texts below are from © https://subsaga.com/bbc/documentaries/science/the-story-of-maths/2-the-genius-of-the-east.html
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From measuring time to understanding our position in the universe,
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from mapping the Earth to navigating the seas,
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from man’s earliest inventions to today’s advanced technologies,
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mathematics has been the pivot on which human life depends.
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The first steps of man’s mathematical journey
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were taken by the ancient cultures of Egypt, Mesopotamia and Greece -
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cultures which created the basic language of number and calculation.
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But when ancient Greece fell into decline,
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mathematical progress juddered to a halt.
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But that was in the West.
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In the East, mathematics would reach dynamic new heights.
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But in the West, much of this mathematical heritage
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has been conveniently forgotten or shaded from view.
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Due credit has not been given to the great mathematical breakthroughs
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that ultimately changed the world we live in.
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This is the untold story of the mathematics of the East
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that would transform the West and give birth to the modern world.
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The Great Wall of China stretches for thousands of miles.
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Nearly 2,000 years in the making, this vast, defensive wall
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was begun in 220BC to protect China’s growing empire.
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The Great Wall of China is an amazing feat of engineering
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built over rough and high countryside.
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As soon as they started building,
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the ancient Chinese realised they had to make calculations
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about distances, angles of elevation and amounts of material.
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So perhaps it isn’t surprising that this inspired
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some very clever mathematics to help build Imperial China.
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At the heart of ancient Chinese mathematics
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was an incredibly simple number system
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which laid the foundations for the way we count in the West today.
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When a mathematician wanted to do a sum, he would use small bamboo rods.
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These rods were arranged to represent the numbers one to nine.
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They were then placed in columns,
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each column representing units, tens,
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hundreds, thousands and so on.
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So the number 924 was represented by putting
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the symbol 4 in the units column, the symbol 2 in the tens column
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and the symbol 9 in the hundreds column.
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This is what we call a decimal place-value system,
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and it’s very similar to the one we use today.
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We too use numbers from one to nine, and we use their position
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to indicate whether it’s units, tens, hundreds or thousands.
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But the power of these rods is that it makes calculations very quick.
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In fact, the way the ancient Chinese did their calculations
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is very similar to the way we learn today in school.
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Not only were the ancient Chinese
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the first to use a decimal place-value system,
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but they did so over 1,000 years before we adopted it in the West.
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But they only used it when calculating with the rods.
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When writing the numbers down,
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the ancient Chinese didn’t use the place-value system.
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Instead, they used a far more laborious method,
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in which special symbols stood for tens, hundreds, thousands and so on.
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So the number 924 would be written out
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as nine hundreds, two tens and four.
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Not quite so efficient.
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The problem was
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that the ancient Chinese didn’t have a concept of zero.
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They didn’t have a symbol for zero. It just didn’t exist as a number.
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Using the counting rods,
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they would use a blank space where today we would write a zero.
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The problem came with trying to write down this number, which is why
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they had to create these new symbols for tens, hundreds and thousands.
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Without a zero, the written number was extremely limited.
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But the absence of zero didn’t stop
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the ancient Chinese from making giant mathematical steps.
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In fact, there was a widespread fascination
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with number in ancient China.
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According to legend, the first sovereign of China,
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the Yellow Emperor, had one of his deities
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create mathematics in 2800BC,
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believing that number held cosmic significance. And to this day,
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the Chinese still believe in the mystical power of numbers.
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Odd numbers are seen as male, even numbers, female.
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The number four is to be avoided at all costs.
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The number eight brings good fortune.
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And the ancient Chinese were drawn to patterns in numbers,
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developing their own rather early version of sudoku.
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It was called the magic square.
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Legend has it that thousands of years ago, Emperor Yu was visited
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by a sacred turtle that came out of the depths of the Yellow River.
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On its back were numbers
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arranged into a magic square, a little like this.
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In this square,
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which was regarded as having great religious significance,
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all the numbers in each line - horizontal, vertical and diagonal -
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all add up to the same number - 15.
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Now, the magic square may be no more than a fun puzzle,
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but it shows the ancient Chinese fascination
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with mathematical patterns, and it wasn’t too long
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before they were creating even bigger magic squares
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with even greater magical and mathematical powers.
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But mathematics also played
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a vital role in the running of the emperor’s court.
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The calendar and the movement of the planets
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were of the utmost importance to the emperor,
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influencing all his decisions, even down to the way his day was planned,
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so astronomers became prized members of the imperial court,
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and astronomers were always mathematicians.
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Everything in the emperor’s life was governed by the calendar,
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and he ran his affairs with mathematical precision.
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The emperor even got his mathematical advisors
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to come up with a system to help him sleep his way
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through the vast number of women he had in his harem.
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Never one to miss a trick, the mathematical advisors decided
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to base the harem on a mathematical idea called a geometric progression.
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Maths has never had such a fun purpose!
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Legend has it that in the space of 15 nights,
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the emperor had to sleep with 121 women…
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..the empress,
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three senior consorts,
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nine wives,
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27 concubines
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and 81 slaves.
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The mathematicians would soon have realised
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that this was a geometric progression - a series of numbers
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in which you get from one number to the next
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by multiplying the same number each time - in this case, three.
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Each group of women is three times as large as the previous group,
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so the mathematicians could quickly draw up a rota to ensure that,
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in the space of 15 nights,
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the emperor slept with every woman in the harem.
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The first night was reserved for the empress.
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The next was for the three senior consorts.
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The nine wives came next,
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and then the 27 concubines were chosen in rotation, nine each night.
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And then finally, over a period of nine nights,
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the 81 slaves were dealt with in groups of nine.
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Being the emperor certainly required stamina,
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a bit like being a mathematician,
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but the object is clear -
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to procure the best possible imperial succession.
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The rota ensured that the emperor
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slept with the ladies of highest rank closest to the full moon,
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when their yin, their female force,
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would be at its highest and be able to match his yang, or male force.
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The emperor’s court wasn’t alone in its dependence on mathematics.
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It was central to the running of the state.
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Ancient China was a vast and growing empire with a strict legal code,
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widespread taxation
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and a standardised system of weights, measures and money.
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The empire needed
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a highly trained civil service, competent in mathematics.
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And to educate these civil servants was a mathematical textbook,
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probably written in around 200BC - the Nine Chapters.
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The book is a compilation of 246 problems
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in practical areas such as trade, payment of wages and taxes.
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And at the heart of these problems lies
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one of the central themes of mathematics, how to solve equations.
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Equations are a little bit like cryptic crosswords.
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You’re given a certain amount of information
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about some unknown numbers, and from that information
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you’ve got to deduce what the unknown numbers are.
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For example, with my weights and scales,
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I’ve found out that one plum…
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..together with three peaches
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weighs a total of 15g.
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But…
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..two plums
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together with one peach
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weighs a total of 10g.
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From this information, I can deduce what a single plum weighs
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and a single peach weighs, and this is how I do it.
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If I take the first set of scales,
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one plum and three peaches weighing 15g,
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and double it, I get two plums and six peaches weighing 30g.
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If I take this and subtract from it the second set of scales -
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that’s two plums and a peach weighing 10g -
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I’m left with an interesting result -
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no plums.
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Having eliminated the plums,
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I’ve discovered that five peaches weighs 20g,
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so a single peach weighs 4g,
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and from this I can deduce that the plum weighs 3g.
