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0:00
Our world is made up of patterns and sequences. They're all around us. Day becomes night. Animals travel across the earth in ever-changing formations. Landscapes are constantly altering. One of the reasons mathematics began was because we needed to find a way of making sense of these natural patterns.
0:27
The most basic concepts of maths - space and quantity - are hard-wired into our brains. Even animals have a sense of distance and number, assessing when their pack is outnumbered, and whether to fight or fly, calculating whether their prey is within striking distance. Understanding maths is the difference between life and death.
0:56
But it was man who took these basic concepts and started to build upon these foundations. At some point, humans started to spot patterns, to make connections, to count and to order the world around them. With this, a whole new mathematical universe began to emerge.
1:20
This is the River Nile. It's been the lifeline of Egypt for millennia.
1:26
I've come here because it's where some of the first signs of mathematics as we know it today emerged.
1:34
People abandoned nomadic life and began settling here as early as 6000BC. The conditions were perfect for farming.
1:47
The most important event for Egyptian agriculture each year was the flooding of the Nile. So this was used as a marker to start each new year. Egyptians did record what was going on over periods of time, so in order to establish a calendar like this, you need to count how many days, for example, happened in-between lunar phases, or how many days happened in-between two floodings of the Nile.
2:19
Recording the patterns for the seasons was essential, not only to their management of the land, but also their religion. The ancient Egyptians who settled on the Nile banks believed it was the river god, Hapy, who flooded the river each year. And in return for the life-giving water, the citizens offered a portion of the yield as a thanksgiving.
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As settlements grew larger, it became necessary to find ways to administer them. Areas of land needed to be calculated, crop yields predicted, taxes charged and collated. In short, people needed to count and measure.
2:59
The Egyptians used their bodies to measure the world, and it's how their units of measurements evolved. A palm was the width of a hand, a cubit an arm length from elbow to fingertips. Land cubits, strips of land measuring a cubit by 100, were used by the pharaoh's surveyors to calculate areas.
3:23
There's a very strong link between bureaucracy and the development of mathematics in ancient Egypt. And we can see this link right from the beginning, from the invention of the number system, throughout Egyptian history, really. For the Old Kingdom, the only evidence we have are metrological systems, that is measurements for areas, for length. This points to a bureaucratic need to develop such things. It was vital to know the area of a farmer's land so he could be taxed accordingly.
3:55
Or if the Nile robbed him of part of his land, so he could request a rebate. It meant that the pharaoh's surveyors were often calculating the area of irregular parcels of land. It was the need to solve such practical problems that made them the earliest mathematical innovators.
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The Egyptians needed some way to record the results of their calculations.
4:25
Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo, I was on the hunt for those that recorded some of the first numbers in history. They were difficult to track down.
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But I did find them in the end.
4:45
The Egyptians were using a decimal system, motivated by the 10 fingers on our hands. The sign for one was a stroke, 10, a heel bone, 100, a coil of rope, and 1,000, a Lotus plant. How much is this T-shirt? Er 25. 25!Yes!So that would be 2 knee bones and 5 strokes. So you're not gonna charge me anything up here?Here, one million! One million?My God! This one million. One million, yeah, that's pretty big!
5:20
The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed.
5:27
They had no concept of a place value, so one stroke could only represent one unit, not 100 or 1,000. Although you can write a million with just one character, rather than the seven that we use, if you want to write a million minus one, then the poor old Egyptian scribe has got to write nine strokes, nine heel bones, nine coils of rope, and so on, a total of 54 characters.
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Despite the drawback of this number system, the Egyptians were brilliant problem solvers.
6:01
We know this because of the few records that have survived. The Egyptian scribes used sheets of papyrus to record their mathematical discoveries. This delicate material made from reeds decayed over time and many secrets perished with it. But there's one revealing document that has survived. The Rhind Mathematical Papyrus is the most important document we have today for Egyptian mathematics. We get a good overview of what types of problems the Egyptians would have dealt with in their mathematics.
6:37
We also get explicitly stated how multiplications and divisions were carried out.
6:44
The papyri show how to multiply two large numbers together. But to illustrate the method, let's take two smaller numbers. Let's do three times six. The scribe would take the first number, three, and put it in one column.
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In the second column, he would place the number one. Then he would double the numbers in each column, so three becomes six...
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..and six would become 12.
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And then in the second column, one would become two, and two becomes four.
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Now, here's the really clever bit. The scribe wants to multiply three by six. So he takes the powers of two in the second column, which add up to six. That's two plus four. Then he moves back to the first column, and just takes those rows corresponding to the two and the four. So that's six and the 12. He adds those together to get the answer of 18. But for me, the most striking thing about this method is that the scribe has effectively written that second number in binary.
8:00
Six is one lot of four, one lot of two, and no units. Which is 1-1-0. The Egyptians have understood the power of binary over 3,000 years before the mathematician and philosopher Leibniz would reveal their potential. Today, the whole technological world depends on the same principles that were used in ancient Egypt.
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The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC. Its problems are concerned with finding solutions to everyday situations. Several of the problems mention bread and beer, which isn't surprising as Egyptian workers were paid in food and drink. One is concerned with how to divide nine loaves equally between 10 people, without a fight breaking out. I've got nine loaves of bread here. I'm gonna take five of them and cut them into halves.
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Of course, nine people could shave a 10th off their loaf and give the pile of crumbs to the 10th person. But the Egyptians developed a far more elegant solution - take the next four and divide those into thirds.
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But two of the thirds I am now going to cut into fifths, so each piece will be one fifteenth.
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Each person then gets one half and one third and one fifteenth. It is through such seemingly practical problems that we start to see a more abstract mathematics developing. Suddenly, new numbers are on the scene - fractions - and it isn't too long before the Egyptians are exploring the mathematics of these numbers.
9:48
Fractions are clearly of practical importance to anyone dividing quantities for trade in the market. To log these transactions, the Egyptians developed notation which recorded these new numbers.
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One of the earliest representations of these fractions came from a hieroglyph which had great mystical significance. It's called the Eye of Horus. Horus was an Old Kingdom god, depicted as half man, half falcon.
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According to legend, Horus' father was killed by his other son, Seth. Horus was determined to avenge the murder. During one particularly fierce battle, Seth ripped out Horus' eye, tore it up and scattered it over Egypt. But the gods were looking favourably on Horus. They gathered up the scattered pieces and reassembled the eye.
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Each part of the eye represented a different fraction. Each one, half the fraction before. Although the original eye represented a whole unit, the reassembled eye is 1/64 short. Although the Egyptians stopped at 1/64, implicit in this picture is the possibility of adding more fractions, halving them each time, the sum getting closer and closer to one, but never quite reaching it. This is the first hint of something called a geometric series,
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and it appears at a number of points in the Rhind Papyrus. But the concept of infinite series would remain hidden until the mathematicians of Asia discovered it centuries later.
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Having worked out a system of numbers, including these new fractions, it was time for the Egyptians to apply their knowledge to understanding shapes that they encountered day to day. These shapes were rarely regular squares or rectangles, and in the Rhind Papyrus, we find the area of a more organic form, the circle. What is astounding in the calculation of the area of the circle is its exactness, really. How they would have found their method is open to speculation, because the texts we have
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do not show us the methods how they were found. This calculation is particularly striking because it depends on seeing how the shape of the circle can be approximated by shapes that the Egyptians already understood. The Rhind Papyrus states that a circular field with a diameter of nine units is close in area to a square with sides of eight. But how would this relationship have been discovered? My favourite theory sees the answer in the ancient game of mancala.
