:41:42and was named after the title of his book Al-jabr W'al-muqabala,
0:41:42 > 0:41:46or Calculation By Restoration Or Reduction.
0:41:50 > 0:41:56Algebra is the grammar that underlies the way that numbers work.
0:41:56 > 0:41:58It's a language that explains the patterns
0:41:58 > 0:42:01that lie behind the behaviour of numbers.
0:42:01 > 0:42:05It's a bit like a code for running a computer program.
0:42:05 > 0:42:09The code will work whatever the numbers you feed in to the program.
0:42:11 > 0:42:14For example, mathematicians might have discovered
0:42:14 > 0:42:16that if you take a number and square it,
0:42:16 > 0:42:19that's always one more than if you'd taken
0:42:19 > 0:42:22the numbers either side and multiplied those together.
0:42:22 > 0:42:25For example, five times five is 25,
0:42:25 > 0:42:29which is one more than four times six - 24.
0:42:29 > 0:42:33Six times six is always one more than five times seven and so on.
0:42:33 > 0:42:34But how can you be sure
0:42:34 > 0:42:38that this is going to work whatever numbers you take?
0:42:38 > 0:42:41To explain the pattern underlying these calculations,
0:42:41 > 0:42:43let's use the dyeing holes in this tannery.
0:42:51 > 0:42:56If we take a square of 25 holes, running five by five,
0:42:56 > 0:43:00and take one row of five away and add it to the bottom,
0:43:00 > 0:43:03we get six by four with one left over.
0:43:05 > 0:43:09But however many holes there are on the side of the square,
0:43:09 > 0:43:12we can always move one row of holes down in a similar way
0:43:12 > 0:43:16to be left with a rectangle of holes with one left over.
0:43:18 > 0:43:20Algebra was a huge breakthrough.
0:43:20 > 0:43:22Here was a new language
0:43:22 > 0:43:25to be able to analyse the way that numbers worked.
0:43:25 > 0:43:27Previously, the Indians and the Chinese
0:43:27 > 0:43:30had considered very specific problems,
0:43:30 > 0:43:33but Al-Khwarizmi went from the specific to the general.
0:43:33 > 0:43:37He developed systematic ways to be able to analyse problems
0:43:37 > 0:43:40so that the solutions would work whatever the numbers that you took.
0:43:40 > 0:43:44This language is used across the mathematical world today.
0:43:46 > 0:43:50Al-Khwarizmi's great breakthrough came when he applied algebra
0:43:50 > 0:43:52to quadratic equations -
0:43:52 > 0:43:55that is equations including numbers to the power of two.
0:43:55 > 0:43:58The ancient Mesopotamians had devised
0:43:58 > 0:44:02a cunning method to solve particular quadratic equations,
0:44:02 > 0:44:06but it was Al-Khwarizmi's abstract language of algebra
0:44:06 > 0:44:10that could finally express why this method always worked.
0:44:11 > 0:44:14This was a great conceptual leap
0:44:14 > 0:44:17and would ultimately lead to a formula that could be used to solve
0:44:17 > 0:44:22any quadratic equation, whatever the numbers involved.
0:44:30 > 0:44:32The next mathematical Holy Grail
0:44:32 > 0:44:37was to find a general method that could solve all cubic equations -
0:44:37 > 0:44:40equations including numbers to the power of three.
0:44:57 > 0:45:00It was an 11th-century Persian mathematician
0:45:00 > 0:45:04who took up the challenge of cracking the problem of the cubic.
0:45:08 > 0:45:11His name was Omar Khayyam, and he travelled widely
0:45:11 > 0:45:15across the Middle East, calculating as he went.
0:45:17 > 0:45:21But he was famous for another, very different, reason.
0:45:21 > 0:45:24Khayyam was a celebrated poet,
0:45:24 > 0:45:28author of the great epic poem the Rubaiyat.
0:45:30 > 0:45:35It may seem a bit odd that a poet was also a master mathematician.
0:45:35 > 0:45:38After all, the combination doesn't immediately spring to mind.
0:45:38 > 0:45:42But there's quite a lot of similarity between the disciplines.