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The ancient Chinese went on to apply similar methods
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to larger and larger numbers of unknowns,
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using it to solve increasingly complicated equations.
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What’s extraordinary is
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that this particular system of solving equations
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didn’t appear in the West until the beginning of the 19th century.
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In 1809, while analysing a rock called Pallas in the asteroid belt,
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Carl Friedrich Gauss,
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who would become known as the prince of mathematics,
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rediscovered this method
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which had been formulated in ancient China centuries earlier.
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Once again, ancient China streets ahead of Europe.
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But the Chinese were to go on to solve
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even more complicated equations involving far larger numbers.
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In what’s become known as the Chinese remainder theorem,
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the Chinese came up with a new kind of problem.
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In this, we know the number that’s left
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when the equation’s unknown number is divided by a given number -
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say, three, five or seven.
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Of course, this is a fairly abstract mathematical problem,
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but the ancient Chinese still couched it in practical terms.
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So a woman in the market has a tray of eggs,
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but she doesn’t know how many eggs she’s got.
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What she does know is that if she arranges them in threes,
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she has one egg left over.
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If she arranges them in fives, she gets two eggs left over.
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But if she arranged them in rows of seven,
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she found she had three eggs left over.
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The ancient Chinese found a systematic way to calculate
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that the smallest number of eggs she could have had in the tray is 52.
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But the more amazing thing is that you can capture
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such a large number, like 52,
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by using these small numbers like three, five and seven.
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This way of looking at numbers
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would become a dominant theme over the last two centuries.
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By the 6th century AD, the Chinese remainder theorem was being used
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in ancient Chinese astronomy to measure planetary movement.
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But today it still has practical uses.
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Internet cryptography encodes numbers using mathematics
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that has its origins in the Chinese remainder theorem.
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By the 13th century,
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mathematics was long established on the curriculum,
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with over 30 mathematics schools scattered across the country.
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The golden age of Chinese maths had arrived.
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And its most important mathematician was called Qin Jiushao.
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Legend has it that Qin Jiushao was something of a scoundrel.
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He was a fantastically corrupt imperial administrator
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who crisscrossed China, lurching from one post to another.
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Repeatedly sacked for embezzling government money,
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he poisoned anyone who got in his way.
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Qin Jiushao was reputedly described as
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as violent as a tiger or a wolf
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and as poisonous as a scorpion or a viper
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so, not surprisingly, he made a fierce warrior.
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For ten years, he fought against the invading Mongols,
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but for much of that time he was complaining that his military life
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took him away from his true passion.
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No, not corruption, but mathematics.
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Qin started trying to solve equations
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that grew out of trying to measure the world around us.
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Quadratic equations involve numbers
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that are squared, or to the power of two - say, five times five.
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The ancient Mesopotamians
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had already realised that these equations
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were perfect for measuring flat, two-dimensional shapes,
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like Tiananmen Square.
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But Qin was interested
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in more complicated equations - cubic equations.
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These involve numbers which are cubed,
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or to the power of three - say, five times five times five,
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and they were perfect for capturing three-dimensional shapes,
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like Chairman Mao’s mausoleum.
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00:17:23,240 –> 00:17:26,000
Qin found a way of solving cubic equations,
248
00:17:26,000 –> 00:17:28,560
and this is how it worked.
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00:17:32,400 –> 00:17:34,440
Say Qin wants to know
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00:17:34,440 –> 00:17:37,600
the exact dimensions of Chairman Mao’s mausoleum.
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He knows the volume of the building
252
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and the relationships between the dimensions.
253
00:17:47,000 –> 00:17:49,320
In order to get his answer,
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Qin uses what he knows to produce a cubic equation.
255
00:17:53,880 –> 00:17:57,800
He then makes an educated guess at the dimensions.
256
00:17:57,800 –> 00:18:01,520
Although he’s captured a good proportion of the mausoleum,
257
00:18:01,520 –> 00:18:03,600
there are still bits left over.
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00:18:05,080 –> 00:18:09,040
Qin takes these bits and creates a new cubic equation.
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00:18:09,040 –> 00:18:11,120
He can now refine his first guess
260
00:18:11,120 –> 00:18:15,200
by trying to find a solution to this new cubic equation, and so on.
261
00:18:18,320 –> 00:18:21,960
Each time he does this, the pieces he’s left with
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00:18:21,960 –> 00:18:26,440
get smaller and smaller and his guesses get better and better.
263
00:18:28,120 –> 00:18:31,640
What’s striking is that Qin’s method for solving equations
264
00:18:31,640 –> 00:18:34,880
wasn’t discovered in the West until the 17th century,
265
00:18:34,880 –> 00:18:39,360
when Isaac Newton came up with a very similar approximation method.
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The power of this technique is
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00:18:41,840 –> 00:18:46,000
that it can be applied to even more complicated equations.
268
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Qin even used his techniques to solve an equation
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00:18:49,720 –> 00:18:51,960
involving numbers up to the power of ten.
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00:18:51,960 –> 00:18:56,000
This was extraordinary stuff - highly complex mathematics.
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00:18:58,400 –> 00:19:00,800
Qin may have been years ahead of his time,
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00:19:00,800 –> 00:19:03,120
but there was a problem with his technique.
273
00:19:03,120 –> 00:19:05,960
It only gave him an approximate solution.
274
00:19:05,960 –> 00:19:09,880
That might be good enough for an engineer - not for a mathematician.
275
00:19:09,880 –> 00:19:13,440
Mathematics is an exact science. We like things to be precise,
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00:19:13,440 –> 00:19:16,320
and Qin just couldn’t come up with a formula
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00:19:16,320 –> 00:19:19,840
to give him an exact solution to these complicated equations.
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00:19:27,840 –> 00:19:30,280
China had made great mathematical leaps,
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but the next great mathematical breakthroughs were to happen
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in a country lying to the southwest of China -
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a country that had a rich mathematical tradition
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that would change the face of maths for ever.
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00:20:13,840 –> 00:20:18,560
India’s first great mathematical gift lay in the world of number.
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Like the Chinese, the Indians had discovered the mathematical benefits
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00:20:22,640 –> 00:20:24,560
of the decimal place-value system
286
00:20:24,560 –> 00:20:28,520
and were using it by the middle of the 3rd century AD.
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00:20:30,600 –> 00:20:34,200
It’s been suggested that the Indians learned the system
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00:20:34,200 –> 00:20:38,800
from Chinese merchants travelling in India with their counting rods,
289
00:20:38,800 –> 00:20:42,640
or they may well just have stumbled across it themselves.
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00:20:42,640 –> 00:20:46,120
It’s all such a long time ago that it’s shrouded in mystery.
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00:20:48,320 –> 00:20:51,840
We may never know how the Indians came up with their number system,
292
00:20:51,840 –> 00:20:54,880
but we do know that they refined and perfected it,
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00:20:54,880 –> 00:20:58,800
creating the ancestors for the nine numerals used across the world now.
294
00:20:58,800 –> 00:21:01,480
Many rank the Indian system of counting
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00:21:01,480 –> 00:21:05,040
as one of the greatest intellectual innovations of all time,
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00:21:05,040 –> 00:21:09,200
developing into the closest thing we could call a universal language.
297
00:21:27,120 –> 00:21:29,600
But there was one number missing,
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00:21:29,600 –> 00:21:33,440
and it was the Indians who would introduce it to the world.
