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Throughout history, humankind has struggled
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to understand the fundamental workings of the material world.
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We’ve endeavoured to discover the rules and patterns that determine the qualities
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of the objects that surround us, and their complex relationship to us and each other.
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Over thousands of years, societies all over the world have found that one discipline
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above all others yields certain knowledge
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about the underlying realities of the physical world.
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That discipline is mathematics.
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I’m Marcus Du Sautoy, and I’m a mathematician.
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I see myself as a pattern searcher, hunting down the hidden structures
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that lie behind the apparent chaos and complexity of the world around us.
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In my search for pattern and order, I draw upon the work of the great mathematicians
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who’ve gone before me, people belonging to cultures across the globe,
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whose innovations created the language the universe is written in.
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I want to take you on a journey through time and space, and track the growth of mathematics
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from its awakening to the sophisticated subject we know today.
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Using computer generated imagery, we will explore
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the trailblazing discoveries that allowed the earliest civilisations
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to understand the world mathematical.
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This is the story of maths.
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Our world is made up of patterns and sequences.
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They’re all around us.
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Day becomes night.
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Animals travel across the earth in ever-changing formations.
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Landscapes are constantly altering.
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One of the reasons mathematics began was because we needed to find a way
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of making sense of these natural patterns.
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The most basic concepts of maths - space and quantity -
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are hard-wired into our brains.
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Even animals have a sense of distance and number,
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assessing when their pack is outnumbered, and whether to fight or fly,
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calculating whether their prey is within striking distance.
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Understanding maths is the difference between life and death.
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But it was man who took these basic concepts
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and started to build upon these foundations.
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At some point, humans started to spot patterns,
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to make connections, to count and to order the world around them.
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With this, a whole new mathematical universe began to emerge.
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This is the River Nile.
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It’s been the lifeline of Egypt for millennia.
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I’ve come here because it’s where some of the first signs
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of mathematics as we know it today emerged.
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People abandoned nomadic life and began settling here as early as 6000BC.
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The conditions were perfect for farming.
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The most important event for Egyptian agriculture each year was the flooding of the Nile.
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So this was used as a marker to start each new year.
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Egyptians did record what was going on over periods of time,
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so in order to establish a calendar like this,
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you need to count how many days, for example,
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happened in-between lunar phases,
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or how many days happened in-between two floodings of the Nile.
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Recording the patterns for the seasons was essential,
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not only to their management of the land, but also their religion.
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The ancient Egyptians who settled on the Nile banks
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believed it was the river god, Hapy, who flooded the river each year.
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And in return for the life-giving water,
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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.
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Areas of land needed to be calculated, crop yields predicted,
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taxes charged and collated.
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In short, people needed to count and measure.
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The Egyptians used their bodies to measure the world,
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and it’s how their units of measurements evolved.
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A palm was the width of a hand,
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a cubit an arm length from elbow to fingertips.
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Land cubits, strips of land measuring a cubit by 100,
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were used by the pharaoh’s surveyors to calculate areas.
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There’s a very strong link between bureaucracy
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and the development of mathematics in ancient Egypt.
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And we can see this link right from the beginning,
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from the invention of the number system,
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throughout Egyptian history, really.
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For the Old Kingdom, the only evidence we have
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are metrological systems, that is measurements for areas, for length.
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This points to a bureaucratic need to develop such things.
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It was vital to know the area of a farmer’s land so he could be taxed accordingly.
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Or if the Nile robbed him of part of his land, so he could request a rebate.
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It meant that the pharaoh’s surveyors were often calculating
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the area of irregular parcels of land.
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It was the need to solve such practical problems
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that made them the earliest mathematical innovators.
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The Egyptians needed some way to record the results of their calculations.
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Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo,
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I was on the hunt for those that recorded some of the first numbers in history.
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They were difficult to track down.
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But I did find them in the end.
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The Egyptians were using a decimal system, motivated by the 10 fingers on our hands.
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The sign for one was a stroke,
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10, a heel bone, 100, a coil of rope, and 1,000, a Lotus plant.
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How much is this T-shirt?
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Er, 25.
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- 25!
- Yes!
- So that would be 2 knee bones and 5 strokes.
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- So you’re not gonna charge me anything up here?
- Here, one million!
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- One million?
- My God!
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This one million.
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One million, yeah, that’s pretty big!
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The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed.
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They had no concept of a place value,
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so one stroke could only represent one unit,
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not 100 or 1,000.
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Although you can write a million with just one character,
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rather than the seven that we use, if you want to write a million minus one,
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then the poor old Egyptian scribe has got to write nine strokes,
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nine heel bones, nine coils of rope, and so on,
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a total of 54 characters.
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Despite the drawback of this number system, the Egyptians were brilliant problem solvers.
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We know this because of the few records that have survived.
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The Egyptian scribes used sheets of papyrus
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to record their mathematical discoveries.
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This delicate material made from reeds decayed over time
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and many secrets perished with it.
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But there’s one revealing document that has survived.
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The Rhind Mathematical Papyrus is the most important document
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we have today for Egyptian mathematics.
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We get a good overview of what types of problems
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the Egyptians would have dealt with in their mathematics.
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We also get explicitly stated how multiplications and divisions were carried out.
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The papyri show how to multiply two large numbers together.
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But to illustrate the method, let’s take two smaller numbers.
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Let’s do three times six.
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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.
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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,
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and two becomes four.
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Now, here’s the really clever bit.
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The scribe wants to multiply three by six.
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So he takes the powers of two in the second column,
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which add up to six. That’s two plus four.
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Then he moves back to the first column, and just takes
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those rows corresponding to the two and the four.
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So that’s six and the 12.
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He adds those together to get the answer of 18.
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But for me, the most striking thing about this method
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is that the scribe has effectively written that second number in binary.
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Six is one lot of four, one lot of two, and no units.
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Which is 1-1-0.
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The Egyptians have understood the power of binary over 3,000 years
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before the mathematician and philosopher Leibniz would reveal their potential.
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Today, the whole technological world depends on the same principles
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that were used in ancient Egypt.
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The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC.
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Its problems are concerned with finding solutions to everyday situations.
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Several of the problems mention bread and beer,
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which isn’t surprising as Egyptian workers were paid in food and drink.
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One is concerned with how to divide nine loaves
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equally between 10 people, without a fight breaking out.
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I’ve got nine loaves of bread here.
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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
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and give the pile of crumbs to the 10th person.
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But the Egyptians developed a far more elegant solution -
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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,
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so each piece will be one fifteenth.
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Each person then gets one half
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and one third
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and one fifteenth.
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It is through such seemingly practical problems
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that we start to see a more abstract mathematics developing.
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Suddenly, new numbers are on the scene - fractions -
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and it isn’t too long before the Egyptians are exploring the mathematics of these numbers.
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Fractions are clearly of practical importance to anyone dividing quantities for trade in the market.
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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
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came from a hieroglyph which had great mystical significance.
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It’s called the Eye of Horus.
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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.
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Horus was determined to avenge the murder.
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During one particularly fierce battle,
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Seth ripped out Horus’ eye, tore it up and scattered it over Egypt.
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But the gods were looking favourably on Horus.
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They gathered up the scattered pieces and reassembled the eye.
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Each part of the eye represented a different fraction.
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Each one, half the fraction before.
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Although the original eye represented a whole unit,
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the reassembled eye is 1/64 short.
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Although the Egyptians stopped at 1/64,
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implicit in this picture
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is the possibility of adding more fractions,
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halving them each time, the sum getting closer and closer to one,
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but never quite reaching it.