12:39
Mancala boards were found carved on the roofs of temples. Each player starts with an equal number of stones, and the object of the game is to move them round the board, capturing your opponent's counters on the way. As the players sat around waiting to make their next move, perhaps one of them realised that sometimes the balls fill the circular holes of the mancala board in a rather nice way. He might have gone on to experiment with trying to make larger circles.
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Perhaps he noticed that 64 stones, the square of 8, can be used to make a circle with diameter nine stones. By rearranging the stones, the circle has been approximated by a square. And because the area of a circle ispi times the radius squared, the Egyptian calculation gives us the first accurate value for pi. The area of the circle is 64. Divide this by the radius squared, in this case 4.5 squared, and you get a value for pi. So 64 divided by 4.5 squared is 3.16,
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just a little under two hundredths away from its true value. But the really brilliant thing is, the Egyptians are using these smaller shapes to capture the larger shape.
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But there's one imposing andmajestic symbol of Egyptian mathematics we haven't attempted to unravel yet - the pyramid. I've seen so many pictures that I couldn't believe I'd be impressed by them. But meeting them face to face, you understand why they're called one of the Seven Wonders of the Ancient World. They're simply breathtaking. And how much more impressive they must have been in their day, when the sides were as smooth as glass, reflecting the desert sun.
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To me it looks like there might be mirror pyramidshiding underneath the desert, which would complete the shapes to make perfectlysymmetrical octahedrons. Sometimes, in the shimmer of the desert heat, you can almost see these shapes.
14:45
It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician. The pyramids are just a little short to create these perfect shapes, but some have suggested another important mathematical concept might be hidden inside the proportions of the Great Pyramid - the golden ratio. Two lengths are in the golden ratio, if the relationship of the longest to the shortest is the same as the sum of the two to the longest side.
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Such a ratio has been associated with the perfect proportions one finds all over the natural world, as well as in the work of artists, architects and designers for millennia.
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Whether the architects of the pyramids were conscious of this important mathematical idea, or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know. For me, the most impressive thing about the pyramidsis the mathematical brilliance that went into making them, including the first inkling of one of the great theorems of the ancient world, Pythagoras' theorem.
15:55
In order to get perfect right-angled corners on their buildings and pyramids, the Egyptians would haveused a rope with knots tied in it. At some point, the Egyptians realised that if they took a triangle with sides marked with three knots, four knots and five knots, it guaranteed them aperfect right-angle. This is because three squared, plus four squared, is equal to five squared. So we've got a perfect Pythagorean triangle.
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In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle. But I'm pretty sure that the Egyptians hadn't got this sweeping generalisation of their 3, 4, 5 triangle. We would not expect to find the general proof because this is not the style of Egyptian mathematics. Every problem was solved using concrete numbers and then if a verification would be carried out at the end, it would use the result
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and these concrete, given numbers, there's no general proof within the Egyptian mathematical texts.
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It would be some 2,000 years before the Greeks and Pythagoras would prove that all right-angled triangles shared certain properties. This wasn't the only mathematical idea that theEgyptians would anticipate. In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume of a pyramid with its peak sliced off,which shows the first hint ofcalculus at work. For a culture like Egypt that is famous for its pyramids, you would expect problems like this
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to have been a regular feature within the mathematical texts. The calculation of the volume of a truncated pyramid is one of the most advanced bits, according to our modern standards of mathematics, that we have from ancient Egypt. The architects and engineers would certainly have wanted such a formula to calculate the amount of materials required to build it. But it's a mark of the sophistication of Egyptian mathematics that they were ableto produce such a beautiful method.
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To understand how they derived their formula, start with a pyramid built such that the highest point sits directly over one corner. Three of these can be put together to make a rectangular box, so the volume of the skewed pyramid is a third the volume of the box. That is, the height, times the length, times the width, divided by three. Now comes an argument which shows the very first hints of the calculus at work,
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thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory. Suppose you could cut the pyramid into slices, you could then slide the layers across to make the more symmetrical pyramid you see in Giza. However, the volume of the pyramid has not changed, despite the rearrangement of the layers. So the same formula works.
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The Egyptians were amazing innovators, and their ability to generate new mathematics was staggering. For me, they revealed the power of geometry and numbers, and made the first moves towards some of the exciting mathematical discoveries to come. But there was another civilisation that had mathematics to rival that of Egypt. And we know much more about their achievements.
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This is Damascus, over 5,000 years old, and still vibrant and bustling today. It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt. The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC. In order to expand and run their empire, they became masters of managing and manipulating numbers. We have law codes for instance that tell us about the way society is ordered.
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The people we know most about are the scribes, the professionally literate and numerate people who kept the records for the wealthy families and for the temples and palaces. Scribe schools existed from around 2500BC. Aspiring scribes were sent there as children, and learned how to read, write and work with numbers. Scribe records were kept on clay tablets, which allowed the Babylonians to manage and advance their empire.
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However, many of the tablets we have today aren't official documents, but children's exercises. It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics. So, this is a geometrical textbook from about the 18th century BC. I hope you can see that there are lots of pictures on it. And underneath each picture is a text that sets a problem about the picture. So for instance this one here says, I drew a square, 60 units long,
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and inside it, I drew four circles - what are their areas? This little tablet here was written 1,000 years at least later than the tablet here, but has a very interesting relationship. It also has four circles on, in a square, roughly drawn, but this isn't a textbook, it's a school exercise. The adult scribe who's teaching the student is being given this as an example of completed homework or something like that.
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Like the Egyptians, the Babylonians appeared interested in solving practical problems to do with measuring and weighing. The Babylonian solutions to these problems are written like mathematical recipes. A scribe would simply follow and record a set of instructions to get a result. Here's an example of the kind of problem they'd solve. I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them. Instead, I'm gonna take four times their weight and add them to the scales.
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Now I'm gonna add 20 gin. Gin was the ancient Babylonian measure of weight. I'm gonna take half of everything here and then add it again... That's two bundles, and ten gin. Everything on this side is equal to one mana. One mana was 60 gin. And here, we have one of the first mathematical equations in history, everything on this side is equal to one mana. But how much does the bundle of cinnamon sticks weigh? Without any algebraic language, they were able to manipulate
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the quantities to be able to prove that the cinnamon sticks weighed five gin. In my mind, it's this kind of problem which gives mathematics a bit of a bad name. You can blame those ancient Babylonians for all those tortuous problems you had at school. But the ancient Babylonian scribes excelled at this kind of problem. Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60.
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The Babylonians invented their number system, like the Egyptians, by using their fingers. But instead of counting through the 10 fingers on their hand, Babylonians found a moreintriguing way to count body parts. They used the 12 knuckles on one hand, and the five fingers on the other to be able to count 12 times 5, ie 60 different numbers. So for example, this number would have been 2 lots of 12, 24, and then, 1, 2, 3, 4, 5, to make 29.
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The number 60 had another powerful property. It can be perfectly divided in a multitude of ways. Here are 60 beans. I can arrange them in 2 rows of 30.
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3 rows of 20. 4 rows of 15. 5 rows of 12. Or 6 rows of 10. The divisibility of 60 makes it a perfect base in which to do arithmetic. The base 60 system was so successful, we still use elements of it today. Every time we want to tell the time, we recognise units of 60 - 60 seconds in a minute, 60 minutes in an hour. But the most important feature of the Babylonians' number system was that it recognised place value.