299
00:21:39,960 –> 00:21:44,400
The earliest known recording of this number dates from the 9th century,
300
00:21:44,400 –> 00:21:48,080
though it was probably in practical use for centuries before.
301
00:21:49,720 –> 00:21:53,560
This strange new numeral is engraved on the wall
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00:21:53,560 –> 00:21:57,360
of small temple in the fort of Gwalior in central India.
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00:22:01,480 –> 00:22:05,400
So here we are in one of the holy sites of the mathematical world,
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00:22:05,400 –> 00:22:08,840
and what I’m looking for is in this inscription on the wall.
305
00:22:09,800 –> 00:22:12,600
Up here are some numbers, and…
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00:22:12,600 –> 00:22:14,880
here’s the new number.
307
00:22:14,880 –> 00:22:16,880
It’s zero.
308
00:22:21,600 –> 00:22:25,720
It’s astonishing to think that before the Indians invented it,
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00:22:25,720 –> 00:22:28,120
there was no number zero.
310
00:22:28,120 –> 00:22:31,280
To the ancient Greeks, it simply hadn’t existed.
311
00:22:31,280 –> 00:22:35,520
To the Egyptians, the Mesopotamians and, as we’ve seen, the Chinese,
312
00:22:35,520 –> 00:22:39,720
zero had been in use but as a placeholder, an empty space
313
00:22:39,720 –> 00:22:42,040
to show a zero inside a number.
314
00:22:45,320 –> 00:22:48,400
The Indians transformed zero from a mere placeholder
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00:22:48,400 –> 00:22:51,320
into a number that made sense in its own right -
316
00:22:51,320 –> 00:22:54,280
a number for calculation, for investigation.
317
00:22:54,280 –> 00:22:58,480
This brilliant conceptual leap would revolutionise mathematics.
318
00:23:02,400 –> 00:23:06,760
Now, with just ten digits - zero to nine - it was suddenly possible
319
00:23:06,760 –> 00:23:09,760
to capture astronomically large numbers
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00:23:09,760 –> 00:23:12,040
in an incredibly efficient way.
321
00:23:15,040 –> 00:23:18,360
But why did the Indians make this imaginative leap?
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00:23:18,360 –> 00:23:20,560
Well, we’ll never know for sure,
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00:23:20,560 –> 00:23:24,520
but it’s possible that the idea and symbol that the Indians use for zero
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00:23:24,520 –> 00:23:27,720
came from calculations they did with stones in the sand.
325
00:23:27,720 –> 00:23:31,040
When stones were removed from the calculation,
326
00:23:31,040 –> 00:23:33,800
a small, round hole was left in its place,
327
00:23:33,800 –> 00:23:37,160
representing the movement from something to nothing.
328
00:23:39,800 –> 00:23:44,120
But perhaps there is also a cultural reason for the invention of zero.
329
00:23:44,120 –> 00:23:47,680
HORNS BLOW AND DRUMS BANG
330
00:23:47,680 –> 00:23:50,600
METALLIC BEATING
331
00:23:53,040 –> 00:23:57,520
For the ancient Indians, the concepts of nothingness and eternity
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00:23:57,520 –> 00:24:00,440
lay at the very heart of their belief system.
333
00:24:04,920 –> 00:24:07,360
BELL CLANGS AND SILENCE FALLS
334
00:24:09,880 –> 00:24:13,880
In the religions of India, the universe was born from nothingness,
335
00:24:13,880 –> 00:24:17,000
and nothingness is the ultimate goal of humanity.
336
00:24:17,000 –> 00:24:18,840
So it’s perhaps not surprising
337
00:24:18,840 –> 00:24:22,680
that a culture that so enthusiastically embraced the void
338
00:24:22,680 –> 00:24:25,880
should be happy with the notion of zero.
339
00:24:25,880 –> 00:24:30,080
The Indians even used the word for the philosophical idea of the void,
340
00:24:30,080 –> 00:24:33,920
shunya, to represent the new mathematical term “zero”.
341
00:24:47,280 –> 00:24:52,680
In the 7th century, the brilliant Indian mathematician Brahmagupta
342
00:24:52,680 –> 00:24:55,680
proved some of the essential properties of zero.
343
00:25:01,480 –> 00:25:04,320
Brahmagupta’s rules about calculating with zero
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00:25:04,320 –> 00:25:08,280
are taught in schools all over the world to this day.
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00:25:09,240 –> 00:25:12,240
One plus zero equals one.
346
00:25:13,280 –> 00:25:16,640
One minus zero equals one.
347
00:25:16,640 –> 00:25:19,920
One times zero is equal to zero.
348
00:25:24,120 –> 00:25:28,680
But Brahmagupta came a cropper when he tried to do one divided by zero.
349
00:25:28,680 –> 00:25:31,880
After all, what number times zero equals one?
350
00:25:31,880 –> 00:25:35,760
It would require a new mathematical concept, that of infinity,
351
00:25:35,760 –> 00:25:38,000
to make sense of dividing by zero,
352
00:25:38,000 –> 00:25:41,920
and the breakthrough was made by a 12th-century Indian mathematician
353
00:25:41,920 –> 00:25:45,040
called Bhaskara II, and it works like this.
354
00:25:45,040 –> 00:25:51,200
If I take a fruit and I divide it into halves, I get two pieces,
355
00:25:51,200 –> 00:25:54,080
so one divided by a half is two.
356
00:25:54,080 –> 00:25:57,480
If I divide it into thirds, I get three pieces.
357
00:25:57,480 –> 00:26:00,920
So when I divide it into smaller and smaller fractions,
358
00:26:00,920 –> 00:26:04,640
I get more and more pieces, so ultimately,
359
00:26:04,640 –> 00:26:06,600
when I divide by a piece
360
00:26:06,600 –> 00:26:10,400
which is of zero size, I’ll have infinitely many pieces.
361
00:26:10,400 –> 00:26:14,560
So for Bhaskara, one divided by zero is infinity.
362
00:26:22,880 –> 00:26:26,680
But the Indians would go further in their calculations with zero.
363
00:26:27,840 –> 00:26:31,920
For example, if you take three from three and get zero,
364
00:26:31,920 –> 00:26:35,240
what happens when you take four from three?
365
00:26:35,240 –> 00:26:37,480
It looks like you have nothing,
366
00:26:37,480 –> 00:26:39,720
but the Indians recognised that this
367
00:26:39,720 –> 00:26:43,720
was a new sort of nothing - negative numbers.
368
00:26:43,720 –> 00:26:47,440
The Indians called them “debts”, because they solved equations like,
369
00:26:47,440 –> 00:26:51,040
“If I have three batches of material and take four away,
370
00:26:51,040 –> 00:26:53,200
“how many have I left?”
371
00:26:56,880 –> 00:26:58,840
This may seem odd and impractical,
372
00:26:58,840 –> 00:27:01,400
but that was the beauty of Indian mathematics.
373
00:27:01,400 –> 00:27:04,680
Their ability to come up with negative numbers and zero
374
00:27:04,680 –> 00:27:08,080
was because they thought of numbers as abstract entities.
375
00:27:08,080 –> 00:27:11,360
They weren’t just for counting and measuring pieces of cloth.
376
00:27:11,360 –> 00:27:15,000
They had a life of their own, floating free of the real world.