0:45:42 > 0:45:45Poetry, with its rhyming structure and rhythmic patterns,
0:45:45 > 0:45:49resonates strongly with constructing a logical mathematical proof.
0:45:53 > 0:45:55Khayyam's major mathematical work
0:45:55 > 0:46:02was devoted to finding the general method to solve all cubic equations.
0:46:02 > 0:46:04Rather than looking at particular examples,
0:46:04 > 0:46:08Khayyam carried out a systematic analysis of the problem,
0:46:08 > 0:46:11true to the algebraic spirit of Al-Khwarizmi.
0:46:13 > 0:46:16Khayyam's analysis revealed for the first time
0:46:16 > 0:46:19that there were several different sorts of cubic equation.
0:46:19 > 0:46:21But he was still very influenced
0:46:21 > 0:46:24by the geometric heritage of the Greeks.
0:46:24 > 0:46:27He couldn't separate the algebra from the geometry.
0:46:27 > 0:46:30In fact, he wouldn't even consider equations in higher degrees,
0:46:30 > 0:46:33because they described objects in more than three dimensions,
0:46:33 > 0:46:35something he saw as impossible.
0:46:35 > 0:46:37Although the geometry allowed him
0:46:37 > 0:46:40to analyse these cubic equations to some extent,
0:46:40 > 0:46:43he still couldn't come up with a purely algebraic solution.
0:46:45 > 0:46:51It would be another 500 years before mathematicians could make the leap
0:46:51 > 0:46:54and find a general solution to the cubic equation.
0:46:56 > 0:47:01And that leap would finally be made in the West - in Italy.
0:47:15 > 0:47:18During the centuries in which China, India and the Islamic empire
0:47:18 > 0:47:20had been in the ascendant,
0:47:20 > 0:47:24Europe had fallen under the shadow of the Dark Ages.
0:47:26 > 0:47:30All intellectual life, including the study of mathematics, had stagnated.
0:47:35 > 0:47:41But by the 13th century, things were beginning to change.
0:47:41 > 0:47:46Led by Italy, Europe was starting to explore and trade with the East.
0:47:46 > 0:47:51With that contact came the spread of Eastern knowledge to the West.
0:47:51 > 0:47:53It was the son of a customs official
0:47:53 > 0:47:56that would become Europe's first great medieval mathematician.
0:47:56 > 0:48:00As a child, he travelled around North Africa with his father,
0:48:00 > 0:48:03where he learnt about the developments of Arabic mathematics
0:48:03 > 0:48:06and especially the benefits of the Hindu-Arabic numerals.
0:48:06 > 0:48:08When he got home to Italy he wrote a book
0:48:08 > 0:48:10that would be hugely influential
0:48:10 > 0:48:13in the development of Western mathematics.
0:48:29 > 0:48:31That mathematician was Leonardo of Pisa,
0:48:31 > 0:48:34better known as Fibonacci,
0:48:35 > 0:48:37and in his Book Of Calculating,
0:48:37 > 0:48:40Fibonacci promoted the new number system,
0:48:40 > 0:48:44demonstrating how simple it was compared to the Roman numerals
0:48:44 > 0:48:47that were in use across Europe.
0:48:47 > 0:48:52Calculations were far easier, a fact that had huge consequences
0:48:52 > 0:48:55for anyone dealing with numbers -
0:48:55 > 0:48:59pretty much everyone, from mathematicians to merchants.
0:48:59 > 0:49:02But there was widespread suspicion of these new numbers.
0:49:02 > 0:49:06Old habits die hard, and the authorities just didn't trust them.
0:49:06 > 0:49:09Some believed that they would be more open to fraud -
0:49:09 > 0:49:11that you could tamper with them.
0:49:11 > 0:49:14Others believed that they'd be so easy to use for calculations
0:49:14 > 0:49:17that it would empower the masses, taking authority away
0:49:17 > 0:49:21from the intelligentsia who knew how to use the old sort of numbers.
0:49:27 > 0:49:31The city of Florence even banned them in 1299,
0:49:31 > 0:49:34but over time, common sense prevailed,
0:49:34 > 0:49:37the new system spread throughout Europe,
0:49:37 > 0:49:40and the old Roman system slowly became defunct.