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Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording, the position of each Babylonian number records the power of 60.
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Instead of inventing new symbols for bigger and bigger numbers, they would write 1-1-1, so this number would be 3,661.
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The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky.
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The Babylonians' calendar was based on the cycles of the moon. They needed a way of recording astronomically large numbers. Month by month, year by year, these cycles were recorded. From about 800BC, there were complete lists of lunar eclipses. The Babylonian system of measurement was quite sophisticated at that time. They had a system of angular measurement, 360 degrees in a full circle, each degree was divided into 60 minutes, a minute was further divided into 60 seconds.
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So they had a regular system for measurement, and it was in perfect harmony with their number system, so it's well suited not only for observation but also for calculation. But in order to calculate and cope with these large numbers, the Babylonians needed to invent a new symbol. And in so doing, they prepared the ground for one of the great breakthroughs in the history of mathematics - zero. In the early days, the Babylonians, in order to mark an empty place in
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the middle of a number, would simply leave a blank space. So they needed a way of representing nothing in the middle of a number. So they used a sign, as a sort of breathing marker, a punctuation mark, and it comes to mean zero in the middle of a number. This was the first time zero, in any form, had appeared in the mathematical universe. But it would be over a 1,000 years before this little place holder would become a number in its own right.
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Having established such a sophisticated system of numbers, they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.
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Babylonian engineers and surveyors found ingenious ways of accessing water, and channelling it to the crop fields. Yet again, they used mathematics to come up with solutions. The Orontes valley in Syria is still an agricultural hub, and the old methods of irrigation are being exploited today, just as they were thousands of years ago. Many of the problems in Babylonian mathematics are concerned with measuring land, and it's here we see for the first time
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the use of quadratic equations, one of the greatest legacies of Babylonian mathematics. Quadratic equations involve things where the unknown quantity you're trying to identify is multiplied by itself. We call this squaring because it gives the area of a square, and it's in the context of calculating the area of land that thesequadratic equations naturally arise.
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Here's a typical problem. If a field has an area of 55 units and one side is six units longer than the other, how long is the shorter side?
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The Babylonian solution was to reconfigure the field as a square. Cut three units off the end and move this round. Now, there's a three-by-three piece missing, so let's add this in. The area of the field has increased by nine units. This makes the new area 64. So the sides of the square are eight units. The problem-solver knows that they've added three to this side. So, the original length must be five.
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It may not look like it, but this is one of the first quadratic equations in history.
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In modern mathematics, I would use the symbolic language of algebra to solve this problem. The amazing feat of the Babylonians is that they were using these geometric games to find the value, without any recourse to symbols or formulas. The Babylonians were enjoying problem-solving for its own sake. They were falling in love with mathematics.
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The Babylonians' fascination with numbers soon found a place in their leisure time, too. They were avid game-players. The Babylonians and their descendants have been playing a version of backgammon for over 5,000 years. The Babylonians played board games, from very posh board games in royal tombs to little bits of board games found in schools, to board games scratched on the entrances of palaces, so that the guardsman must have played when they were bored,
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and they used dice to move their counters round. People who played games were using numbers in their leisure time to try and outwit their opponent, doing mental arithmetic very fast, and so they were calculating in their leisure time, without even thinking about it as being mathematical hard work.
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Now's my chance. 'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.' It's up to you.Six... I need to move something. 'But it wasn't as easy as I thought.' Ah! What the hell was that? Yeah.This is one, this is two. Now you're in trouble. So I can't move anything. You cannot move these. Oh gosh.
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There you go.
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Three and four. 'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.' Yeah.
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Put it there. Good game.
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The Babylonians are recognised as one of the first cultures to use symmetrical mathematical shapes to make dice, but there is more heated debates about whether they might also have been the first to discover the secrets of another important shape. The right-angled triangle.
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We've already seen how the Egyptians use a 3-4-5 right-angled triangle. But what the Babylonians knew about this shape and others like it is much more sophisticated. This is the most famous and controversial ancient tablet we have. It's called Plimpton 322.
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Many mathematicians are convinced it shows the Babylonians could well have known the principle regarding right-angled triangles, that the square on the diagonal is the sum of the squares on the sides, and known it centuries before the Greeks claimed it.
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This is a copy of arguably the most famous Babylonian tablet, which is Plimpton 322, and these numbers here reflect the width or height of a triangle, this being the diagonal, the other side would be over here, and the square of this column plus the square of the number in this column equals the square of the diagonal. They are arranged in an order of steadily decreasing angle, on a very uniform basis, showing that somebody had a lot of understanding of how the numbers fit together.
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Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths. It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem, and it's a conclusion that generations of historians have been seduced by. But there could be a much simpler explanation for the sets of three numbers which fulfil Pythagoras' theorem. It's not a systematic explanation of Pythagorean triples, it's simply
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a mathematics teacher doing some quite complicated calculations, but in order to produce some very simple numbers, in order to set his students problems about right-angled triangles, and in that sense it's about Pythagorean triples only incidentally.
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The most valuable clues to what they understood could lie elsewhere. This small school exercise tablet is nearly 4,000 years old and reveals just what the Babylonians did know about right-angled triangles. It uses a principle of Pythagoras' theorem to find the value of an astounding new number.
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Drawn along the diagonal is a really very good approximation to the square root of two, and so that shows us that it was known and used in school environments. Why's this important? Because the square root of two is what we now call an irrational number, that is, if we write it out in decimals, or even in sexigesimal places, it doesn't end, the numbers go on forever after the decimal point. The implications of this calculation are far-reaching.
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Firstly, it means the Babylonians knew something of Pythagoras' theorem 1,000 years before Pythagoras. Secondly, the fact that they can calculate this number to an accuracy of four decimal places shows an amazing arithmetic facility, as well as a passion for mathematical detail.
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The Babylonians' mathematical dexterity was astounding, and for nearly 2,000 years they spearheaded intellectual progress in the ancient world. But when their imperial power began to wane, so did their intellectual vigour.
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By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.
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This is Palmyra in central Syria, a once-great city built by the Greeks.
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The mathematical expertise needed to build structures with such geometric perfection is impressive.
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Just like the Babylonians before them, the Greeks were passionate about mathematics.
35:59
The Greeks were clever colonists. They took the best from the civilisations they invaded to advance their own power and influence, but they were soon making contributions themselves. In my opinion, their greatest innovation was to do with a shift in the mind. What they initiated would influence humanity for centuries. They gave us the power of proof. Somehow they decided that they had to have a deductive system for their mathematics
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and the typical deductive system was to begin with certain axioms, which you assume are true. It's as if you assume a certain theorem is true without proving it. And then, using logical methods and very careful steps, from these axioms you prove theorems and from those theorems you prove more theorems, and it just snowballs.
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Proof is what gives mathematics its strength. It's the power or proof which means that the discoveries of the Greeks are as true today as they were 2,000 years ago. I needed to head west into the heart of the old Greek empire to learn more.
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For me, Greek mathematics has always been heroic and romantic.
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I'm on my way to Samos, less than a mile from the Turkish coast. This place has become synonymous with the birth of Greek mathematics, and it's down to the legend of one man.
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His name is Pythagoras. The legends that surround his life and work have contributed to the celebrity status he has gained over the last 2,000 years. He's credited, rightly or wrongly, with beginning the transformation from mathematics as a tool for accounting to the analytic subject we recognise today.