377
00:27:15,000 –> 00:27:19,000
This led to an explosion of mathematical ideas.
378
00:27:30,880 –> 00:27:34,560
The Indians’ abstract approach to mathematics soon revealed
379
00:27:34,560 –> 00:27:38,440
a new side to the problem of how to solve quadratic equations.
380
00:27:38,440 –> 00:27:42,000
That is equations including numbers to the power of two.
381
00:27:43,520 –> 00:27:47,520
Brahmagupta’s understanding of negative numbers allowed him to see
382
00:27:47,520 –> 00:27:50,720
that quadratic equations always have two solutions,
383
00:27:50,720 –> 00:27:52,600
one of which could be negative.
384
00:27:55,120 –> 00:27:57,040
Brahmagupta went even further,
385
00:27:57,040 –> 00:28:00,000
solving quadratic equations with two unknowns,
386
00:28:00,000 –> 00:28:04,040
a question which wouldn’t be considered in the West until 1657,
387
00:28:04,040 –> 00:28:05,920
when French mathematician Fermat
388
00:28:05,920 –> 00:28:08,600
challenged his colleagues with the same problem.
389
00:28:08,600 –> 00:28:11,760
Little did he know that they’d been beaten to a solution
390
00:28:11,760 –> 00:28:14,680
by Brahmagupta 1,000 years earlier.
391
00:28:20,000 –> 00:28:24,640
Brahmagupta was beginning to find abstract ways of solving equations,
392
00:28:24,640 –> 00:28:27,800
but astonishingly, he was also developing
393
00:28:27,800 –> 00:28:31,120
a new mathematical language to express that abstraction.
394
00:28:32,440 –> 00:28:36,640
Brahmagupta was experimenting with ways of writing his equations down,
395
00:28:36,640 –> 00:28:40,120
using the initials of the names of different colours
396
00:28:40,120 –> 00:28:42,680
to represent unknowns in his equations.
397
00:28:44,640 –> 00:28:47,400
A new mathematical language was coming to life,
398
00:28:47,400 –> 00:28:49,840
which would ultimately lead to the x’s and y’s
399
00:28:49,840 –> 00:28:52,880
which fill today’s mathematical journals.
400
00:29:07,160 –> 00:29:10,840
But it wasn’t just new notation that was being developed.
401
00:29:13,200 –> 00:29:15,840
Indian mathematicians were responsible for making
402
00:29:15,840 –> 00:29:19,560
fundamental new discoveries in the theory of trigonometry.
403
00:29:22,400 –> 00:29:26,640
The power of trigonometry is that it acts like a dictionary,
404
00:29:26,640 –> 00:29:29,880
translating geometry into numbers and back.
405
00:29:29,880 –> 00:29:33,120
Although first developed by the ancient Greeks,
406
00:29:33,120 –> 00:29:35,720
it was in the hands of the Indian mathematicians
407
00:29:35,720 –> 00:29:37,760
that the subject truly flourished.
408
00:29:37,760 –> 00:29:42,280
At its heart lies the study of right-angled triangles.
409
00:29:44,520 –> 00:29:48,000
In trigonometry, you can use this angle here
410
00:29:48,000 –> 00:29:52,240
to find the ratios of the opposite side to the longest side.
411
00:29:52,240 –> 00:29:55,000
There’s a function called the sine function
412
00:29:55,000 –> 00:29:58,040
which, when you input the angle, outputs the ratio.
413
00:29:58,040 –> 00:30:01,720
So for example in this triangle, the angle is about 30 degrees,
414
00:30:01,720 –> 00:30:05,720
so the output of the sine function is a ratio of one to two,
415
00:30:05,720 –> 00:30:10,320
telling me that this side is half the length of the longest side.
416
00:30:12,800 –> 00:30:16,800
The sine function enables you to calculate distances
417
00:30:16,800 –> 00:30:21,080
when you’re not able to make an accurate measurement.
418
00:30:21,080 –> 00:30:25,160
To this day, it’s used in architecture and engineering.
419
00:30:25,160 –> 00:30:28,000
The Indians used it to survey the land around them,
420
00:30:28,000 –> 00:30:32,840
navigate the seas and, ultimately, chart the depths of space itself.
421
00:30:34,800 –> 00:30:37,760
It was central to the work of observatories,
422
00:30:37,760 –> 00:30:39,600
like this one in Delhi,
423
00:30:39,600 –> 00:30:42,480
where astronomers would study the stars.
424
00:30:42,480 –> 00:30:45,000
The Indian astronomers could use trigonometry
425
00:30:45,000 –> 00:30:48,120
to work out the relative distance between Earth and the moon
426
00:30:48,120 –> 00:30:49,560
and Earth and the sun.
427
00:30:49,560 –> 00:30:53,360
You can only make the calculation when the moon is half full,
428
00:30:53,360 –> 00:30:56,560
because that’s when it’s directly opposite the sun,
429
00:30:56,560 –> 00:31:01,080
so that the sun, moon and Earth create a right-angled triangle.
430
00:31:02,640 –> 00:31:04,480
Now, the Indians could measure
431
00:31:04,480 –> 00:31:07,800
that the angle between the sun and the observatory
432
00:31:07,800 –> 00:31:09,640
was one-seventh of a degree.
433
00:31:10,880 –> 00:31:14,160
The sine function of one-seventh of a degree
434
00:31:14,160 –> 00:31:18,080
gives me the ratio of 400:1.
435
00:31:18,080 –> 00:31:23,240
This means the sun is 400 times further from Earth than the moon is.
436
00:31:23,240 –> 00:31:25,120
So using trigonometry,
437
00:31:25,120 –> 00:31:28,400
the Indian mathematicians could explore the solar system
438
00:31:28,400 –> 00:31:31,440
without ever having to leave the surface of the Earth.
439
00:31:39,000 –> 00:31:42,600
The ancient Greeks had been the first to explore the sine function,
440
00:31:42,600 –> 00:31:46,960
listing precise values for some angles,
441
00:31:46,960 –> 00:31:50,600
but they couldn’t calculate the sines of every angle.
442
00:31:50,600 –> 00:31:55,120
The Indians were to go much further, setting themselves a mammoth task.
443
00:31:55,120 –> 00:31:57,200
The search was on to find a way
444
00:31:57,200 –> 00:32:01,200
to calculate the sine function of any angle you might be given.
445
00:32:17,920 –> 00:32:21,440
The breakthrough in the search for the sine function of every angle
446
00:32:21,440 –> 00:32:24,480
would be made here in Kerala in south India.
447
00:32:24,480 –> 00:32:27,560
In the 15th century, this part of the country
448
00:32:27,560 –> 00:32:31,360
became home to one of the most brilliant schools of mathematicians
449
00:32:31,360 –> 00:32:33,160
to have ever worked.
450
00:32:35,280 –> 00:32:38,560
Their leader was called Madhava, and he was to make
451
00:32:38,560 –> 00:32:42,320
some extraordinary mathematical discoveries.
452
00:32:45,120 –> 00:32:49,080
The key to Madhava’s success was the concept of the infinite.
453
00:32:49,080 –> 00:32:52,680
Madhava discovered that you could add up infinitely many things
454
00:32:52,680 –> 00:32:54,520
with dramatic effects.