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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.
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But the concept of infinite series would remain hidden
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until the mathematicians of Asia discovered it centuries later.
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Having worked out a system of numbers, including these new fractions,
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it was time for the Egyptians to apply their knowledge
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to understanding shapes that they encountered day to day.
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These shapes were rarely regular squares or rectangles,
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and in the Rhind Papyrus, we find the area of a more organic form, the circle.
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What is astounding in the calculation
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of the area of the circle is its exactness, really.
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How they would have found their method is open to speculation,
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because the texts we have
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do not show us the methods how they were found.
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This calculation is particularly striking because it depends
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on seeing how the shape of the circle
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can be approximated by shapes that the Egyptians already understood.
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The Rhind Papyrus states that a circular field
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with a diameter of nine units
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is close in area to a square with sides of eight.
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But how would this relationship have been discovered?
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My favourite theory sees the answer in the ancient game of mancala.
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Mancala boards were found carved on the roofs of temples.
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Each player starts with an equal number of stones,
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and the object of the game is to move them round the board,
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capturing your opponent’s counters on the way.
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As the players sat around waiting to make their next move,
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perhaps one of them realised that sometimes the balls fill the circular holes
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of the mancala board in a rather nice way.
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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,
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can be used to make a circle with diameter nine stones.
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By rearranging the stones, the circle has been approximated by a square.
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And because the area of a circle is pi times the radius squared,
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the Egyptian calculation gives us the first accurate value for pi.
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The area of the circle is 64. Divide this by the radius squared,
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in this case 4.5 squared, and you get a value for pi.
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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.
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But the really brilliant thing is, the Egyptians
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are using these smaller shapes to capture the larger shape.
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But there’s one imposing and majestic symbol of Egyptian
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mathematics we haven’t attempted to unravel yet -
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the pyramid.
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I’ve seen so many pictures that I couldn’t believe I’d be impressed by them.
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But meeting them face to face, you understand why they’re called
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one of the Seven Wonders of the Ancient World.
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They’re simply breathtaking.
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And how much more impressive they must have been in their day,
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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 pyramids hiding underneath the desert,
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which would complete the shapes to make perfectly symmetrical octahedrons.
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Sometimes, in the shimmer of the desert heat, you can almost see these shapes.
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It’s the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician.
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The pyramids are just a little short to create these perfect shapes,
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but some have suggested another important mathematical concept
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00:16:51,160 –> 00:16:57,120
might be hidden inside the proportions of the Great Pyramid - the golden ratio.
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00:16:57,120 –> 00:17:01,880
Two lengths are in the golden ratio, if the relationship of the longest
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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
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all over the natural world, as well as in the work of artists,
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architects and designers for millennia.
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Whether the architects of the pyramids were conscious of this important mathematical idea,
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or were instinctively drawn to it because of its satisfying aesthetic properties, we’ll never know.
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For me, the most impressive thing about the pyramids is the mathematical brilliance
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00:17:37,040 –> 00:17:40,600
that went into making them, including the first inkling
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of one of the great theorems of the ancient world, Pythagoras’ theorem.
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In order to get perfect right-angled corners on their buildings
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and pyramids, the Egyptians would have used a rope with knots tied in it.
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At some point, the Egyptians realised that if they took a triangle with sides
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marked with three knots, four knots and five knots, it guaranteed them a perfect right-angle.
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This is because three squared, plus four squared, is equal to five squared.
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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.
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00:18:20,960 –> 00:18:23,600
But I’m pretty sure that the Egyptians hadn’t got
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00:18:23,600 –> 00:18:28,480
this sweeping generalisation of their 3, 4, 5 triangle.
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00:18:28,480 –> 00:18:32,240
We would not expect to find the general proof
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00:18:32,240 –> 00:18:35,720
because this is not the style of Egyptian mathematics.
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00:18:35,720 –> 00:18:39,320
Every problem was solved using concrete numbers and then
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00:18:39,320 –> 00:18:43,760
if a verification would be carried out at the end, it would use the result
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00:18:43,760 –> 00:18:45,720
and these concrete, given numbers,
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there’s no general proof within the Egyptian mathematical texts.
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00:18:50,960 –> 00:18:54,080
It would be some 2,000 years before the Greeks and Pythagoras
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would prove that all right-angled triangles shared certain properties.
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This wasn’t the only mathematical idea that the Egyptians would anticipate.
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00:19:03,640 –> 00:19:10,160
In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume
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of a pyramid with its peak sliced off, which shows the first hint of calculus at work.
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00:19:16,120 –> 00:19:22,920
For a culture like Egypt that is famous for its pyramids, you would expect problems like this
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00:19:22,920 –> 00:19:26,560
to have been a regular feature within the mathematical texts.
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The calculation of the volume of a truncated pyramid is one of the most
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00:19:31,280 –> 00:19:36,480
advanced bits, according to our modern standards of mathematics,
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that we have from ancient Egypt.
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00:19:39,080 –> 00:19:43,120
The architects and engineers would certainly have wanted such a formula
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00:19:43,120 –> 00:19:46,760
to calculate the amount of materials required to build it.
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00:19:46,760 –> 00:19:49,000
But it’s a mark of the sophistication
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00:19:49,000 –> 00:19:53,760
of Egyptian mathematics that they were able to produce such a beautiful method.
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00:19:59,760 –> 00:20:03,760
To understand how they derived their formula, start with a pyramid
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00:20:03,760 –> 00:20:08,480
built such that the highest point sits directly over one corner.
283
00:20:08,480 –> 00:20:13,080
Three of these can be put together to make a rectangular box,
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00:20:13,080 –> 00:20:18,240
so the volume of the skewed pyramid is a third the volume of the box.
285
00:20:18,240 –> 00:20:24,280
That is, the height, times the length, times the width, divided by three.
286
00:20:24,280 –> 00:20:29,320
Now comes an argument which shows the very first hints of the calculus at work,
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00:20:29,320 –> 00:20:35,320
thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.
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00:20:35,320 –> 00:20:39,640
Suppose you could cut the pyramid into slices, you could then slide
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00:20:39,640 –> 00:20:44,960
the layers across to make the more symmetrical pyramid you see in Giza.
290
00:20:44,960 –> 00:20:49,720
However, the volume of the pyramid has not changed, despite the rearrangement of the layers.
291
00:20:49,720 –> 00:20:52,120
So the same formula works.
292
00:20:55,360 –> 00:20:58,880
The Egyptians were amazing innovators,
293
00:20:58,880 –> 00:21:02,080
and their ability to generate new mathematics was staggering.
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00:21:02,080 –> 00:21:07,320
For me, they revealed the power of geometry and numbers, and made the first moves
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00:21:07,320 –> 00:21:11,760
towards some of the exciting mathematical discoveries to come.
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00:21:11,760 –> 00:21:15,960
But there was another civilisation that had mathematics to rival that of Egypt.
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And we know much more about their achievements.
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00:21:24,280 –> 00:21:27,880
This is Damascus, over 5,000 years old,
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00:21:27,880 –> 00:21:31,280
and still vibrant and bustling today.
300
00:21:31,280 –> 00:21:36,840
It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.
301
00:21:36,840 –> 00:21:43,720
The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.
302
00:21:43,720 –> 00:21:51,120
In order to expand and run their empire, they became masters of managing and manipulating numbers.
303
00:21:51,120 –> 00:21:53,920
We have law codes for instance that tell us
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00:21:53,920 –> 00:21:56,200
about the way society is ordered.