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Pythagoras is a controversial figure. Because he left no mathematical writings, many have questioned whether he indeed solved any of the theorems attributed to him. He founded a school in Samos in the sixth century BC, but his teachings were considered suspect and the Pythagoreans a bizarre sect.
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There is good evidence that there were schools of Pythagoreans, and they may have looked more like sects than what we associate with philosophical schools, because they didn't just share knowledge, they also shared a way of life. There may have been communal living and they all seemed to have been involved in the politics of their cities. One feature that makes them unusual in the ancient world is that they included women.
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But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians - the properties of right-angled triangles. What's known as Pythagoras' theorem states that if you take any right-angled triangle, build squares on all the sides, then the area of the largest square is equal to the sum of the squares on the two smaller sides.
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It's at this point for me that mathematics is born and a gulf opens up between the other sciences, and the proof is as simple as it is devastating in its implications. Place four copies of the right-angled triangle on top of this surface. The square that you now see has sides equal to the hypotenuse of the triangle. By sliding these triangles around, we see how we can break the area of the large square up into the sum of two smaller squares,
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whose sides are given by the two short sides of the triangle. In other words, the square on the hypotenuse is equal to the sum of the squares on the other sides. Pythagoras' theorem.
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It illustrates one of the characteristic themes of Greek mathematics - the appeal to beautiful arguments in geometry rather than a reliance on number.
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Pythagoras may have fallen out of favour and many of the discoveries accredited to him have been contested recently, but there's one mathematical theory that I'm loath to take away from him. It's to do with music and the discoveryof the harmonic series.
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The story goes that, walking past a blacksmith's one day, Pythagoras heard anvils being struck, and noticed how the notes being produced sounded in perfect harmony. He believed that there must be some rational explanation to make sense of why the notes sounded so appealing. The answer was mathematics.
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Experimenting with a stringed instrument, Pythagoras discovered that the intervals between harmonious musical notes were always represented as whole-number ratios.
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And here's how he might have constructed his theory.
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First, play a note on the open string. MAN PLAYS NOTE Next, take half the length.
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The note almost sounds the same as the first note. In fact it's an octave higher, but the relationship is so strong, we give these notes the same name. Now take a third the length.
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We get another note which sounds harmonious next to the first two, but take a length of string which is not in a whole-number ratio and all we get is dissonance.
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According to legend, Pythagoras was so excited by this discovery that he concluded the whole universe was built from numbers. But he and his followers were in for a rather unsettling challenge to their world view and it came about as a result of the theorem which bears Pythagoras' name.
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Legend has it, one of his followers, a mathematician called Hippasus, set out to find the length of the diagonal for a right-angled triangle with two sides measuring one unit. Pythagoras' theorem implied that the length of the diagonal was a number whose square was two. The Pythagoreans assumed that the answer would be a fraction, but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it. Eventually he realised his mistake.
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It was the assumption that the value was a fraction at all which was wrong. The value of the square root of two was the number that the Babylonians etched into the Yale tablet. However, they didn't recognise the special character of this number. But Hippasus did. It was an irrational number.
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The discovery of this new number, and others like it, is akin to an explorer
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Indeed, the importance Plato attached to geometry is encapsulated in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here."
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Plato proposed that the universe could be crystallised into five regular symmetrical shapes. These shapes, which we now call the Platonic solids, were composed of regular polygons, assembled to create three-dimensional symmetrical objects. The tetrahedron represented fire. The icosahedron, made from 20 triangles, represented water. The stable cube was Earth. The eight-faced octahedron was air. And the fifth Platonic solid, the dodecahedron, made out of 12 pentagons, was reserved for the shape
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that captured Plato's view of the universe.
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Plato's theory would have a seismic influence and continued to inspire mathematicians and astronomers for over 1,500 years. In addition to the breakthroughs made in the Academy, mathematical triumphs were also emerging from the edge of the Greek empire, and owed as much to the mathematical heritage of the Egyptians as the Greeks. Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,
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and its famous library soon gained a reputation to rival Plato's academy. The kings of Alexandria were prepared to invest in the arts and culture, in technology, mathematics, grammar, because patronage for cultural pursuits was one way of showing that you were a more prestigious ruler, and had a better entitlement to greatness.
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The old library and its precious contents were destroyed when the Muslims conquered Egypt in the 7th Century. But its spirit is alive in a new building.
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Today, the library remains a place of discovery and scholarship.
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Mathematicians and philosophers flocked to Alexandria, driven by their thirst for knowledge and the pursuit of excellence. The patrons of the library were the first professional scientists, individuals who were paid for their devotion to research. But of all those early pioneers, my hero is the enigmatic Greek mathematician Euclid.
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We know very little about Euclid's life, but his greatest achievements were as a chronicler of mathematics. Around 300 BC, he wrote the most important text book of all time - The Elements. In The Elements, we find the culmination of the mathematical revolution which had taken place in Greece.
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It's built on a series of mathematical assumptions, called axioms. For example, a line can be drawn between any two points. From these axioms, logical deductions are made and mathematical theorems established.
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The Elements contains formulas for calculating the volumes of cones and cylinders, proofs about geometric series, perfect numbers and primes. The climax of The Elements is a proof that there are only five Platonic solids.
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For me, this last theorem captures the power of mathematics. It's one thing to build five symmetrical solids, quite another to come up with a watertight, logical argument for why there can't be a sixth. The Elements unfolds like a wonderful, logical mystery novel. But this is a story which transcends time. Scientific theories get knocked down, from one generation to the next, but the theorems in The Elements are as true today as they were 2,000 years ago.
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When you stop and think about it, it's really amazing. It's the same theorems that we teach. We may teach them in a slightly different way, we may organise them differently, but it's Euclidean geometry that is still valid, and even in higher mathematics, when you go to higher dimensional spaces, you're still using Euclidean geometry.
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Alexandria must have been an inspiring place for the ancient scholars, and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port. One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes. He would become a mathematical visionary. The best Greek mathematicians, they were always pushing the limits, pushing the envelope. So Archimedes... did what he could with polygons, with solids.
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He then moved on to centres of gravity. He then moved on to the spiral. This instinct to try and mathematise everything is something that I see as a legacy.
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One of Archimedes' specialities was weapons of mass destruction. They were used against the Romans when they invaded his home of Syracuse in 212 BC. He also designed mirrors, which harnessed the power of the sun, to set the Roman ships on fire. But to Archimedes, these endeavours were mere amusements in geometry. He had loftier ambitions.
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Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake and not for the ignoble trade of engineering or the sordid quest for profit. One of his finest investigations into pure mathematics was to produce formulas to calculate the areas of regular shapes.
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Archimedes' method was to capture new shapes by using shapes he already understood. So, for example, to calculate the area of a circle, he would enclose it inside a triangle, and then by doubling the number of sides on the triangle, the enclosing shape would get closer and closer to the circle. Indeed, we sometimes call a circle a polygon with an infinite number of sides. But by estimating the area of the circle, Archimedes is, in fact,
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getting a value for pi, the most important number in mathematics.
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However, it was calculating the volumes of solid objects where Archimedes excelled. He found a way to calculate the volume of a sphere by slicing it up and approximating each slice as a cylinder. He then added up the volumes of the slices to get an approximate value for the sphere. But his act of genius was to see what happens if you make the slices thinner and thinner. In the limit, the approximation becomes an exact calculation.