455
00:32:54,520 –> 00:32:57,840
Previous cultures had been nervous of these infinite sums,
456
00:32:57,840 –> 00:33:00,320
but Madhava was happy to play with them.
457
00:33:00,320 –> 00:33:02,880
For example, here’s how one can be made up
458
00:33:02,880 –> 00:33:05,320
by adding infinitely many fractions.
459
00:33:06,840 –> 00:33:11,200
I’m heading from zero to one on my boat,
460
00:33:11,200 –> 00:33:15,440
but I can split my journey up into infinitely many fractions.
461
00:33:15,440 –> 00:33:18,200
So I can get to a half,
462
00:33:18,200 –> 00:33:21,920
then I can sail on a quarter,
463
00:33:21,920 –> 00:33:24,920
then an eighth, then a sixteenth, and so on.
464
00:33:24,920 –> 00:33:29,320
The smaller the fractions I move, the nearer to one I get,
465
00:33:29,320 –> 00:33:33,720
but I’ll only get there once I’ve added up infinitely many fractions.
466
00:33:36,040 –> 00:33:38,160
Physically and philosophically,
467
00:33:38,160 –> 00:33:41,640
it seems rather a challenge to add up infinitely many things,
468
00:33:41,640 –> 00:33:45,680
but the power of mathematics is to make sense of the impossible.
469
00:33:45,680 –> 00:33:47,240
By producing a language
470
00:33:47,240 –> 00:33:49,600
to articulate and manipulate the infinite,
471
00:33:49,600 –> 00:33:52,480
you can prove that after infinitely many steps
472
00:33:52,480 –> 00:33:54,440
you’ll reach your destination.
473
00:33:57,640 –> 00:34:01,880
Such infinite sums are called infinite series, and Madhava
474
00:34:01,880 –> 00:34:04,520
was doing a lot of research into the connections
475
00:34:04,520 –> 00:34:07,560
between these series and trigonometry.
476
00:34:08,560 –> 00:34:12,200
First, he realised that he could use the same principle
477
00:34:12,200 –> 00:34:14,840
of adding up infinitely many fractions to capture
478
00:34:14,840 –> 00:34:19,360
one of the most important numbers in mathematics - pi.
479
00:34:20,880 –> 00:34:25,680
Pi is the ratio of the circle’s circumference to its diameter.
480
00:34:25,680 –> 00:34:29,880
It’s a number that appears in all sorts of mathematics,
481
00:34:29,880 –> 00:34:32,360
but is especially useful for engineers,
482
00:34:32,360 –> 00:34:36,600
because any measurements involving curves soon require pi.
483
00:34:38,200 –> 00:34:42,800
So for centuries, mathematicians searched for a precise value for pi.
484
00:34:48,320 –> 00:34:52,320
It was in 6th-century India that the mathematician Aryabhata
485
00:34:52,320 –> 00:34:57,160
gave a very accurate approximation for pi - namely 3.1416.
486
00:34:57,160 –> 00:34:58,840
He went on to use this
487
00:34:58,840 –> 00:35:02,000
to make a measurement of the circumference of the Earth,
488
00:35:02,000 –> 00:35:05,480
and he got it as 24,835 miles,
489
00:35:05,480 –> 00:35:09,800
which, amazingly, is only 70 miles away from its true value.
490
00:35:09,800 –> 00:35:12,360
But it was in Kerala in the 15th century
491
00:35:12,360 –> 00:35:15,240
that Madhava realised he could use infinity
492
00:35:15,240 –> 00:35:17,680
to get an exact formula for pi.
493
00:35:21,200 –> 00:35:24,800
By successively adding and subtracting different fractions,
494
00:35:24,800 –> 00:35:28,320
Madhava could hone in on an exact formula for pi.
495
00:35:30,640 –> 00:35:34,160
First, he moved four steps up the number line.
496
00:35:34,160 –> 00:35:36,520
That took him way past pi.
497
00:35:38,040 –> 00:35:41,080
So next he took four-thirds of a step,
498
00:35:41,080 –> 00:35:44,400
or one-and-one-third steps, back.
499
00:35:44,400 –> 00:35:46,560
Now he’d come too far the other way.
500
00:35:47,800 –> 00:35:51,520
So he headed forward four-fifths of a step.
501
00:35:51,520 –> 00:35:56,320
Each time, he alternated between four divided by the next odd number.
502
00:36:03,040 –> 00:36:06,160
He zigzagged up and down the number line,
503
00:36:06,160 –> 00:36:08,640
getting closer and closer to pi.
504
00:36:08,640 –> 00:36:12,000
He discovered that if you went through all the odd numbers,
505
00:36:12,000 –> 00:36:15,520
infinitely many of them, you would hit pi exactly.
506
00:36:19,920 –> 00:36:22,640
I was taught at university that this formula for pi
507
00:36:22,640 –> 00:36:26,480
was discovered by the 17th-century German mathematician Leibniz,
508
00:36:26,480 –> 00:36:29,880
but amazingly, it was actually discovered here in Kerala
509
00:36:29,880 –> 00:36:31,760
two centuries earlier by Madhava.
510
00:36:31,760 –> 00:36:34,360
He went on to use the same sort of mathematics
511
00:36:34,360 –> 00:36:36,280
to get infinite-series expressions
512
00:36:36,280 –> 00:36:38,640
for the sine formula in trigonometry.
513
00:36:38,640 –> 00:36:42,080
And the wonderful thing is that you can use these formulas now
514
00:36:42,080 –> 00:36:46,040
to calculate the sine of any angle to any degree of accuracy.
515
00:36:56,760 –> 00:37:00,520
It seems incredible that the Indians made these discoveries
516
00:37:00,520 –> 00:37:03,400
centuries before Western mathematicians.
517
00:37:06,160 –> 00:37:10,760
And it says a lot about our attitude in the West to non-Western cultures
518
00:37:10,760 –> 00:37:14,720
that we nearly always claim their discoveries as our own.
519
00:37:14,720 –> 00:37:18,760
What is clear is the West has been very slow to give due credit
520
00:37:18,760 –> 00:37:22,320
to the major breakthroughs made in non-Western mathematics.
521
00:37:22,320 –> 00:37:25,520
Madhava wasn’t the only mathematician to suffer this way.
522
00:37:25,520 –> 00:37:28,600
As the West came into contact more and more with the East
523
00:37:28,600 –> 00:37:30,480
during the 18th and 19th centuries,
524
00:37:30,480 –> 00:37:33,120
there was a widespread dismissal and denigration
525
00:37:33,120 –> 00:37:35,200
of the cultures they were colonising.
526
00:37:35,200 –> 00:37:38,000
The natives, it was assumed, couldn’t have anything
527
00:37:38,000 –> 00:37:40,240
of intellectual worth to offer the West.
528
00:37:40,240 –> 00:37:43,160
It’s only now, at the beginning of the 21st century,
529
00:37:43,160 –> 00:37:45,880
that history is being rewritten.
530
00:37:45,880 –> 00:37:49,880
But Eastern mathematics was to have a major impact in Europe,
531
00:37:49,880 –> 00:37:53,040
thanks to the development of one of the major powers
532
00:37:53,040 –> 00:37:54,720
of the medieval world.
533
00:38:17,440 –> 00:38:20,960
In the 7th century, a new empire began to spread
534
00:38:20,960 –> 00:38:23,200
across the Middle East.