305
00:21:56,200 –> 00:22:00,120
The people we know most about are the scribes, the professionally literate
306
00:22:00,120 –> 00:22:05,280
and numerate people who kept the records for the wealthy families and for the temples and palaces.
307
00:22:05,280 –> 00:22:10,320
Scribe schools existed from around 2500BC.
308
00:22:10,320 –> 00:22:17,240
Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.
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00:22:17,240 –> 00:22:20,120
Scribe records were kept on clay tablets,
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00:22:20,120 –> 00:22:24,200
which allowed the Babylonians to manage and advance their empire.
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00:22:24,200 –> 00:22:31,000
However, many of the tablets we have today aren’t official documents, but children’s exercises.
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00:22:31,000 –> 00:22:37,640
It’s these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.
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00:22:37,640 –> 00:22:42,440
So, this is a geometrical textbook from about the 18th century BC.
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00:22:42,440 –> 00:22:44,920
I hope you can see that there are lots of pictures on it.
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00:22:44,920 –> 00:22:49,160
And underneath each picture is a text that sets a problem about the picture.
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00:22:49,160 –> 00:22:55,800
So for instance this one here says, I drew a square, 60 units long,
317
00:22:55,800 –> 00:23:01,200
and inside it, I drew four circles - what are their areas?
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00:23:01,200 –> 00:23:07,240
This little tablet here was written 1,000 years at least later than the tablet here,
319
00:23:07,240 –> 00:23:10,120
but has a very interesting relationship.
320
00:23:10,120 –> 00:23:12,520
It also has four circles on,
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00:23:12,520 –> 00:23:17,280
in a square, roughly drawn, but this isn’t a textbook, it’s a school exercise.
322
00:23:17,280 –> 00:23:21,400
The adult scribe who’s teaching the student is being given this
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00:23:21,400 –> 00:23:25,320
as an example of completed homework or something like that.
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00:23:26,440 –> 00:23:29,560
Like the Egyptians, the Babylonians appeared interested
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00:23:29,560 –> 00:23:32,920
in solving practical problems to do with measuring and weighing.
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00:23:32,920 –> 00:23:37,400
The Babylonian solutions to these problems are written like mathematical recipes.
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00:23:37,400 –> 00:23:43,000
A scribe would simply follow and record a set of instructions to get a result.
328
00:23:43,000 –> 00:23:47,760
Here’s an example of the kind of problem they’d solve.
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00:23:47,760 –> 00:23:51,760
I’ve got a bundle of cinnamon sticks here, but I’m not gonna weigh them.
330
00:23:51,760 –> 00:23:56,440
Instead, I’m gonna take four times their weight and add them to the scales.
331
00:23:58,040 –> 00:24:04,640
Now I’m gonna add 20 gin. Gin was the ancient Babylonian measure of weight.
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00:24:04,640 –> 00:24:07,960
I’m gonna take half of everything here and then add it again…
333
00:24:07,960 –> 00:24:10,280
That’s two bundles, and ten gin.
334
00:24:10,280 –> 00:24:16,320
Everything on this side is equal to one mana. One mana was 60 gin.
335
00:24:16,320 –> 00:24:20,280
And here, we have one of the first mathematical equations in history,
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00:24:20,280 –> 00:24:23,160
everything on this side is equal to one mana.
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00:24:23,160 –> 00:24:26,200
But how much does the bundle of cinnamon sticks weigh?
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00:24:26,200 –> 00:24:29,480
Without any algebraic language, they were able to manipulate
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00:24:29,480 –> 00:24:35,200
the quantities to be able to prove that the cinnamon sticks weighed five gin.
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00:24:35,200 –> 00:24:40,560
In my mind, it’s this kind of problem which gives mathematics a bit of a bad name.
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00:24:40,560 –> 00:24:45,040
You can blame those ancient Babylonians for all those tortuous problems you had at school.
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00:24:45,040 –> 00:24:50,200
But the ancient Babylonian scribes excelled at this kind of problem.
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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.
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00:25:05,320 –> 00:25:08,520
But instead of counting through the 10 fingers on their hand,
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00:25:08,520 –> 00:25:11,480
Babylonians found a more intriguing way to count body parts.
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00:25:11,480 –> 00:25:14,000
They used the 12 knuckles on one hand,
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00:25:14,000 –> 00:25:16,400
and the five fingers on the other to be able to count
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12 times 5, ie 60 different numbers.
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00:25:20,520 –> 00:25:25,000
So for example, this number would have been 2 lots of 12, 24,
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and then, 1, 2, 3, 4, 5, to make 29.
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The number 60 had another powerful property.
353
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It can be perfectly divided in a multitude of ways.
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00:25:39,360 –> 00:25:41,360
Here are 60 beans.
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00:25:41,360 –> 00:25:44,800
I can arrange them in 2 rows of 30.
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3 rows of 20.
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4 rows of 15.
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5 rows of 12.
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Or 6 rows of 10.
360
00:25:59,320 –> 00:26:04,560
The divisibility of 60 makes it a perfect base in which to do arithmetic.
361
00:26:04,560 –> 00:26:11,000
The base 60 system was so successful, we still use elements of it today.
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00:26:11,000 –> 00:26:15,080
Every time we want to tell the time, we recognise units of 60 -
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00:26:15,080 –> 00:26:19,040
60 seconds in a minute, 60 minutes in an hour.
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00:26:19,040 –> 00:26:24,800
But the most important feature of the Babylonians’ number system was that it recognised place value.
365
00:26:24,800 –> 00:26:30,200
Just as our decimal numbers count how many lots of tens, hundreds and thousands you’re recording,
366
00:26:30,200 –> 00:26:34,320
the position of each Babylonian number records the power of 60.
367
00:26:41,360 –> 00:26:44,440
Instead of inventing new symbols for bigger and bigger numbers,
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00:26:44,440 –> 00:26:50,440
they would write 1-1-1, so this number would be 3,661.
369
00:26:54,000 –> 00:26:59,680
The catalyst for this discovery was the Babylonians’ desire to chart the course of the night sky.
370
00:27:07,400 –> 00:27:10,840
The Babylonians’ calendar was based on the cycles of the moon.
371
00:27:10,840 –> 00:27:15,200
They needed a way of recording astronomically large numbers.
372
00:27:15,200 –> 00:27:19,560
Month by month, year by year, these cycles were recorded.
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00:27:19,560 –> 00:27:25,720
From about 800BC, there were complete lists of lunar eclipses.
374
00:27:25,720 –> 00:27:30,480
The Babylonian system of measurement was quite sophisticated at that time.
375
00:27:30,480 –> 00:27:32,840
They had a system of angular measurement,
376
00:27:32,840 –> 00:27:36,960
360 degrees in a full circle, each degree was divided
377
00:27:36,960 –> 00:27:41,920
into 60 minutes, a minute was further divided into 60 seconds.
378
00:27:41,920 –> 00:27:48,560
So they had a regular system for measurement, and it was in perfect harmony with their number system,
379
00:27:48,560 –> 00:27:52,200
so it’s well suited not only for observation but also for calculation.
380
00:27:52,200 –> 00:27:56,360
But in order to calculate and cope with these large numbers,
381
00:27:56,360 –> 00:28:00,720
the Babylonians needed to invent a new symbol.
382
00:28:00,720 –> 00:28:03,760
And in so doing, they prepared the ground for one of the great
383
00:28:03,760 –> 00:28:06,880
breakthroughs in the history of mathematics - zero.