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But it was Archimedes' commitment to mathematics that would be his undoing.
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Archimedes was contemplating a problem about circles traced in the sand. When a Roman soldier accosted him, Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem. But the Roman soldier was not interested in Archimedes' problem and killed him on the spot. Even in death, Archimedes' devotion to mathematics was unwavering.
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By the middle of the 1st Century BC, the Romans had tightened their grip on the old Greek empire. They were less smitten with the beauty of mathematics and were more concerned with its practical applications. This pragmatic attitude signalled the beginning of the end for the great library of Alexandria. But one mathematician was determined to keep the legacy of the Greeks alive. Hypatia was exceptional, a female mathematician, and a pagan in the piously Christian Roman empire.
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Hypatia was very prestigious and very influential in her time. She was a teacher with a lot of students, a lot of followers. She was politically influential in Alexandria. So it's this combination of... high knowledge and high prestige that may have made her a figure of hatred for... the Christian mob.
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One morning during Lent, Hypatia was dragged off her chariot by a zealous Christian mob and taken to a church. There, she was tortured and brutally murdered.
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The dramatic circumstances of her life and death fascinated later generations. Sadly, her cult status eclipsed her mathematical achievements. She was, in fact, a brilliant teacher and theorist, and her death dealt a final blow to the Greek mathematical heritage of Alexandria.
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My travels have taken me on a fascinating journey to uncover the passion and innovation of the world's earliest mathematicians. It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece that are the foundations on which my subject is built today. But this is just the beginning of my mathematical odyssey. The next leg of my journey lies east, in the depths of Asia, where mathematicians scaled even greater heights in pursuit of knowledge.
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With this new era came a new language of algebra and numbers, better suited to telling the next chapter in the story of maths.
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You can learn more about the story of maths with the Open University at...
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This is what we call a decimal place-value system, and it's very similar to the one we use today. We too use numbers from one to nine, and we use their position to indicate whether it's units, tens, hundreds or thousands. But the power of these rods is that it makes calculations very quick. In fact, the way the ancient Chinese did their calculations is very similar to the way we learn today in school.
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Not only were the ancient Chinese the first to use a decimal place-value system, but they did so over 1,000 years before we adopted it in the West. But they only used it when calculating with the rods. When writing the numbers down, the ancient Chinese didn't use the place-value system.
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Instead, they used a far more laborious method, in which special symbols stood for tens, hundreds, thousands and so on.
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So the number 924 would be written out as nine hundreds, two tens and four. Not quite so efficient.
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The problem was that the ancient Chinese didn't have a concept of zero. They didn't have a symbol for zero. It just didn't exist as a number. Using the counting rods, they would use a blank space where today we would write a zero. The problem came with trying to write down this number, which is why they had to create these new symbols for tens, hundreds and thousands. Without a zero, the written number was extremely limited.
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But the absence of zero didn't stop the ancient Chinese from making giant mathematical steps. In fact, there was a widespread fascination with number in ancient China. According to legend, the first sovereign of China, the Yellow Emperor, had one of his deities create mathematics in 2800BC, believing that number held cosmic significance. And to this day, the Chinese still believe in the mystical power of numbers.
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Odd numbers are seen as male, even numbers, female. The number four is to be avoided at all costs. The number eight brings good fortune.
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And the ancient Chinese were drawn to patterns in numbers, 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 by a sacred turtle that came out of the depths of the Yellow River. On its back were numbers arranged into a magic square, a little like this.
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In this square, which was regarded as having great religious significance, all the numbers in each line - horizontal, vertical and diagonal - all add up to the same number - 15.
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Now, the magic square may be no more than a fun puzzle, but it shows the ancient Chinese fascination with mathematical patterns, and it wasn't too long before they were creating even bigger magic squares with even greater magical and mathematical powers.
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But mathematics also played a vital role in the running of the emperor's court. The calendar and the movement of the planets were of the utmost importance to the emperor, influencing all his decisions, even down to the way his day was planned, so astronomers became prized members of the imperial court, and astronomers were always mathematicians.
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Everything in the emperor's life was governed by the calendar, and he ran his affairs with mathematical precision. The emperor even got his mathematical advisors to come up with a system to help him sleep his way through the vast number of women he had in his harem. Never one to miss a trick, the mathematical advisors decided to base the harem on a mathematical idea called a geometric progression. Maths has never had such a fun purpose! 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, three senior consorts, nine wives, 27 concubines and 81 slaves.
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The mathematicians would soon have realised that this was a geometric progression - a series of numbers in which you get from one number to the next 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, so the mathematicians could quickly draw up a rota to ensure that, in the space of 15 nights, the emperor slept with every woman in the harem.
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The first night was reserved for the empress. The next was for the three senior consorts. The nine wives came next, and then the 27 concubines were chosen in rotation, nine each night. And then finally, over a period of nine nights, the 81 slaves were dealt with in groups of nine.
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Being the emperor certainly required stamina, a bit like being a mathematician, but the object is clear - to procure the best possible imperial succession. The rota ensured that the emperor slept with the ladies of highest rank closest to the full moon, when their yin, their female force, 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. It was central to the running of the state. Ancient China was a vast and growing empire with a strict legal code, widespread taxation and a standardised system of weights, measures and money.
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The empire needed a highly trained civil service, competent in mathematics.
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And to educate these civil servants was a mathematical textbook, probably written in around 200BC - the Nine Chapters.
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The book is a compilation of 246 problems in practical areas such as trade, payment of wages and taxes.
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And at the heart of these problems lies one of the central themes of mathematics, how to solve equations.
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Equations are a little bit like cryptic crosswords. You're given a certain amount of information about some unknown numbers, and from that information you've got to deduce what the unknown numbers are. For example, with my weights and scales, I've found out that one plum...
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..together with three peaches weighs a total of 15 grams.
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But...
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..two plums together with one peach weighs a total of 10g. From this information, I can deduce what a single plum weighs and a single peach weighs, and this is how I do it.
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If I take the first set of scales, one plum and three peaches weighing 15g, 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 - that's two plums and a peach weighing 10g - I'm left with an interesting result - no plums. Having eliminated the plums, I've discovered that five peaches weighs 20g, so a single peach weighs 4g, and from this I can deduce that the plum weighs 3g. The ancient Chinese went on to apply similar methods to larger and larger numbers of unknowns, using it to solve increasingly complicated equations. What's extraordinary is
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that this particular system of solving equations didn't appear in the West until the beginning of the 19th century. In 1809, while analysing a rock called Pallas in the asteroid belt, Carl Friedrich Gauss, who would become known as the prince of mathematics, rediscovered this method which had been formulated in ancient China centuries earlier. Once again, ancient China streets ahead of Europe.
1:03:58
But the Chinese were to go on to solve even more complicated equations involving far larger numbers. In what's become known as the Chinese remainder theorem, the Chinese came up with a new kind of problem. In this, we know the number that's left when the equation's unknown number is divided by a given number - say, three, five or seven.
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Of course, this is a fairly abstract mathematical problem, but the ancient Chinese still couched it in practical terms.