535
00:38:23,200 –> 00:38:25,680
The teachings of the Prophet Mohammed
536
00:38:25,680 –> 00:38:28,560
inspired a vast and powerful Islamic empire
537
00:38:28,560 –> 00:38:30,920
which soon stretched from India in the east
538
00:38:30,920 –> 00:38:35,160
to here in Morocco in the west.
539
00:38:41,960 –> 00:38:46,480
And at the heart of this empire lay a vibrant intellectual culture.
540
00:38:51,400 –> 00:38:56,160
A great library and centre of learning was established in Baghdad.
541
00:38:56,160 –> 00:38:59,640
Called the House of Wisdom, its teaching spread
542
00:38:59,640 –> 00:39:01,840
throughout the Islamic empire,
543
00:39:01,840 –> 00:39:05,080
reaching schools like this one here in Fez.
544
00:39:05,080 –> 00:39:08,360
Subjects studied included astronomy, medicine,
545
00:39:08,360 –> 00:39:10,240
chemistry, zoology
546
00:39:10,240 –> 00:39:11,920
and mathematics.
547
00:39:13,480 –> 00:39:18,160
The Muslim scholars collected and translated many ancient texts,
548
00:39:18,160 –> 00:39:20,600
effectively saving them for posterity.
549
00:39:20,600 –> 00:39:23,880
In fact, without their intervention, we may never have known
550
00:39:23,880 –> 00:39:27,480
about the ancient cultures of Egypt, Babylon, Greece and India.
551
00:39:27,480 –> 00:39:30,440
But the scholars at the House of Wisdom weren’t content
552
00:39:30,440 –> 00:39:33,360
simply with translating other people’s mathematics.
553
00:39:33,360 –> 00:39:36,080
They wanted to create a mathematics of their own,
554
00:39:36,080 –> 00:39:37,920
to push the subject forward.
555
00:39:42,080 –> 00:39:46,080
Such intellectual curiosity was actively encouraged
556
00:39:46,080 –> 00:39:49,320
in the early centuries of the Islamic empire.
557
00:39:51,320 –> 00:39:54,880
The Koran asserted the importance of knowledge.
558
00:39:54,880 –> 00:39:58,640
Learning was nothing less than a requirement of God.
559
00:40:01,720 –> 00:40:05,400
In fact, the needs of Islam demanded mathematical skill.
560
00:40:05,400 –> 00:40:07,920
The devout needed to calculate the time of prayer
561
00:40:07,920 –> 00:40:10,640
and the direction of Mecca to pray towards,
562
00:40:10,640 –> 00:40:13,640
and the prohibition of depicting the human form
563
00:40:13,640 –> 00:40:15,520
meant that they had to use
564
00:40:15,520 –> 00:40:18,520
much more geometric patterns to cover their buildings.
565
00:40:18,520 –> 00:40:22,080
The Muslim artists discovered all the different sorts of symmetry
566
00:40:22,080 –> 00:40:26,320
that you can depict on a two-dimensional wall.
567
00:40:34,000 –> 00:40:37,040
The director of the House of Wisdom in Baghdad
568
00:40:37,040 –> 00:40:40,400
was a Persian scholar called Muhammad Al-Khwarizmi.
569
00:40:43,520 –> 00:40:48,440
Al-Khwarizmi was an exceptional mathematician who was responsible
570
00:40:48,440 –> 00:40:52,680
for introducing two key mathematical concepts to the West.
571
00:40:52,680 –> 00:40:55,680
Al-Khwarizmi recognised the incredible potential
572
00:40:55,680 –> 00:40:57,520
that the Hindu numerals had
573
00:40:57,520 –> 00:41:00,480
to revolutionise mathematics and science.
574
00:41:00,480 –> 00:41:03,040
His work explaining the power of these numbers
575
00:41:03,040 –> 00:41:06,000
to speed up calculations and do things effectively
576
00:41:06,000 –> 00:41:09,400
was so influential that it wasn’t long before they were adopted
577
00:41:09,400 –> 00:41:13,240
as the numbers of choice amongst the mathematicians of the Islamic world.
578
00:41:13,240 –> 00:41:16,000
In fact, these numbers have now become known
579
00:41:16,000 –> 00:41:18,320
as the Hindu-Arabic numerals.
580
00:41:18,320 –> 00:41:21,360
These numbers - one to nine and zero -
581
00:41:21,360 –> 00:41:25,160
are the ones we use today all over the world.
582
00:41:29,680 –> 00:41:34,640
But Al-Khwarizmi was to create a whole new mathematical language.
583
00:41:36,280 –> 00:41:38,240
It was called algebra
584
00:41:38,240 –> 00:41:42,760
and was named after the title of his book Al-jabr W’al-muqabala,
585
00:41:42,760 –> 00:41:46,120
or Calculation By Restoration Or Reduction.
586
00:41:50,960 –> 00:41:56,080
Algebra is the grammar that underlies the way that numbers work.
587
00:41:56,080 –> 00:41:58,480
It’s a language that explains the patterns
588
00:41:58,480 –> 00:42:01,640
that lie behind the behaviour of numbers.
589
00:42:01,640 –> 00:42:05,560
It’s a bit like a code for running a computer program.
590
00:42:05,560 –> 00:42:09,240
The code will work whatever the numbers you feed in to the program.
591
00:42:11,040 –> 00:42:14,680
For example, mathematicians might have discovered
592
00:42:14,680 –> 00:42:16,960
that if you take a number and square it,
593
00:42:16,960 –> 00:42:19,240
that’s always one more than if you’d taken
594
00:42:19,240 –> 00:42:22,240
the numbers either side and multiplied those together.
595
00:42:22,240 –> 00:42:25,440
For example, five times five is 25,
596
00:42:25,440 –> 00:42:29,360
which is one more than four times six - 24.
597
00:42:29,360 –> 00:42:33,160
Six times six is always one more than five times seven and so on.
598
00:42:33,160 –> 00:42:34,880
But how can you be sure
599
00:42:34,880 –> 00:42:38,080
that this is going to work whatever numbers you take?
600
00:42:38,080 –> 00:42:41,040
To explain the pattern underlying these calculations,
601
00:42:41,040 –> 00:42:43,320
let’s use the dyeing holes in this tannery.
602
00:42:51,280 –> 00:42:56,520
If we take a square of 25 holes, running five by five,
603
00:42:56,520 –> 00:43:00,760
and take one row of five away and add it to the bottom,
604
00:43:00,760 –> 00:43:03,640
we get six by four with one left over.
605
00:43:05,880 –> 00:43:09,440
But however many holes there are on the side of the square,
606
00:43:09,440 –> 00:43:12,320
we can always move one row of holes down in a similar way
607
00:43:12,320 –> 00:43:16,240
to be left with a rectangle of holes with one left over.
608
00:43:18,880 –> 00:43:20,960
Algebra was a huge breakthrough.
609
00:43:20,960 –> 00:43:22,680
Here was a new language
610
00:43:22,680 –> 00:43:25,720
to be able to analyse the way that numbers worked.
611
00:43:25,720 –> 00:43:27,880
Previously, the Indians and the Chinese
612
00:43:27,880 –> 00:43:30,120
had considered very specific problems,
613
00:43:30,120 –> 00:43:33,600
but Al-Khwarizmi went from the specific to the general.