384
00:28:06,880 –> 00:28:11,240
In the early days, the Babylonians, in order to mark an empty place in
385
00:28:11,240 –> 00:28:14,640
the middle of a number, would simply leave a blank space.
386
00:28:14,640 –> 00:28:19,960
So they needed a way of representing nothing in the middle of a number.
387
00:28:19,960 –> 00:28:25,360
So they used a sign, as a sort of breathing marker, a punctuation mark,
388
00:28:25,360 –> 00:28:28,480
and it comes to mean zero in the middle of a number.
389
00:28:28,480 –> 00:28:31,680
This was the first time zero, in any form,
390
00:28:31,680 –> 00:28:35,440
had appeared in the mathematical universe.
391
00:28:35,440 –> 00:28:42,000
But it would be over a 1,000 years before this little place holder would become a number in its own right.
392
00:28:50,600 –> 00:28:53,920
Having established such a sophisticated system of numbers,
393
00:28:53,920 –> 00:28:59,720
they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.
394
00:29:02,080 –> 00:29:06,400
Babylonian engineers and surveyors found ingenious ways of
395
00:29:06,400 –> 00:29:10,400
accessing water, and channelling it to the crop fields.
396
00:29:10,400 –> 00:29:15,760
Yet again, they used mathematics to come up with solutions.
397
00:29:15,760 –> 00:29:19,200
The Orontes valley in Syria is still an agricultural hub,
398
00:29:19,200 –> 00:29:26,320
and the old methods of irrigation are being exploited today, just as they were thousands of years ago.
399
00:29:26,320 –> 00:29:29,160
Many of the problems in Babylonian mathematics
400
00:29:29,160 –> 00:29:34,360
are concerned with measuring land, and it’s here we see for the first time
401
00:29:34,360 –> 00:29:39,920
the use of quadratic equations, one of the greatest legacies of Babylonian mathematics.
402
00:29:39,920 –> 00:29:43,560
Quadratic equations involve things where the unknown quantity
403
00:29:43,560 –> 00:29:46,920
you’re trying to identify is multiplied by itself.
404
00:29:46,920 –> 00:29:49,880
We call this squaring because it gives the area of a square,
405
00:29:49,880 –> 00:29:53,040
and it’s in the context of calculating the area of land
406
00:29:53,040 –> 00:29:55,960
that these quadratic equations naturally arise.
407
00:30:01,320 –> 00:30:03,280
Here’s a typical problem.
408
00:30:03,280 –> 00:30:06,160
If a field has an area of 55 units
409
00:30:06,160 –> 00:30:10,640
and one side is six units longer than the other,
410
00:30:10,640 –> 00:30:12,560
how long is the shorter side?
411
00:30:14,200 –> 00:30:18,640
The Babylonian solution was to reconfigure the field as a square.
412
00:30:18,640 –> 00:30:21,920
Cut three units off the end
413
00:30:21,920 –> 00:30:24,760
and move this round.
414
00:30:24,760 –> 00:30:29,920
Now, there’s a three-by-three piece missing, so let’s add this in.
415
00:30:29,920 –> 00:30:34,640
The area of the field has increased by nine units.
416
00:30:34,640 –> 00:30:38,040
This makes the new area 64.
417
00:30:38,040 –> 00:30:41,880
So the sides of the square are eight units.
418
00:30:41,880 –> 00:30:45,320
The problem-solver knows that they’ve added three to this side.
419
00:30:45,320 –> 00:30:49,520
So, the original length must be five.
420
00:30:50,520 –> 00:30:55,600
It may not look like it, but this is one of the first quadratic equations in history.
421
00:30:57,400 –> 00:31:02,400
In modern mathematics, I would use the symbolic language of algebra to solve this problem.
422
00:31:02,400 –> 00:31:07,400
The amazing feat of the Babylonians is that they were using these geometric games to find the value,
423
00:31:07,400 –> 00:31:10,200
without any recourse to symbols or formulas.
424
00:31:10,200 –> 00:31:13,920
The Babylonians were enjoying problem-solving for its own sake.
425
00:31:13,920 –> 00:31:17,960
They were falling in love with mathematics.
426
00:31:29,080 –> 00:31:34,080
The Babylonians’ fascination with numbers soon found a place in their leisure time, too.
427
00:31:34,080 –> 00:31:35,960
They were avid game-players.
428
00:31:35,960 –> 00:31:38,760
The Babylonians and their descendants have been playing
429
00:31:38,760 –> 00:31:43,160
a version of backgammon for over 5,000 years.
430
00:31:43,160 –> 00:31:45,840
The Babylonians played board games,
431
00:31:45,840 –> 00:31:52,200
from very posh board games in royal tombs to little bits of board games found in schools,
432
00:31:52,200 –> 00:31:56,280
to board games scratched on the entrances of palaces,
433
00:31:56,280 –> 00:32:00,520
so that the guardsmen must have played when they were bored,
434
00:32:00,520 –> 00:32:03,760
and they used dice to move their counters round.
435
00:32:04,880 –> 00:32:09,800
People who played games were using numbers in their leisure time to try and outwit their opponent,
436
00:32:09,800 –> 00:32:12,680
doing mental arithmetic very fast,
437
00:32:12,680 –> 00:32:17,280
and so they were calculating in their leisure time,
438
00:32:17,280 –> 00:32:21,000
without even thinking about it as being mathematical hard work.
439
00:32:23,320 –> 00:32:24,600
Now’s my chance.
440
00:32:24,600 –> 00:32:30,000
‘I hadn’t played backgammon for ages but I reckoned my maths would give me a fighting chance.’
441
00:32:30,000 –> 00:32:33,560
- It’s up to you.
- Six… I need to move something.
442
00:32:33,560 –> 00:32:36,560
‘But it wasn’t as easy as I thought.’
443
00:32:36,560 –> 00:32:38,680
Ah! What the hell was that?
444
00:32:38,680 –> 00:32:42,440
- Yeah.
- This is one, this is two.
445
00:32:42,440 –> 00:32:44,200
Now you’re in trouble.
446
00:32:44,200 –> 00:32:47,800
- So I can’t move anything.
- You cannot move these.
447
00:32:47,800 –> 00:32:49,200
Oh, gosh.
448
00:32:50,520 –> 00:32:52,320
There you go.
449
00:32:53,320 –> 00:32:54,960
Three and four.
450
00:32:54,960 –> 00:33:00,720
‘Just like the ancient Babylonians, my opponents were masters of tactical mathematics.’
451
00:33:00,720 –> 00:33:02,120
Yeah.
452
00:33:03,120 –> 00:33:05,840
Put it there. Good game.
453
00:33:07,120 –> 00:33:10,080
The Babylonians are recognised as one of the first cultures
454
00:33:10,080 –> 00:33:13,840
to use symmetrical mathematical shapes to make dice,
455
00:33:13,840 –> 00:33:17,440
but there is more heated debate about whether they might also
456
00:33:17,440 –> 00:33:20,920
have been the first to discover the secrets of another important shape.
457
00:33:20,920 –> 00:33:24,040
The right-angled triangle.
458
00:33:27,000 –> 00:33:32,360
We’ve already seen how the Egyptians use a 3-4-5 right-angled triangle.
459
00:33:32,360 –> 00:33:37,600
But what the Babylonians knew about this shape and others like it is much more sophisticated.
460
00:33:37,600 –> 00:33:42,120
This is the most famous and controversial ancient tablet we have.