1:04:33
So a woman in the market has a tray of eggs, but she doesn't know how many eggs she's got. What she does know is that if she arranges them in threes, she has one egg left over. If she arranges them in fives, she gets two eggs left over. But if she arranged them in rows of seven, she found she had three eggs left over. The ancient Chinese found a systematic way to calculate 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 such a large number, like 52, by using these small numbers like three, five and seven. This way of looking at numbers 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 in ancient Chinese astronomy to measure planetary movement. But today it still has practical uses. Internet cryptography encodes numbers using mathematics that has its origins in the Chinese remainder theorem.
1:05:54
By the 13th century, mathematics was long established on the curriculum, with over 30 mathematics schools scattered across the country. 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. He was a fantastically corrupt imperial administrator who crisscrossed China, lurching from one post to another. Repeatedly sacked for embezzling government money, he poisoned anyone who got in his way.
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Qin Jiushao was reputedly described as as violent as a tiger or a wolf and as poisonous as a scorpion or a viper so, not surprisingly, he made a fierce warrior. For ten years, he fought against the invading Mongols, but for much of that time he was complaining that his military life took him away from his true passion. No, not corruption, but mathematics.
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Qin started trying to solve equations that grew out of trying to measure the world around us. Quadratic equations involve numbers that are squared, or to the power of two - say, five times five. The ancient Mesopotamians had already realised that these equations were perfect for measuring flat, two-dimensional shapes, like Tiananmen Square.
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But Qin was interested in more complicated equations - cubic equations.
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These involve numbers which are cubed, or to the power of three - say, five times five times five, and they were perfect for capturing three-dimensional shapes, like Chairman Mao's mausoleum.
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Qin found a way of solving cubic equations, and this is how it worked.
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Say Qin wants to know the exact dimensions of Chairman Mao's mausoleum.
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He knows the volume of the building and the relationships between the dimensions.
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In order to get his answer, Qin uses what he knows to produce a cubic equation. He then makes an educated guess at the dimensions. Although he's captured a good proportion of the mausoleum, there are still bits left over.
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Qin takes these bits and creates a new cubic equation. He can now refine his first guess by trying to find a solution to this new cubic equation, and so on.
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Each time he does this, the pieces he's left with get smaller and smaller and his guesses get better and better.
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What's striking is that Qin's method for solving equations wasn't discovered in the West until the 17th century, when Isaac Newton came up with a very similar approximation method. The power of this technique is that it can be applied to even more complicated equations. Qin even used his techniques to solve an equation involving numbers up to the power of ten. This was extraordinary stuff - highly complex mathematics.
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Qin may have been years ahead of his time, but there was a problem with his technique. It only gave him an approximate solution. That might be good enough for an engineer - not for a mathematician. Mathematics is an exact science. We like things to be precise, and Qin just couldn't come up with a formula to give him an exact solution to these complicated equations.
1:10:04
China had made great mathematical leaps, but the next great mathematical breakthroughs were to happen in a country lying to the southwest of China - a country that had a rich mathematical tradition that would change the face of maths for ever.
1:10:50
India's first great mathematical gift lay in the world of number. Like the Chinese, the Indians had discovered the mathematical benefits of the decimal place-value system and were using it by the middle of the 3rd century AD.
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It's been suggested that the Indians learned the system from Chinese merchants travelling in India with their counting rods, or they may well just have stumbled across it themselves. It's all such a long time ago that it's shrouded in mystery.
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We may never know how the Indians came up with their number system, but we do know that they refined and perfected it, creating the ancestors for the nine numerals used across the world now. Many rank the Indian system of counting as one of the greatest intellectual innovations of all time, developing into the closest thing we could call a universal language.
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But there was one number missing, and it was the Indians who would introduce it to the world.
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The earliest known recording of this number dates from the 9th century, though it was probably in practical use for centuries before.
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This strange new numeral is engraved on the wall of small temple in the fort of Gwalior in central India.
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So here we are in one of the holy sites of the mathematical world, and what I'm looking for is in this inscription on the wall. Up here are some numbers, and... here's the new number. It's zero.
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It's astonishing to think that before the Indians invented it, there was no number zero. To the ancient Greeks, it simply hadn't existed. To the Egyptians, the Mesopotamians and, as we've seen, the Chinese, zero had been in use but as a placeholder, an empty space to show a zero inside a number.
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The Indians transformed zero from a mere placeholder into a number that made sense in its own right - a number for calculation, for investigation. This brilliant conceptual leap would revolutionise mathematics.
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Now, with just ten digits - zero to nine - it was suddenly possible to capture astronomically large numbers in an incredibly efficient way.
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But why did the Indians make this imaginative leap? Well, we'll never know for sure, but it's possible that the idea and symbol that the Indians use for zero came from calculations they did with stones in the sand. When stones were removed from the calculation, a small, round hole was left in its place, representing the movement from something to nothing.
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But perhaps there is also a cultural reason for the invention of zero. HORNS BLOW AND DRUMS BANG METALLIC BEATING
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For the ancient Indians, the concepts of nothingness and eternity lay at the very heart of their belief system.
1:14:42
BELL CLANGS AND SILENCE FALLS
1:14:46
In the religions of India, the universe was born from nothingness, and nothingness is the ultimate goal of humanity. So it's perhaps not surprising that a culture that so enthusiastically embraced the void should be happy with the notion of zero. The Indians even used the word for the philosophical idea of the void, shunya, to represent the new mathematical term "zero".
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In the 7th century, the brilliant Indian mathematician Brahmagupta proved some of the essential properties of zero.
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Brahmagupta's rules about calculating with zero are taught in schools all over the world to this day. One plus zero equals one.
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One minus zero equals one. One times zero is equal to zero.
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But Brahmagupta came a cropper when he tried to do one divided by zero. After all, what number times zero equals one? It would require a new mathematical concept, that of infinity, to make sense of dividing by zero, and the breakthrough was made by a 12th-century Indian mathematician called Bhaskara II, and it works like this. If I take a fruit and I divide it into halves, I get two pieces, so one divided by a half is two. If I divide it into thirds, I get three pieces.
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So when I divide it into smaller and smaller fractions, I get more and more pieces, so ultimately, when I divide by a piece which is of zero size, I'll have infinitely many pieces. So for Bhaskara, one divided by zero is infinity.
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But the Indians would go further in their calculations with zero.
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For example, if you take three from three and get zero, what happens when you take four from three? It looks like you have nothing, but the Indians recognised that this was a new sort of nothing - negative numbers. The Indians called them "debts", because they solved equations like, "If I have three batches of material and take four away, "how many have I left?"
1:17:33
This may seem odd and impractical, but that was the beauty of Indian mathematics. Their ability to come up with negative numbers and zero was because they thought of numbers as abstract entities. They weren't just for counting and measuring pieces of cloth. They had a life of their own, floating free of the real world. This led to an explosion of mathematical ideas.
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The Indians' abstract approach to mathematics soon revealed a new side to the problem of how to solve quadratic equations. That is equations including numbers to the power of two.
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Brahmagupta's understanding of negative numbers allowed him to see that quadratic equations always have two solutions, one of which could be negative.
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Brahmagupta went even further, solving quadratic equations with two unknowns, a question which wouldn't be considered in the West until 1657, when French mathematician Fermat challenged his colleagues with the same problem. Little did he know that they'd been beaten to a solution by Brahmagupta 1,000 years earlier.
1:18:57
Brahmagupta was beginning to find abstract ways of solving equations, but astonishingly, he was also developing a new mathematical language to express that abstraction.
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Brahmagupta was experimenting with ways of writing his equations down, using the initials of the names of different colours to represent unknowns in his equations.