614
00:43:33,600 –> 00:43:37,200
He developed systematic ways to be able to analyse problems
615
00:43:37,200 –> 00:43:40,800
so that the solutions would work whatever the numbers that you took.
616
00:43:40,800 –> 00:43:44,560
This language is used across the mathematical world today.
617
00:43:46,080 –> 00:43:50,800
Al-Khwarizmi’s great breakthrough came when he applied algebra
618
00:43:50,800 –> 00:43:52,480
to quadratic equations -
619
00:43:52,480 –> 00:43:55,560
that is equations including numbers to the power of two.
620
00:43:55,560 –> 00:43:58,360
The ancient Mesopotamians had devised
621
00:43:58,360 –> 00:44:02,120
a cunning method to solve particular quadratic equations,
622
00:44:02,120 –> 00:44:06,240
but it was Al-Khwarizmi’s abstract language of algebra
623
00:44:06,240 –> 00:44:10,000
that could finally express why this method always worked.
624
00:44:11,600 –> 00:44:14,200
This was a great conceptual leap
625
00:44:14,200 –> 00:44:17,920
and would ultimately lead to a formula that could be used to solve
626
00:44:17,920 –> 00:44:22,160
any quadratic equation, whatever the numbers involved.
627
00:44:30,480 –> 00:44:32,440
The next mathematical Holy Grail
628
00:44:32,440 –> 00:44:37,040
was to find a general method that could solve all cubic equations -
629
00:44:37,040 –> 00:44:40,640
equations including numbers to the power of three.
630
00:44:57,920 –> 00:45:00,640
It was an 11th-century Persian mathematician
631
00:45:00,640 –> 00:45:04,000
who took up the challenge of cracking the problem of the cubic.
632
00:45:08,440 –> 00:45:11,960
His name was Omar Khayyam, and he travelled widely
633
00:45:11,960 –> 00:45:15,600
across the Middle East, calculating as he went.
634
00:45:17,520 –> 00:45:21,440
But he was famous for another, very different, reason.
635
00:45:21,440 –> 00:45:24,080
Khayyam was a celebrated poet,
636
00:45:24,080 –> 00:45:28,040
author of the great epic poem the Rubaiyat.
637
00:45:30,920 –> 00:45:35,120
It may seem a bit odd that a poet was also a master mathematician.
638
00:45:35,120 –> 00:45:38,560
After all, the combination doesn’t immediately spring to mind.
639
00:45:38,560 –> 00:45:42,200
But there’s quite a lot of similarity between the disciplines.
640
00:45:42,200 –> 00:45:45,560
Poetry, with its rhyming structure and rhythmic patterns,
641
00:45:45,560 –> 00:45:49,520
resonates strongly with constructing a logical mathematical proof.
642
00:45:53,000 –> 00:45:55,320
Khayyam’s major mathematical work
643
00:45:55,320 –> 00:46:02,040
was devoted to finding the general method to solve all cubic equations.
644
00:46:02,040 –> 00:46:04,600
Rather than looking at particular examples,
645
00:46:04,600 –> 00:46:08,640
Khayyam carried out a systematic analysis of the problem,
646
00:46:08,640 –> 00:46:11,920
true to the algebraic spirit of Al-Khwarizmi.
647
00:46:13,760 –> 00:46:16,280
Khayyam’s analysis revealed for the first time
648
00:46:16,280 –> 00:46:19,480
that there were several different sorts of cubic equation.
649
00:46:19,480 –> 00:46:21,560
But he was still very influenced
650
00:46:21,560 –> 00:46:24,320
by the geometric heritage of the Greeks.
651
00:46:24,320 –> 00:46:27,080
He couldn’t separate the algebra from the geometry.
652
00:46:27,080 –> 00:46:30,440
In fact, he wouldn’t even consider equations in higher degrees,
653
00:46:30,440 –> 00:46:33,840
because they described objects in more than three dimensions,
654
00:46:33,840 –> 00:46:35,640
something he saw as impossible.
655
00:46:35,640 –> 00:46:37,520
Although the geometry allowed him
656
00:46:37,520 –> 00:46:40,120
to analyse these cubic equations to some extent,
657
00:46:40,120 –> 00:46:43,280
he still couldn’t come up with a purely algebraic solution.
658
00:46:45,800 –> 00:46:51,400
It would be another 500 years before mathematicians could make the leap
659
00:46:51,400 –> 00:46:54,720
and find a general solution to the cubic equation.
660
00:46:56,240 –> 00:47:01,400
And that leap would finally be made in the West - in Italy.
661
00:47:15,400 –> 00:47:18,880
During the centuries in which China, India and the Islamic empire
662
00:47:18,880 –> 00:47:20,520
had been in the ascendant,
663
00:47:20,520 –> 00:47:24,760
Europe had fallen under the shadow of the Dark Ages.
664
00:47:26,280 –> 00:47:30,560
All intellectual life, including the study of mathematics, had stagnated.
665
00:47:35,760 –> 00:47:41,400
But by the 13th century, things were beginning to change.
666
00:47:41,400 –> 00:47:46,680
Led by Italy, Europe was starting to explore and trade with the East.
667
00:47:46,680 –> 00:47:51,120
With that contact came the spread of Eastern knowledge to the West.
668
00:47:51,120 –> 00:47:53,120
It was the son of a customs official
669
00:47:53,120 –> 00:47:56,640
that would become Europe’s first great medieval mathematician.
670
00:47:56,640 –> 00:48:00,240
As a child, he travelled around North Africa with his father,
671
00:48:00,240 –> 00:48:03,440
where he learnt about the developments of Arabic mathematics
672
00:48:03,440 –> 00:48:06,720
and especially the benefits of the Hindu-Arabic numerals.
673
00:48:06,720 –> 00:48:08,760
When he got home to Italy he wrote a book
674
00:48:08,760 –> 00:48:10,640
that would be hugely influential
675
00:48:10,640 –> 00:48:13,240
in the development of Western mathematics.
676
00:48:29,320 –> 00:48:31,800
That mathematician was Leonardo of Pisa,
677
00:48:31,800 –> 00:48:34,440
better known as Fibonacci,
678
00:48:35,480 –> 00:48:37,920
and in his Book Of Calculating,
679
00:48:37,920 –> 00:48:40,720
Fibonacci promoted the new number system,
680
00:48:40,720 –> 00:48:44,080
demonstrating how simple it was compared to the Roman numerals
681
00:48:44,080 –> 00:48:47,560
that were in use across Europe.
682
00:48:47,560 –> 00:48:52,640
Calculations were far easier, a fact that had huge consequences
683
00:48:52,640 –> 00:48:55,080
for anyone dealing with numbers -
684
00:48:55,080 –> 00:48:59,920
pretty much everyone, from mathematicians to merchants.
685
00:48:59,920 –> 00:49:02,640
But there was widespread suspicion of these new numbers.
686
00:49:02,640 –> 00:49:06,320
Old habits die hard, and the authorities just didn’t trust them.
687
00:49:06,320 –> 00:49:09,200
Some believed that they would be more open to fraud -
688
00:49:09,200 –> 00:49:11,040
that you could tamper with them.
689
00:49:11,040 –> 00:49:14,520
Others believed that they’d be so easy to use for calculations
690
00:49:14,520 –> 00:49:17,800
that it would empower the masses, taking authority away
691
00:49:17,800 –> 00:49:21,800
from the intelligentsia who knew how to use the old sort of numbers.