461
00:33:42,120 –> 00:33:44,480
It’s called Plimpton 322.
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00:33:45,480 –> 00:33:49,080
Many mathematicians are convinced it shows the Babylonians
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00:33:49,080 –> 00:33:53,360
could well have known the principle regarding right-angled triangles,
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00:33:53,360 –> 00:33:57,400
that the square on the diagonal is the sum of the squares on the sides,
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00:33:57,400 –> 00:34:00,280
and known it centuries before the Greeks claimed it.
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00:34:01,880 –> 00:34:06,320
This is a copy of arguably the most famous Babylonian tablet,
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00:34:06,320 –> 00:34:08,040
which is Plimpton 322,
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00:34:08,040 –> 00:34:12,680
and these numbers here reflect the width or height of a triangle,
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00:34:12,680 –> 00:34:17,520
this being the diagonal, the other side would be over here,
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00:34:17,520 –> 00:34:19,880
and the square of this column
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00:34:19,880 –> 00:34:23,280
plus the square of the number in this column
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00:34:23,280 –> 00:34:26,360
equals the square of the diagonal.
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00:34:26,360 –> 00:34:31,120
They are arranged in an order of steadily decreasing angle,
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00:34:31,120 –> 00:34:34,000
on a very uniform basis, showing that somebody
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00:34:34,000 –> 00:34:38,600
had a lot of understanding of how the numbers fit together.
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00:34:44,680 –> 00:34:50,800
Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.
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00:34:50,800 –> 00:34:56,160
It’s tempting to think that the Babylonians were the first custodians of Pythagoras’ theorem,
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00:34:56,160 –> 00:35:01,200
and it’s a conclusion that generations of historians have been seduced by.
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00:35:01,200 –> 00:35:03,960
But there could be a much simpler explanation
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00:35:03,960 –> 00:35:07,760
for the sets of three numbers which fulfil Pythagoras’ theorem.
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00:35:07,760 –> 00:35:12,800
It’s not a systematic explanation of Pythagorean triples, it’s simply
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00:35:12,800 –> 00:35:17,640
a mathematics teacher doing some quite complicated calculations,
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00:35:17,640 –> 00:35:21,160
but in order to produce some very simple numbers,
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00:35:21,160 –> 00:35:26,120
in order to set his students problems about right-angled triangles,
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00:35:26,120 –> 00:35:31,000
and in that sense it’s about Pythagorean triples only incidentally.
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00:35:33,480 –> 00:35:39,040
The most valuable clues to what they understood could lie elsewhere.
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00:35:39,040 –> 00:35:43,360
This small school exercise tablet is nearly 4,000 years old
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00:35:43,360 –> 00:35:48,800
and reveals just what the Babylonians did know about right-angled triangles.
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00:35:48,800 –> 00:35:54,360
It uses a principle of Pythagoras’ theorem to find the value of an astounding new number.
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00:35:57,920 –> 00:36:05,000
Drawn along the diagonal is a really very good approximation to the square root of two,
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00:36:05,000 –> 00:36:10,880
and so that shows us that it was known and used in school environments.
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00:36:10,880 –> 00:36:12,880
Why’s this important?
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00:36:12,880 –> 00:36:18,440
Because the square root of two is what we now call an irrational number,
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00:36:18,440 –> 00:36:23,960
that is, if we write it out in decimals, or even in sexigesimal places,
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00:36:23,960 –> 00:36:28,360
it doesn’t end, the numbers go on forever after the decimal point.
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00:36:29,640 –> 00:36:33,640
The implications of this calculation are far-reaching.
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00:36:33,640 –> 00:36:37,920
Firstly, it means the Babylonians knew something of Pythagoras’ theorem
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00:36:37,920 –> 00:36:39,800
1,000 years before Pythagoras.
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00:36:39,800 –> 00:36:45,560
Secondly, the fact that they can calculate this number to an accuracy of four decimal places
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00:36:45,560 –> 00:36:50,600
shows an amazing arithmetic facility, as well as a passion for mathematical detail.
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00:36:52,200 –> 00:36:56,440
The Babylonians’ mathematical dexterity was astounding,
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00:36:56,440 –> 00:37:03,080
and for nearly 2,000 years they spearheaded intellectual progress in the ancient world.
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00:37:03,080 –> 00:37:08,280
But when their imperial power began to wane, so did their intellectual vigour.
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00:37:16,400 –> 00:37:23,280
By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.
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00:37:25,200 –> 00:37:31,000
This is Palmyra in central Syria, a once-great city built by the Greeks.
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00:37:33,800 –> 00:37:41,000
The mathematical expertise needed to build structures with such geometric perfection is impressive.
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00:37:42,120 –> 00:37:48,320
Just like the Babylonians before them, the Greeks were passionate about mathematics.
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00:37:50,520 –> 00:37:53,080
The Greeks were clever colonists.
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00:37:53,080 –> 00:37:56,280
They took the best from the civilisations they invaded
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00:37:56,280 –> 00:37:58,720
to advance their own power and influence,
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00:37:58,720 –> 00:38:01,880
but they were soon making contributions themselves.
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00:38:01,880 –> 00:38:07,080
In my opinion, their greatest innovation was to do with a shift in the mind.
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00:38:07,080 –> 00:38:11,560
What they initiated would influence humanity for centuries.
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00:38:11,560 –> 00:38:14,520
They gave us the power of proof.
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00:38:14,520 –> 00:38:18,200
Somehow they decided that they had to have a deductive system
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00:38:18,200 –> 00:38:19,640
for their mathematics
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00:38:19,640 –> 00:38:21,800
and the typical deductive system
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00:38:21,800 –> 00:38:25,720
was to begin with certain axioms, which you assume are true.
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00:38:25,720 –> 00:38:29,080
It’s as if you assume a certain theorem is true without proving it.
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00:38:29,080 –> 00:38:34,600
And then, using logical methods and very careful steps,
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00:38:34,600 –> 00:38:37,480
from these axioms you prove theorems
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00:38:37,480 –> 00:38:42,400
and from those theorems you prove more theorems, and it just snowballs.
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00:38:43,520 –> 00:38:47,000
Proof is what gives mathematics its strength.
524
00:38:47,000 –> 00:38:51,360
It’s the power of proof which means that the discoveries of the Greeks
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00:38:51,360 –> 00:38:55,480
are as true today as they were 2,000 years ago.
526
00:38:55,480 –> 00:39:01,120
I needed to head west into the heart of the old Greek empire to learn more.
527
00:39:08,720 –> 00:39:14,000
For me, Greek mathematics has always been heroic and romantic.
528
00:39:15,280 –> 00:39:20,240
I’m on my way to Samos, less than a mile from the Turkish coast.
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00:39:20,240 –> 00:39:25,000
This place has become synonymous with the birth of Greek mathematics,
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00:39:25,000 –> 00:39:27,920
and it’s down to the legend of one man.
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00:39:31,000 –> 00:39:33,120
His name is Pythagoras.
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00:39:33,120 –> 00:39:36,520
The legends that surround his life and work have contributed
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00:39:36,520 –> 00:39:40,320
to the celebrity status he has gained over the last 2,000 years.
534
00:39:40,320 –> 00:39:44,960
He’s credited, rightly or wrongly, with beginning the transformation
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00:39:44,960 –> 00:39:50,240
from mathematics as a tool for accounting to the analytic subject we recognise today.
536
00:39:54,160 –> 00:39:57,160
Pythagoras is a controversial figure.