0:49:40 > 0:49:46At last, the Hindu-Arabic numerals, zero to nine, had triumphed.
0:49:48 > 0:49:51Today Fibonacci is best known for the discovery of some numbers,
0:49:51 > 0:49:55now called the Fibonacci sequence, that arose when he was trying
0:49:55 > 0:49:58to solve a riddle about the mating habits of rabbits.
0:49:58 > 0:50:01Suppose a farmer has a pair of rabbits.
0:50:01 > 0:50:03Rabbits take two months to reach maturity,
0:50:03 > 0:50:07and after that they give birth to another pair of rabbits each month.
0:50:07 > 0:50:09So the problem was how to determine
0:50:09 > 0:50:12how many pairs of rabbits there will be in any given month.
0:50:14 > 0:50:20Well, during the first month you have one pair of rabbits,
0:50:20 > 0:50:24and since they haven't matured, they can't reproduce.
0:50:24 > 0:50:28During the second month, there is still only one pair.
0:50:28 > 0:50:32But at the beginning of the third month, the first pair
0:50:32 > 0:50:36reproduces for the first time, so there are two pairs of rabbits.
0:50:36 > 0:50:38At the beginning of the fourth month,
0:50:38 > 0:50:40the first pair reproduces again,
0:50:40 > 0:50:45but the second pair is not mature enough, so there are three pairs.
0:50:46 > 0:50:50In the fifth month, the first pair reproduces
0:50:50 > 0:50:53and the second pair reproduces for the first time,
0:50:53 > 0:50:58but the third pair is still too young, so there are five pairs.
0:50:58 > 0:51:00The mating ritual continues,
0:51:00 > 0:51:02but what you soon realise is
0:51:02 > 0:51:05the number of pairs of rabbits you have in any given month
0:51:05 > 0:51:09is the sum of the pairs of rabbits that you have had
0:51:09 > 0:51:13in each of the two previous months, so the sequence goes...
0:51:13 > 0:51:171...1...2...3...
0:51:17 > 0:51:215...8...13...
0:51:21 > 0:51:2621...34...55...and so on.
0:51:26 > 0:51:29The Fibonacci numbers are nature's favourite numbers.
0:51:29 > 0:51:31It's not just rabbits that use them.
0:51:31 > 0:51:35The number of petals on a flower is invariably a Fibonacci number.
0:51:35 > 0:51:39They run up and down pineapples if you count the segments.
0:51:39 > 0:51:42Even snails use them to grow their shells.
0:51:42 > 0:51:46Wherever you find growth in nature, you find the Fibonacci numbers.
0:51:51 > 0:51:54But the next major breakthrough in European mathematics
0:51:54 > 0:51:58wouldn't happen until the early 16th century.
0:51:58 > 0:52:00It would involve
0:52:00 > 0:52:04finding the general method that would solve all cubic equations,
0:52:04 > 0:52:08and it would happen here in the Italian city of Bologna.
0:52:10 > 0:52:14The University of Bologna was the crucible
0:52:14 > 0:52:17of European mathematical thought at the beginning of the 16th century.
0:52:20 > 0:52:24Pupils from all over Europe flocked here and developed
0:52:24 > 0:52:29a new form of spectator sport - the mathematical competition.
0:52:31 > 0:52:34Large audiences would gather to watch mathematicians
0:52:34 > 0:52:39challenge each other with numbers, a kind of intellectual fencing match.
0:52:39 > 0:52:42But even in this questioning atmosphere
0:52:42 > 0:52:46it was believed that some problems were just unsolvable.
0:52:46 > 0:52:51It was generally assumed that finding a general method
0:52:51 > 0:52:54to solve all cubic equations was impossible.
0:52:54 > 0:52:58But one scholar was to prove everyone wrong.
0:53:01 > 0:53:03His name was Tartaglia,
0:53:03 > 0:53:05but he certainly didn't look
0:53:05 > 0:53:08the heroic architect of a new mathematics.
0:53:08 > 0:53:11At the age of 12, he'd been slashed across the face
0:53:11 > 0:53:13with a sabre by a rampaging French army.