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A new mathematical language was coming to life, which would ultimately lead to the x's and y's which fill today's mathematical journals.
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But it wasn't just new notation that was being developed.
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Indian mathematicians were responsible for making fundamental new discoveries in the theory of trigonometry.
1:19:59
The power of trigonometry is that it acts like a dictionary, translating geometry into numbers and back. Although first developed by the ancient Greeks, it was in the hands of the Indian mathematicians that the subject truly flourished. At its heart lies the study of right-angled triangles.
1:20:21
In trigonometry, you can use this angle here to find the ratios of the opposite side to the longest side. There's a function called the sine function which, when you input the angle, outputs the ratio. So for example in this triangle, the angle is about 30 degrees, so the output of the sine function is a ratio of one to two, telling me that this side is half the length of the longest side.
1:20:49
The sine function enables you to calculate distances when you're not able to make an accurate measurement. To this day, it's used in architecture and engineering. The Indians used it to survey the land around them, navigate the seas and, ultimately, chart the depths of space itself.
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It was central to the work of observatories, like this one in Delhi, where astronomers would study the stars. The Indian astronomers could use trigonometry to work out the relative distance between Earth and the moon and Earth and the sun. You can only make the calculation when the moon is half full, because that's when it's directly opposite the sun, so that the sun, moon and Earth create a right-angled triangle.
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Now, the Indians could measure that the angle between the sun and the observatory was one-seventh of a degree.
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The sine function of one-seventh of a degree gives me the ratio of 400:1. This means the sun is 400 times further from Earth than the moon is. So using trigonometry, the Indian mathematicians could explore the solar system without ever having to leave the surface of the Earth.
1:22:16
The ancient Greeks had been the first to explore the sine function, listing precise values for some angles, but they couldn't calculate the sines of every angle. The Indians were to go much further, setting themselves a mammoth task. The search was on to find a way to calculate the sine function of any angle you might be given.
1:22:55
The breakthrough in the search for the sine function of every angle would be made here in Kerala in south India. In the 15th century, this part of the country became home to one of the most brilliant schools of mathematicians to have ever worked.
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Their leader was called Madhava, and he was to make some extraordinary mathematical discoveries.
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The key to Madhava's success was the concept of the infinite. Madhava discovered that you could add up infinitely many things with dramatic effects. Previous cultures had been nervous of these infinite sums, but Madhava was happy to play with them. For example, here's how one can be made up by adding infinitely many fractions.
1:23:44
By successively adding and subtracting different fractions, Madhava could hone in on an exact formula for pi.
1:23:53
First, he moved four steps up the number line. That took him way past pi.
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So next he took four-thirds of a step, or one-and-one-third steps, back. Now he'd come too far the other way.
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So he headed forward four-fifths of a step. Each time, he alternated between four divided by the next odd number.
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He zigzagged up and down the number line, getting closer and closer to pi. He discovered that if you went through all the odd numbers, infinitely many of them, you would hit pi exactly.
1:24:43
I was taught at university that this formula for pi was discovered by the 17th-century German mathematician Leibniz, but amazingly, it was actually discovered here in Kerala two centuries earlier by Madhava. He went on to use the same sort of mathematics to get infinite-series expressions for the sine formula in trigonometry. And the wonderful thing is that you can use these formulas now to calculate the sine of any angle to any degree of accuracy.
1:25:20
It seems incredible that the Indians made these discoveries centuries before Western mathematicians.
1:25:29
And it says a lot about our attitude in the West to non-Western cultures that we nearly always claim their discoveries as our own. What is clear is the West has been very slow to give due credit to the major breakthroughs made in non-Western mathematics. Madhava wasn't the only mathematician to suffer this way. As the West came into contact more and more with the East during the 18th and 19th centuries, there was a widespread dismissal and denigration of the cultures they were colonising.
1:25:58
The natives, it was assumed, couldn't have anything of intellectual worth to offer the West. It's only now, at the beginning of the 21st century, that history is being rewritten. But Eastern mathematics was to have a major impact in Europe, thanks to the development of one of the major powers of the medieval world.
1:26:41
In the 7th century, a new empire began to spread across the Middle East. The teachings of the Prophet Mohammed inspired a vast and powerful Islamic empire which soon stretched from India in the east to here in Morocco in the west.
1:27:05
And at the heart of this empire lay a vibrant intellectual culture.
1:27:14
A great library and centre of learning was established in Baghdad. Called the House of Wisdom, its teaching spread throughout the Islamic empire, reaching schools like this one here in Fez. Subjects studied included astronomy, medicine, chemistry zoology and mathematics.
1:27:37
The Muslim scholars collected and translated many ancient texts, effectively saving them for posterity. In fact, without their intervention, we may never have known about the ancient cultures of Egypt, Babylon, Greece and India. But the scholars at the House of Wisdom weren't content simply with translating other people's mathematics. They wanted to create a mathematics of their own, to push the subject forward.
1:28:05
Such intellectual curiosity was actively encouraged in the early centuries of the Islamic empire.
1:28:15
The Koran asserted the importance of knowledge. Learning was nothing less than a requirement of God.
1:28:24
In fact, the needs of Islam demanded mathematical skill. The devout needed to calculate the time of prayer and the direction of Mecca to pray towards, and the prohibition of depicting the human form meant that they had to use much more geometric patterns to cover their buildings. The Muslim artists discovered all the different sorts of symmetry that you can depict on a two-dimensional wall.
1:28:57
The director of the House of Wisdom in Baghdad was a Persian scholar called Muhammad Al-Khwarizmi.
1:29:07
Al-Khwarizmi was an exceptional mathematician who was responsible for introducing two key mathematical concepts to the West. Al-Khwarizmi recognised the incredible potential that the Hindu numerals had to revolutionise mathematics and science. His work explaining the power of these numbers to speed up calculations and do things effectively was so influential that it wasn't long before they were adopted as the numbers of choice amongst the mathematicians of the Islamic world.
1:29:36
In fact, these numbers have now become known as the Hindu-Arabic numerals. These numbers - one to nine and zero - are the ones we use today all over the world.
1:29:53
But Al-Khwarizmi was to create a whole new mathematical language.
1:29:59
It was called algebra and was named after the title of his book Al-jabr W'al-muqabala, or Calculation By Restoration Or Reduction.
1:30:14
Algebra is the grammar that underlies the way that numbers work. It's a language that explains the patterns that lie behind the behaviour of numbers. It's a bit like a code for running a computer program. The code will work whatever the numbers you feed in to the program.
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For example, mathematicians might have discovered that if you take a number and square it, that's always one more than if you'd taken the numbers either side and multiplied those together. For example, five times five is 25, which is one more than four times six - 24. Six times six is always one more than five times seven and so on. But how can you be sure that this is going to work whatever numbers you take?
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To explain the pattern underlying these calculations, let's use the dyeing holes in this tannery.
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If we take a square of 25 holes, running five by five, and take one row of five away and add it to the bottom, we get six by four with one left over.
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But however many holes there are on the side of the square, we can always move one row of holes down in a similar way to be left with a rectangle of holes with one left over.
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Algebra was a huge breakthrough. Here was a new language to be able to analyse the way that numbers worked. Previously, the Indians and the Chinese had considered very specific problems, but Al-Khwarizmi went from the specific to the general. He developed systematic ways to be able to analyse problems so that the solutions would work whatever the numbers that you took. This language is used across the mathematical world today.