537
00:39:57,160 –> 00:40:00,360
Because he left no mathematical writings, many have questioned
538
00:40:00,360 –> 00:40:04,920
whether he indeed solved any of the theorems attributed to him.
539
00:40:04,920 –> 00:40:07,960
He founded a school in Samos in the sixth century BC,
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00:40:07,960 –> 00:40:13,440
but his teachings were considered suspect and the Pythagoreans a bizarre sect.
541
00:40:14,960 –> 00:40:19,720
There is good evidence that there were schools of Pythagoreans,
542
00:40:19,720 –> 00:40:22,360
and they may have looked more like sects
543
00:40:22,360 –> 00:40:25,920
than what we associate with philosophical schools,
544
00:40:25,920 –> 00:40:30,920
because they didn’t just share knowledge, they also shared a way of life.
545
00:40:30,920 –> 00:40:36,080
There may have been communal living and they all seemed to have been
546
00:40:36,080 –> 00:40:40,000
involved in the politics of their cities.
547
00:40:40,000 –> 00:40:45,440
One feature that makes them unusual in the ancient world is that they included women.
548
00:40:46,560 –> 00:40:52,280
But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians -
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00:40:52,280 –> 00:40:56,040
the properties of right-angled triangles.
550
00:40:56,040 –> 00:40:58,400
What’s known as Pythagoras’ theorem
551
00:40:58,400 –> 00:41:01,360
states that if you take any right-angled triangle,
552
00:41:01,360 –> 00:41:05,320
build squares on all the sides, then the area of the largest square
553
00:41:05,320 –> 00:41:09,320
is equal to the sum of the squares on the two smaller sides.
554
00:41:13,240 –> 00:41:16,680
It’s at this point for me that mathematics is born
555
00:41:16,680 –> 00:41:19,880
and a gulf opens up between the other sciences,
556
00:41:19,880 –> 00:41:24,600
and the proof is as simple as it is devastating in its implications.
557
00:41:24,600 –> 00:41:28,080
Place four copies of the right-angled triangle
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00:41:28,080 –> 00:41:29,840
on top of this surface.
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00:41:29,840 –> 00:41:31,720
The square that you now see
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00:41:31,720 –> 00:41:35,440
has sides equal to the hypotenuse of the triangle.
561
00:41:35,440 –> 00:41:37,600
By sliding these triangles around,
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00:41:37,600 –> 00:41:40,720
we see how we can break the area of the large square up
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00:41:40,720 –> 00:41:43,160
into the sum of two smaller squares,
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00:41:43,160 –> 00:41:47,280
whose sides are given by the two short sides of the triangle.
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00:41:47,280 –> 00:41:52,040
In other words, the square on the hypotenuse is equal to the sum
566
00:41:52,040 –> 00:41:55,840
of the squares on the other sides. Pythagoras’ theorem.
567
00:41:58,040 –> 00:42:02,400
It illustrates one of the characteristic themes of Greek mathematics -
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00:42:02,400 –> 00:42:07,600
the appeal to beautiful arguments in geometry rather than a reliance on number.
569
00:42:11,400 –> 00:42:16,000
Pythagoras may have fallen out of favour and many of the discoveries accredited to him
570
00:42:16,000 –> 00:42:21,840
have been contested recently, but there’s one mathematical theory that I’m loath to take away from him.
571
00:42:21,840 –> 00:42:25,840
It’s to do with music and the discovery of the harmonic series.
572
00:42:27,680 –> 00:42:31,480
The story goes that, walking past a blacksmith’s one day,
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00:42:31,480 –> 00:42:33,800
Pythagoras heard anvils being struck,
574
00:42:33,800 –> 00:42:38,800
and noticed how the notes being produced sounded in perfect harmony.
575
00:42:38,800 –> 00:42:42,240
He believed that there must be some rational explanation
576
00:42:42,240 –> 00:42:46,080
to make sense of why the notes sounded so appealing.
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00:42:46,080 –> 00:42:48,560
The answer was mathematics.
578
00:42:53,480 –> 00:42:58,120
Experimenting with a stringed instrument, Pythagoras discovered that the intervals between
579
00:42:58,120 –> 00:43:02,400
harmonious musical notes were always represented as whole-number ratios.
580
00:43:05,200 –> 00:43:08,160
And here’s how he might have constructed his theory.
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00:43:10,720 –> 00:43:13,600
First, play a note on the open string.
582
00:43:13,600 –> 00:43:15,120
MAN PLAYS NOTE
583
00:43:15,120 –> 00:43:17,040
Next, take half the length.
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00:43:18,960 –> 00:43:22,160
The note almost sounds the same as the first note.
585
00:43:22,160 –> 00:43:27,120
In fact it’s an octave higher, but the relationship is so strong, we give these notes the same name.
586
00:43:27,120 –> 00:43:28,960
Now take a third the length.
587
00:43:31,600 –> 00:43:35,640
We get another note which sounds harmonious next to the first two,
588
00:43:35,640 –> 00:43:41,240
but take a length of string which is not in a whole-number ratio and all we get is dissonance.
589
00:43:46,600 –> 00:43:51,000
According to legend, Pythagoras was so excited by this discovery
590
00:43:51,000 –> 00:43:54,440
that he concluded the whole universe was built from numbers.
591
00:43:54,440 –> 00:44:00,040
But he and his followers were in for a rather unsettling challenge to their world view
592
00:44:00,040 –> 00:44:05,120
and it came about as a result of the theorem which bears Pythagoras’ name.
593
00:44:07,120 –> 00:44:12,400
Legend has it, one of his followers, a mathematician called Hippasus,
594
00:44:12,400 –> 00:44:15,480
set out to find the length of the diagonal
595
00:44:15,480 –> 00:44:19,760
for a right-angled triangle with two sides measuring one unit.
596
00:44:19,760 –> 00:44:25,520
Pythagoras’ theorem implied that the length of the diagonal was a number whose square was two.
597
00:44:25,520 –> 00:44:29,560
The Pythagoreans assumed that the answer would be a fraction,
598
00:44:29,560 –> 00:44:36,000
but when Hippasus tried to express it in this way, no matter how he tried, he couldn’t capture it.
599
00:44:36,000 –> 00:44:38,600
Eventually he realised his mistake.
600
00:44:38,600 –> 00:44:43,320
It was the assumption that the value was a fraction at all which was wrong.
601
00:44:43,320 –> 00:44:49,440
The value of the square root of two was the number that the Babylonians etched into the Yale tablet.
602
00:44:49,440 –> 00:44:53,320
However, they didn’t recognise the special character of this number.
603
00:44:53,320 –> 00:44:55,040
But Hippasus did.
604
00:44:55,040 –> 00:44:57,560
It was an irrational number.
605
00:45:00,880 –> 00:45:04,800
The discovery of this new number, and others like it, is akin to an explorer
606
00:45:04,800 –> 00:45:09,240
discovering a new continent, or a naturalist finding a new species.
607
00:45:09,240 –> 00:45:13,520
But these irrational numbers didn’t fit the Pythagorean world view.
608
00:45:13,520 –> 00:45:19,120
Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy,
609
00:45:19,120 –> 00:45:21,840
but Hippasus let slip the discovery
610
00:45:21,840 –> 00:45:25,600
and was promptly drowned for his attempts to broadcast their research.
611
00:45:27,080 –> 00:45:32,440
But these mathematical discoveries could not be easily suppressed.
612
00:45:32,440 –> 00:45:37,920
Schools of philosophy and science started to flourish all over Greece, building on these foundations.