0:53:13 > 0:53:16The result was a terrible facial scar
0:53:16 > 0:53:19and a devastating speech impediment.
0:53:19 > 0:53:22In fact, Tartaglia was the nickname he'd been given as a child
0:53:22 > 0:53:24and means "the stammerer".
0:53:30 > 0:53:33Shunned by his schoolmates,
0:53:33 > 0:53:37Tartaglia lost himself in mathematics, and it wasn't long
0:53:37 > 0:53:43before he'd found the formula to solve one type of cubic equation.
0:53:43 > 0:53:45But Tartaglia soon discovered
0:53:45 > 0:53:48that he wasn't the only one to believe he'd cracked the cubic.
0:53:48 > 0:53:51A young Italian called Fior was boasting
0:53:51 > 0:53:57that he too held the secret formula for solving cubic equations.
0:53:57 > 0:53:59When news broke about the discoveries
0:53:59 > 0:54:02made by the two mathematicians,
0:54:02 > 0:54:06a competition was arranged to pit them against each other.
0:54:06 > 0:54:10The intellectual fencing match of the century was about to begin.
0:54:17 > 0:54:19The trouble was that Tartaglia
0:54:19 > 0:54:22only knew how to solve one sort of cubic equation,
0:54:22 > 0:54:24and Fior was ready to challenge him
0:54:24 > 0:54:27with questions about a different sort.
0:54:27 > 0:54:29But just a few days before the contest,
0:54:29 > 0:54:32Tartaglia worked out how to solve this different sort,
0:54:32 > 0:54:35and with this new weapon in his arsenal he thrashed his opponent,
0:54:35 > 0:54:38solving all the questions in under two hours.
0:54:41 > 0:54:44Tartaglia went on
0:54:44 > 0:54:48to find the formula to solve all types of cubic equations.
0:54:48 > 0:54:51News soon spread, and a mathematician in Milan
0:54:51 > 0:54:54called Cardano became so desperate to find the solution
0:54:54 > 0:54:59that he persuaded a reluctant Tartaglia to reveal the secret,
0:54:59 > 0:55:01but on one condition -
0:55:01 > 0:55:05that Cardano keep the secret and never publish.
0:55:07 > 0:55:09But Cardano couldn't resist
0:55:09 > 0:55:14discussing Tartaglia's solution with his brilliant student, Ferrari.
0:55:14 > 0:55:16As Ferrari got to grips with Tartaglia's work,
0:55:16 > 0:55:19he realised that he could use it to solve
0:55:19 > 0:55:22the more complicated quartic equation, an amazing achievement.
0:55:22 > 0:55:25Cardano couldn't deny his student his just rewards,
0:55:25 > 0:55:29and he broke his vow of secrecy, publishing Tartaglia's work
0:55:29 > 0:55:32together with Ferrari's brilliant solution of the quartic.
0:55:35 > 0:55:39Poor Tartaglia never recovered and died penniless,
0:55:39 > 0:55:42and to this day, the formula that solves the cubic equation
0:55:42 > 0:55:45is known as Cardano's formula.
0:55:54 > 0:55:57Tartaglia may not have won glory in his lifetime,
0:55:57 > 0:56:01but his mathematics managed to solve a problem that had bewildered
0:56:01 > 0:56:05the great mathematicians of China, India and the Arab world.
0:56:07 > 0:56:11It was the first great mathematical breakthrough
0:56:11 > 0:56:13to happen in modern Europe.
0:56:17 > 0:56:20The Europeans now had in their hands the new language of algebra,
0:56:20 > 0:56:24the powerful techniques of the Hindu-Arabic numerals
0:56:24 > 0:56:27and the beginnings of the mastery of the infinite.
0:56:27 > 0:56:28It was time for the Western world
0:56:28 > 0:56:31to start writing its own mathematical stories
0:56:31 > 0:56:33in the language of the East.
0:56:33 > 0:56:35The mathematical revolution was about to begin.
0:56:39 > 0:56:43You can learn more about The Story Of Maths with the Open University
0:56:43 > 0:56:45at open2.net.