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Al-Khwarizmi's great breakthrough came when he applied algebra to quadratic equations - that is equations including numbers to the power of two. The ancient Mesopotamians had devised a cunning method to solve particular quadratic equations, but it was Al-Khwarizmi's abstract language of algebra that could finally express why this method always worked.
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This was a great conceptual leap and would ultimately lead to a formula that could be used to solve any quadratic equation, whatever the numbers involved.
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The next mathematical Holy Grail was to find a general method that could solve all cubic equations - equations including numbers to the power of three.
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It was an 11th-century Persian mathematician who took up the challenge of cracking the problem of the cubic.
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His name was Omar Khayyam, and he travelled widely across the Middle East, calculating as he went.
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But he was famous for another, very different, reason. Khayyam was a celebrated poet, author of the great epic poem the Rubaiyat.
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It may seem a bit odd that a poet was also a master mathematician. After all, the combination doesn't immediately spring to mind. But there's quite a lot of similarity between the disciplines. Poetry, with its rhyming structure and rhythmic patterns, resonates strongly with constructing a logical mathematical proof.
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Khayyam's major mathematical work was devoted to finding the general method to solve all cubic equations. Rather than looking at particular examples, Khayyam carried out a systematic analysis of the problem, true to the algebraic spirit of Al-Khwarizmi.
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Khayyam's analysis revealed for the first time that there were several different sorts of cubic equation. But he was still very influenced by the geometric heritage of the Greeks. He couldn't separate the algebra from the geometry. In fact, he wouldn't even consider equations in higher degrees, because they described objects in more than three dimensions, something he saw as impossible. Although the geometry allowed him to analyse these cubic equations to some extent,
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he still couldn't come up with a purely algebraic solution.
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It would be another 500 years before mathematicians could make the leap and find a general solution to the cubic equation.
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And that leap would finally be made in the West - in Italy.
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During the centuries in which China, India and the Islamic empire had been in the ascendant, Europe had fallen under the shadow of the Dark Ages.
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All intellectual life, including the study of mathematics, had stagnated.
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But by the 13th century, things were beginning to change. Led by Italy, Europe was starting to explore and trade with the East. With that contact came the spread of Eastern knowledge to the West. It was the son of a customs official that would become Europe's first great medieval mathematician. As a child, he travelled around North Africa with his father, where he learnt about the developments of Arabic mathematics
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and especially the benefits of the Hindu-Arabic numerals. When he got home to Italy he wrote a book that would be hugely influential in the development of Western mathematics.
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That mathematician was Leonardo of Pisa, better known as Fibonacci, and in his Book Of Calculating, Fibonacci promoted the new number system, demonstrating how simple it was compared to the Roman numerals that were in use across Europe. Calculations were far easier, a fact that had huge consequences for anyone dealing with numbers - pretty much everyone, from mathematicians to merchants. But there was widespread suspicion of these new numbers.
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Old habits die hard, and the authorities just didn't trust them. Some believed that they would be more open to fraud - that you could tamper with them. Others believed that they'd be so easy to use for calculations that it would empower the masses, taking authority away from the intelligentsia who knew how to use the old sort of numbers.
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The city of Florence even banned them in 1299, but over time, common sense prevailed, the new system spread throughout Europe, and the old Roman system slowly became defunct. At last, the Hindu-Arabic numerals, zero to nine, had triumphed.
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Today Fibonacci is best known for the discovery of some numbers, now called the Fibonacci sequence, that arose when he was trying to solve a riddle about the mating habits of rabbits. Suppose a farmer has a pair of rabbits. Rabbits take two months to reach maturity, and after that they give birth to another pair of rabbits each month. So the problem was how to determine how many pairs of rabbits there will be in any given month.
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Well, during the first month you have one pair of rabbits, and since they haven't matured, they can't reproduce. During the second month, there is still only one pair. But at the beginning of the third month, the first pair reproduces for the first time, so there are two pairs of rabbits. At the beginning of the fourth month, the first pair reproduces again, but the second pair is not mature enough, so there are three pairs.
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In the fifth month, the first pair reproduces and the second pair reproduces for the first time, but the third pair is still too young, so there are five pairs. The mating ritual continues, but what you soon realise is the number of pairs of rabbits you have in any given month is the sum of the pairs of rabbits that you have had in each of the two previous months, so the sequence goes... 1...1...2...3... 5...8...13...
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21...34...55...and so on. The Fibonacci numbers are nature's favourite numbers. It's not just rabbits that use them. The number of petals on a flower is invariably a Fibonacci number. They run up and down pineapples if you count the segments. Even snails use them to grow their shells. Wherever you find growth in nature, you find the Fibonacci numbers.
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But the next major breakthrough in European mathematics wouldn't happen until the early 16th century. It would involve finding the general method that would solve all cubic equations, and it would happen here in the Italian city of Bologna.
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The University of Bologna was the crucible of European mathematical thought at the beginning of the 16th century.
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Pupils from all over Europe flocked here and developed a new form of spectator sport - the mathematical competition.
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Large audiences would gather to watch mathematicians challenge each other with numbers, a kind of intellectual fencing match. But even in this questioning atmosphere it was believed that some problems were just unsolvable. It was generally assumed that finding a general method to solve all cubic equations was impossible. But one scholar was to prove everyone wrong.
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His name was Tartaglia, but he certainly didn't look the heroic architect of a new mathematics. At the age of 12, he'd been slashed across the face with a sabre by a rampaging French army. The result was a terrible facial scar and a devastating speech impediment. In fact, Tartaglia was the nickname he'd been given as a child and means "the stammerer".
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Shunned by his schoolmates, Tartaglia lost himself in mathematics, and it wasn't long before he'd found the formula to solve one type of cubic equation. But Tartaglia soon discovered that he wasn't the only one to believe he'd cracked the cubic. A young Italian called Fior was boasting that he too held the secret formula for solving cubic equations. When news broke about the discoveries made by the two mathematicians, 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 only knew how to solve one sort of cubic equation, and Fior was ready to challenge him with questions about a different sort. But just a few days before the contest, Tartaglia worked out how to solve this different sort, and with this new weapon in his arsenal he thrashed his opponent, solving all the questions in under two hours.
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Tartaglia went on to find the formula to solve all types of cubic equations. News soon spread, and a mathematician in Milan called Cardano became so desperate to find the solution that he persuaded a reluctant Tartaglia to reveal the secret, but on one condition - that Cardano keep the secret and never publish.
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But Cardano couldn't resist discussing Tartaglia's solution with his brilliant student, Ferrari. As Ferrari got to grips with Tartaglia's work, he realised that he could use it to solve the more complicated quartic equation, an amazing achievement. Cardano couldn't deny his student his just rewards, and he broke his vow of secrecy, publishing Tartaglia's work together with Ferrari's brilliant solution of the quartic.
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Poor Tartaglia never recovered and died penniless, and to this day, the formula that solves the cubic equation is known as Cardano's formula.
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Tartaglia may not have won glory in his lifetime, but his mathematics managed to solve a problem that had bewildered the great mathematicians of China, India and the Arab world.
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It was the first great mathematical breakthrough to happen in modern Europe.
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The Europeans now had in their hands the new language of algebra, the powerful techniques of the Hindu-Arabic numerals and the beginnings of the mastery of the infinite. It was time for the Western world to start writing its own mathematical stories in the language of the East. The mathematical revolution was about to begin.
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