613
00:45:37,920 –> 00:45:42,360
The most famous of these was the Academy.
614
00:45:42,360 –> 00:45:47,560
Plato founded this school in Athens in 387 BC.
615
00:45:47,560 –> 00:45:54,040
Although we think of him today as a philosopher, he was one of mathematics’ most important patrons.
616
00:45:54,040 –> 00:45:57,720
Plato was enraptured by the Pythagorean world view
617
00:45:57,720 –> 00:46:02,040
and considered mathematics the bedrock of knowledge.
618
00:46:02,040 –> 00:46:07,200
Some people would say that Plato is the most influential figure
619
00:46:07,200 –> 00:46:10,080
for our perception of Greek mathematics.
620
00:46:10,080 –> 00:46:15,120
He argued that mathematics is an important form of knowledge
621
00:46:15,120 –> 00:46:17,600
and does have a connection with reality.
622
00:46:17,600 –> 00:46:23,480
So by knowing mathematics, we know more about reality.
623
00:46:23,480 –> 00:46:29,240
In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking
624
00:46:29,240 –> 00:46:33,480
the secrets of the universe, a view still held by scientists today.
625
00:46:33,480 –> 00:46:37,480
Indeed, the importance Plato attached to geometry is encapsulated
626
00:46:37,480 –> 00:46:43,960
in the sign that was mounted above the Academy, “Let no-one ignorant of geometry enter here.”
627
00:46:47,520 –> 00:46:53,720
Plato proposed that the universe could be crystallised into five regular symmetrical shapes.
628
00:46:53,720 –> 00:46:56,640
These shapes, which we now call the Platonic solids,
629
00:46:56,640 –> 00:46:59,600
were composed of regular polygons, assembled to create
630
00:46:59,600 –> 00:47:03,080
three-dimensional symmetrical objects.
631
00:47:03,080 –> 00:47:05,720
The tetrahedron represented fire.
632
00:47:05,720 –> 00:47:09,960
The icosahedron, made from 20 triangles, represented water.
633
00:47:09,960 –> 00:47:12,160
The stable cube was Earth.
634
00:47:12,160 –> 00:47:15,880
The eight-faced octahedron was air.
635
00:47:15,880 –> 00:47:19,440
And the fifth Platonic solid, the dodecahedron,
636
00:47:19,440 –> 00:47:22,280
made out of 12 pentagons, was reserved for the shape
637
00:47:22,280 –> 00:47:26,000
that captured Plato’s view of the universe.
638
00:47:29,600 –> 00:47:33,640
Plato’s theory would have a seismic influence and continued to inspire
639
00:47:33,640 –> 00:47:37,400
mathematicians and astronomers for over 1,500 years.
640
00:47:38,360 –> 00:47:41,120
In addition to the breakthroughs made in the Academy,
641
00:47:41,120 –> 00:47:45,040
mathematical triumphs were also emerging from the edge of the Greek empire,
642
00:47:45,040 –> 00:47:51,520
and owed as much to the mathematical heritage of the Egyptians as the Greeks.
643
00:47:51,520 –> 00:47:58,000
Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,
644
00:47:58,000 –> 00:48:04,320
and its famous library soon gained a reputation to rival Plato’s Academy.
645
00:48:04,320 –> 00:48:11,760
The kings of Alexandria were prepared to invest in the arts and culture,
646
00:48:11,760 –> 00:48:14,960
in technology, mathematics, grammar,
647
00:48:14,960 –> 00:48:19,680
because patronage for cultural pursuits
648
00:48:19,680 –> 00:48:27,000
was one way of showing that you were a more prestigious ruler,
649
00:48:27,000 –> 00:48:30,320
and had a better entitlement to greatness.
650
00:48:32,040 –> 00:48:35,360
The old library and its precious contents were destroyed
651
00:48:35,360 –> 00:48:38,240
But its spirit is alive in a new building.
652
00:48:40,240 –> 00:48:44,120
Today, the library remains a place of discovery and scholarship.
653
00:48:48,600 –> 00:48:51,920
Mathematicians and philosophers flocked to Alexandria,
654
00:48:51,920 –> 00:48:55,080
driven by their thirst for knowledge and the pursuit of excellence.
655
00:48:55,080 –> 00:48:59,040
The patrons of the library were the first professional scientists,
656
00:48:59,040 –> 00:49:02,600
individuals who were paid for their devotion to research.
657
00:49:02,600 –> 00:49:04,720
But of all those early pioneers,
658
00:49:04,720 –> 00:49:08,880
my hero is the enigmatic Greek mathematician Euclid.
659
00:49:12,560 –> 00:49:15,120
We know very little about Euclid’s life,
660
00:49:15,120 –> 00:49:19,360
but his greatest achievements were as a chronicler of mathematics.
661
00:49:19,360 –> 00:49:24,600
Around 300 BC, he wrote the most important text book of all time -
662
00:49:24,600 –> 00:49:27,080
The Elements. In The Elements,
663
00:49:27,080 –> 00:49:31,120
we find the culmination of the mathematical revolution
664
00:49:31,120 –> 00:49:32,960
which had taken place in Greece.
665
00:49:34,880 –> 00:49:39,240
It’s built on a series of mathematical assumptions, called axioms.
666
00:49:39,240 –> 00:49:44,000
For example, a line can be drawn between any two points.
667
00:49:44,000 –> 00:49:48,760
From these axioms, logical deductions are made and mathematical theorems established.
668
00:49:51,880 –> 00:49:56,360
The Elements contains formulas for calculating the volumes of cones
669
00:49:56,360 –> 00:49:59,400
and cylinders, proofs about geometric series,
670
00:49:59,400 –> 00:50:02,160
perfect numbers and primes.
671
00:50:02,160 –> 00:50:06,760
The climax of The Elements is a proof that there are only five Platonic solids.
672
00:50:09,560 –> 00:50:14,280
For me, this last theorem captures the power of mathematics.
673
00:50:14,280 –> 00:50:17,080
It’s one thing to build five symmetrical solids,
674
00:50:17,080 –> 00:50:22,600
quite another to come up with a watertight, logical argument for why there can’t be a sixth.
675
00:50:22,600 –> 00:50:26,600
The Elements unfolds like a wonderful, logical mystery novel.
676
00:50:26,600 –> 00:50:29,720
But this is a story which transcends time.
677
00:50:29,720 –> 00:50:33,560
Scientific theories get knocked down, from one generation to the next,
678
00:50:33,560 –> 00:50:39,920
but the theorems in The Elements are as true today as they were 2,000 years ago.
679
00:50:39,920 –> 00:50:43,480
When you stop and think about it, it’s really amazing.
680
00:50:43,480 –> 00:50:45,160
It’s the same theorems that we teach.
681
00:50:45,160 –> 00:50:49,960
We may teach them in a slightly different way, we may organise them differently,
682
00:50:49,960 –> 00:50:54,200
but it’s Euclidean geometry that is still valid,
683
00:50:54,200 –> 00:50:58,320
and even in higher mathematics, when you go to higher dimensional spaces,
684
00:50:58,320 –> 00:51:00,560
you’re still using Euclidean geometry.
685
00:51:02,080 –> 00:51:06,080
Alexandria must have been an inspiring place for the ancient scholars,
686
00:51:06,080 –> 00:51:12,360
and Euclid’s fame would have attracted even more eager, young intellectuals to the Egyptian port.
687
00:51:12,360 –> 00:51:18,680
One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.