692
00:49:27,200 –> 00:49:31,200
The city of Florence even banned them in 1299,
693
00:49:31,200 –> 00:49:34,400
but over time, common sense prevailed,
694
00:49:34,400 –> 00:49:37,200
the new system spread throughout Europe,
695
00:49:37,200 –> 00:49:40,960
and the old Roman system slowly became defunct.
696
00:49:40,960 –> 00:49:46,440
At last, the Hindu-Arabic numerals, zero to nine, had triumphed.
697
00:49:48,360 –> 00:49:51,720
Today Fibonacci is best known for the discovery of some numbers,
698
00:49:51,720 –> 00:49:55,200
now called the Fibonacci sequence, that arose when he was trying
699
00:49:55,200 –> 00:49:58,240
to solve a riddle about the mating habits of rabbits.
700
00:49:58,240 –> 00:50:01,040
Suppose a farmer has a pair of rabbits.
701
00:50:01,040 –> 00:50:03,520
Rabbits take two months to reach maturity,
702
00:50:03,520 –> 00:50:07,240
and after that they give birth to another pair of rabbits each month.
703
00:50:07,240 –> 00:50:09,080
So the problem was how to determine
704
00:50:09,080 –> 00:50:12,560
how many pairs of rabbits there will be in any given month.
705
00:50:14,800 –> 00:50:20,000
Well, during the first month you have one pair of rabbits,
706
00:50:20,000 –> 00:50:24,200
and since they haven’t matured, they can’t reproduce.
707
00:50:24,200 –> 00:50:28,400
During the second month, there is still only one pair.
708
00:50:28,400 –> 00:50:32,000
But at the beginning of the third month, the first pair
709
00:50:32,000 –> 00:50:36,400
reproduces for the first time, so there are two pairs of rabbits.
710
00:50:36,400 –> 00:50:38,720
At the beginning of the fourth month,
711
00:50:38,720 –> 00:50:40,800
the first pair reproduces again,
712
00:50:40,800 –> 00:50:45,160
but the second pair is not mature enough, so there are three pairs.
713
00:50:46,840 –> 00:50:50,000
In the fifth month, the first pair reproduces
714
00:50:50,000 –> 00:50:53,480
and the second pair reproduces for the first time,
715
00:50:53,480 –> 00:50:58,200
but the third pair is still too young, so there are five pairs.
716
00:50:58,200 –> 00:51:00,120
The mating ritual continues,
717
00:51:00,120 –> 00:51:02,240
but what you soon realise is
718
00:51:02,240 –> 00:51:05,760
the number of pairs of rabbits you have in any given month
719
00:51:05,760 –> 00:51:09,400
is the sum of the pairs of rabbits that you have had
720
00:51:09,400 –> 00:51:13,120
in each of the two previous months, so the sequence goes…
721
00:51:13,120 –> 00:51:17,280
1…1…2…3…
722
00:51:17,280 –> 00:51:21,120
5…8…13…
723
00:51:21,120 –> 00:51:26,640
21…34…55…and so on.
724
00:51:26,640 –> 00:51:29,680
The Fibonacci numbers are nature’s favourite numbers.
725
00:51:29,680 –> 00:51:31,600
It’s not just rabbits that use them.
726
00:51:31,600 –> 00:51:35,880
The number of petals on a flower is invariably a Fibonacci number.
727
00:51:35,880 –> 00:51:39,960
They run up and down pineapples if you count the segments.
728
00:51:39,960 –> 00:51:42,960
Even snails use them to grow their shells.
729
00:51:42,960 –> 00:51:46,920
Wherever you find growth in nature, you find the Fibonacci numbers.
730
00:51:51,560 –> 00:51:54,880
But the next major breakthrough in European mathematics
731
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wouldn’t happen until the early 16th century.
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It would involve
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finding the general method that would solve all cubic equations,
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and it would happen here in the Italian city of Bologna.
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The University of Bologna was the crucible
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of European mathematical thought at the beginning of the 16th century.
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Pupils from all over Europe flocked here and developed
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a new form of spectator sport - the mathematical competition.
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Large audiences would gather to watch mathematicians
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challenge each other with numbers, a kind of intellectual fencing match.
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But even in this questioning atmosphere
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it was believed that some problems were just unsolvable.
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It was generally assumed that finding a general method
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to solve all cubic equations was impossible.
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But one scholar was to prove everyone wrong.
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His name was Tartaglia,
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but he certainly didn’t look
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the heroic architect of a new mathematics.
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At the age of 12, he’d been slashed across the face
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with a sabre by a rampaging French army.
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The result was a terrible facial scar
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and a devastating speech impediment.
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In fact, Tartaglia was the nickname he’d been given as a child
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and means “the stammerer”.
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Shunned by his schoolmates,
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Tartaglia lost himself in mathematics, and it wasn’t long
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before he’d found the formula to solve one type of cubic equation.
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But Tartaglia soon discovered
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that he wasn’t the only one to believe he’d cracked the cubic.
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A young Italian called Fior was boasting
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that he too held the secret formula for solving cubic equations.
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When news broke about the discoveries
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00:53:59,840 –> 00:54:02,440
made by the two mathematicians,
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a competition was arranged to pit them against each other.
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The intellectual fencing match of the century was about to begin.
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The trouble was that Tartaglia
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only knew how to solve one sort of cubic equation,
768
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and Fior was ready to challenge him
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with questions about a different sort.
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But just a few days before the contest,
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Tartaglia worked out how to solve this different sort,
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and with this new weapon in his arsenal he thrashed his opponent,
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solving all the questions in under two hours.
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Tartaglia went on
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to find the formula to solve all types of cubic equations.
776
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News soon spread, and a mathematician in Milan
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called Cardano became so desperate to find the solution
778
00:54:54,800 –> 00:54:59,360
that he persuaded a reluctant Tartaglia to reveal the secret,
779
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but on one condition -
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that Cardano keep the secret and never publish.
781
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But Cardano couldn’t resist
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discussing Tartaglia’s solution with his brilliant student, Ferrari.
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As Ferrari got to grips with Tartaglia’s work,
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he realised that he could use it to solve
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the more complicated quartic equation, an amazing achievement.
786
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Cardano couldn’t deny his student his just rewards,
787
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and he broke his vow of secrecy, publishing Tartaglia’s work
788
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together with Ferrari’s brilliant solution of the quartic.
789
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Poor Tartaglia never recovered and died penniless,
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and to this day, the formula that solves the cubic equation
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is known as Cardano’s formula.
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Tartaglia may not have won glory in his lifetime,
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but his mathematics managed to solve a problem that had bewildered
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the great mathematicians of China, India and the Arab world.
795
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It was the first great mathematical breakthrough
796
00:56:11,440 –> 00:56:13,440
to happen in modern Europe.
797
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The Europeans now had in their hands the new language of algebra,
798
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the powerful techniques of the Hindu-Arabic numerals
799
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and the beginnings of the mastery of the infinite.
800
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It was time for the Western world
801
00:56:28,920 –> 00:56:31,400
to start writing its own mathematical stories
802
00:56:31,400 –> 00:56:33,040
in the language of the East.
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The mathematical revolution was about to begin.
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You can learn more about The Story Of Maths with the Open University
805
00:56:43,560 –> 00:56:45,800
at open2.net.
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