688
00:51:19,640 –> 00:51:23,200
He would become a mathematical visionary.
689
00:51:23,200 –> 00:51:28,080
The best Greek mathematicians, they were always pushing the limits,
690
00:51:28,080 –> 00:51:29,560
pushing the envelope.
691
00:51:29,560 –> 00:51:32,200
So, Archimedes…
692
00:51:32,200 –> 00:51:35,200
did what he could with polygons,
693
00:51:35,200 –> 00:51:37,520
with solids.
694
00:51:37,520 –> 00:51:40,360
He then moved on to centres of gravity.
695
00:51:40,360 –> 00:51:44,680
He then moved on to the spiral.
696
00:51:44,680 –> 00:51:50,800
This instinct to try and mathematise everything
697
00:51:50,800 –> 00:51:54,440
is something that I see as a legacy.
698
00:51:55,520 –> 00:52:00,280
One of Archimedes’ specialities was weapons of mass destruction.
699
00:52:00,280 –> 00:52:06,360
They were used against the Romans when they invaded his home of Syracuse in 212 BC.
700
00:52:06,360 –> 00:52:10,200
He also designed mirrors, which harnessed the power of the sun,
701
00:52:10,200 –> 00:52:12,760
to set the Roman ships on fire.
702
00:52:12,760 –> 00:52:17,520
But to Archimedes, these endeavours were mere amusements in geometry.
703
00:52:17,520 –> 00:52:20,280
He had loftier ambitions.
704
00:52:23,040 –> 00:52:29,560
Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake
705
00:52:29,560 –> 00:52:33,800
and not for the ignoble trade of engineering or the sordid quest for profit.
706
00:52:33,800 –> 00:52:37,840
One of his finest investigations into pure mathematics
707
00:52:37,840 –> 00:52:41,840
was to produce formulas to calculate the areas of regular shapes.
708
00:52:43,760 –> 00:52:49,480
Archimedes’ method was to capture new shapes by using shapes he already understood.
709
00:52:49,480 –> 00:52:52,720
So, for example, to calculate the area of a circle,
710
00:52:52,720 –> 00:52:57,920
he would enclose it inside a triangle, and then by doubling the number of sides on the triangle,
711
00:52:57,920 –> 00:53:02,320
the enclosing shape would get closer and closer to the circle.
712
00:53:02,320 –> 00:53:04,360
Indeed, we sometimes call a circle
713
00:53:04,360 –> 00:53:07,360
a polygon with an infinite number of sides.
714
00:53:07,360 –> 00:53:11,200
But by estimating the area of the circle, Archimedes is, in fact,
715
00:53:11,200 –> 00:53:15,480
getting a value for pi, the most important number in mathematics.
716
00:53:16,480 –> 00:53:22,760
However, it was calculating the volumes of solid objects where Archimedes excelled.
717
00:53:22,760 –> 00:53:25,800
He found a way to calculate the volume of a sphere
718
00:53:25,800 –> 00:53:30,280
by slicing it up and approximating each slice as a cylinder.
719
00:53:30,280 –> 00:53:33,120
He then added up the volumes of the slices
720
00:53:33,120 –> 00:53:36,480
to get an approximate value for the sphere.
721
00:53:36,480 –> 00:53:39,440
But his act of genius was to see what happens
722
00:53:39,440 –> 00:53:42,280
if you make the slices thinner and thinner.
723
00:53:42,280 –> 00:53:47,040
In the limit, the approximation becomes an exact calculation.
724
00:53:51,080 –> 00:53:56,040
But it was Archimedes’ commitment to mathematics that would be his undoing.
725
00:53:58,120 –> 00:54:02,960
Archimedes was contemplating a problem about circles traced in the sand.
726
00:54:02,960 –> 00:54:05,600
When a Roman soldier accosted him,
727
00:54:05,600 –> 00:54:11,640
Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.
728
00:54:11,640 –> 00:54:16,920
But the Roman soldier was not interested in Archimedes’ problem and killed him on the spot.
729
00:54:16,920 –> 00:54:21,800
Even in death, Archimedes’ devotion to mathematics was unwavering.
730
00:54:43,360 –> 00:54:46,480
By the middle of the 1st century BC,
731
00:54:46,480 –> 00:54:50,520
the Romans had tightened their grip on the old Greek empire.
732
00:54:50,520 –> 00:54:53,320
They were less smitten with the beauty of mathematics
733
00:54:53,320 –> 00:54:56,640
and were more concerned with its practical applications.
734
00:54:56,640 –> 00:55:02,520
This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.
735
00:55:02,520 –> 00:55:06,760
But one mathematician was determined to keep the legacy of the Greeks alive.
736
00:55:06,760 –> 00:55:11,640
Hypatia was exceptional, a female mathematician,
737
00:55:11,640 –> 00:55:14,800
and a pagan in the piously Christian Roman empire.
738
00:55:16,680 –> 00:55:21,560
Hypatia was very prestigious and very influential in her time.
739
00:55:21,560 –> 00:55:27,440
She was a teacher with a lot of students, a lot of followers.
740
00:55:27,440 –> 00:55:31,680
She was politically influential in Alexandria.
741
00:55:31,680 –> 00:55:34,560
So it’s this combination of…
742
00:55:34,560 –> 00:55:40,840
high knowledge and high prestige that may have made her
743
00:55:40,840 –> 00:55:44,400
a figure of hatred for…
744
00:55:44,400 –> 00:55:46,080
the Christian mob.
745
00:55:51,760 –> 00:55:55,800
One morning during Lent, Hypatia was dragged off her chariot
746
00:55:55,800 –> 00:55:59,840
by a zealous Christian mob and taken to a church.
747
00:55:59,840 –> 00:56:03,560
There, she was tortured and brutally murdered.
748
00:56:06,280 –> 00:56:09,880
The dramatic circumstances of her life and death
749
00:56:09,880 –> 00:56:12,000
fascinated later generations.
750
00:56:12,000 –> 00:56:17,680
Sadly, her cult status eclipsed her mathematical achievements.
751
00:56:17,680 –> 00:56:20,720
She was, in fact, a brilliant teacher and theorist,
752
00:56:20,720 –> 00:56:26,440
and her death dealt a final blow to the Greek mathematical heritage of Alexandria.
753
00:56:33,800 –> 00:56:37,680
My travels have taken me on a fascinating journey to uncover
754
00:56:37,680 –> 00:56:42,880
the passion and innovation of the world’s earliest mathematicians.
755
00:56:42,880 –> 00:56:47,920
It’s the breakthroughs made by those early pioneers of Egypt, Babylon and Greece
756
00:56:47,920 –> 00:56:52,320
that are the foundations on which my subject is built today.
757
00:56:52,320 –> 00:56:55,760
But this is just the beginning of my mathematical odyssey.
758
00:56:55,760 –> 00:56:59,400
The next leg of my journey lies east, in the depths of Asia,
759
00:56:59,400 –> 00:57:02,560
where mathematicians scaled even greater heights
760
00:57:02,560 –> 00:57:04,800
in pursuit of knowledge.
761
00:57:04,800 –> 00:57:08,720
With this new era came a new language of algebra and numbers,
762
00:57:08,720 –> 00:57:12,920
better suited to telling the next chapter in the story of maths.
763
00:57:12,920 –> 00:57:16,600
You can learn more about the story of maths
764
00:57:16,600 –> 00:57:19,840
with the Open University at…
765
00:57:36,040 –> 00:57:39,080
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