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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 ispi 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 andmajestic 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 pyramidshiding underneath the desert,

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which would complete the shapes to make perfectlysymmetrical 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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might be hidden inside the proportions of the Great Pyramid - the golden ratio.

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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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00:17:11,640 --> 00:17:15,600
all over the natural world, as well as in the work of artists,

246
00:17:15,640 --> 00:17:18,480
architects and designers for millennia.

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00:17:22,360 --> 00:17:26,760
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 pyramidsis the mathematical brilliance

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00:17:36,840 --> 00:17:40,360
that went into making them, including the first inkling

251
00:17:40,400 --> 00:17:44,400
of one of the great theorems of the ancient world, Pythagoras' theorem.

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00:17:45,960 --> 00:17:48,920
In order to get perfect right-angled corners on their buildings

253
00:17:48,960 --> 00:17:54,080
and pyramids, the Egyptians would haveused a rope with knots tied in it.

254
00:17:54,120 --> 00:17:57,960
At some point, the Egyptians realised that if they took a triangle with sides

255
00:17:58,000 --> 00:18:05,400
marked with three knots, four knots and five knots, it guaranteed them aperfect right-angle.

256
00:18:05,440 --> 00:18:09,880
This is because three squared, plus four squared, is equal to five squared.

257
00:18:09,920 --> 00:18:12,600
So we've got a perfect Pythagorean triangle.

258
00:18:14,960 --> 00:18:20,720
In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle.

259
00:18:20,760 --> 00:18:23,360
But I'm pretty sure that the Egyptians hadn't got

260
00:18:23,400 --> 00:18:28,240
this sweeping generalisation of their 3, 4, 5 triangle.

261
00:18:28,280 --> 00:18:32,000
We would not expect to find the general proof

262
00:18:32,040 --> 00:18:35,480
because this is not the style of Egyptian mathematics.

263
00:18:35,520 --> 00:18:39,080
Every problem was solved using concrete numbers and then

264
00:18:39,120 --> 00:18:43,520
if a verification would be carried out at the end, it would use the result

265
00:18:43,560 --> 00:18:45,480
and these concrete, given numbers,

266
00:18:45,520 --> 00:18:49,200
there's no general proof within the Egyptian mathematical texts.

267
00:18:50,760 --> 00:18:53,840
It would be some 2,000 years before the Greeks and Pythagoras

268
00:18:53,880 --> 00:18:59,040
would prove that all right-angled triangles shared certain properties.

269
00:18:59,080 --> 00:19:03,400
This wasn't the only mathematical idea that theEgyptians would anticipate.

270
00:19:03,440 --> 00:19:09,920
In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume

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00:19:09,960 --> 00:19:15,880
of a pyramid with its peak sliced off,which shows the first hint ofcalculus at work.

272
00:19:15,920 --> 00:19:22,680
For a culture like Egypt that is famous for its pyramids, you would expect problems like this

273
00:19:22,720 --> 00:19:26,320
to have been a regular feature within the mathematical texts.

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00:19:26,360 --> 00:19:31,040
The calculation of the volume of a truncated pyramid is one of the most

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00:19:31,080 --> 00:19:36,240
advanced bits, according to our modern standards of mathematics,

276
00:19:36,280 --> 00:19:38,840
that we have from ancient Egypt.

277
00:19:38,880 --> 00:19:42,880
The architects and engineers would certainly have wanted such a formula

278
00:19:42,920 --> 00:19:46,520
to calculate the amount of materials required to build it.

279
00:19:46,560 --> 00:19:48,760
But it's a mark of the sophistication

280
00:19:48,800 --> 00:19:53,520
of Egyptian mathematics that they were ableto produce such a beautiful method.

281
00:19:59,560 --> 00:20:03,520
To understand how they derived their formula, start with a pyramid

282
00:20:03,560 --> 00:20:08,240
built such that the highest point sits directly over one corner.

283
00:20:08,280 --> 00:20:12,840
Three of these can be put together to make a rectangular box,

284
00:20:12,880 --> 00:20:18,000
so the volume of the skewed pyramid is a third the volume of the box.

285
00:20:18,040 --> 00:20:24,040
That is, the height, times the length, times the width, divided by three.

286
00:20:24,080 --> 00:20:29,080
Now comes an argument which shows the very first hints of the calculus at work,

287
00:20:29,120 --> 00:20:35,080
thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.

288
00:20:35,120 --> 00:20:39,400
Suppose you could cut the pyramid into slices, you could then slide

289
00:20:39,440 --> 00:20:44,720
the layers across to make the more symmetrical pyramid you see in Giza.

290
00:20:44,760 --> 00:20:49,480
However, the volume of the pyramid has not changed, despite the rearrangement of the layers.

291
00:20:49,520 --> 00:20:51,880
So the same formula works.

292
00:20:55,160 --> 00:20:58,640
The Egyptians were amazing innovators,

293
00:20:58,680 --> 00:21:01,840
and their ability to generate new mathematics was staggering.

294
00:21:01,880 --> 00:21:07,080
For me, they revealed the power of geometry and numbers, and made the first moves

295
00:21:07,120 --> 00:21:11,520
towards some of the exciting mathematical discoveries to come.

296
00:21:11,560 --> 00:21:15,720
But there was another civilisation that had mathematics to rival that of Egypt.

297
00:21:15,760 --> 00:21:19,800
And we know much more about their achievements.

298
00:21:24,080 --> 00:21:27,640
This is Damascus, over 5,000 years old,

299
00:21:27,680 --> 00:21:31,040
and still vibrant and bustling today.

300
00:21:31,080 --> 00:21:36,600
It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.

301
00:21:36,640 --> 00:21:43,480
The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.

302
00:21:43,520 --> 00:21:50,880
In order to expand and run their empire, they became masters of managing and manipulating numbers.

303
00:21:50,920 --> 00:21:53,680
We have law codes for instance that tell us

304
00:21:53,720 --> 00:21:55,960
about the way society is ordered.

305
00:21:56,000 --> 00:21:59,880
The people we know most about are the scribes, the professionally literate

306
00:21:59,920 --> 00:22:05,040
and numerate people who kept the records for the wealthy families and for the temples and palaces.

307
00:22:05,080 --> 00:22:10,080
Scribe schools existed from around 2500BC.

308
00:22:10,120 --> 00:22:17,000
Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.

309
00:22:17,040 --> 00:22:19,880
Scribe records were kept on clay tablets,

310
00:22:19,920 --> 00:22:23,960
which allowed the Babylonians to manage and advance their empire.

311
00:22:24,000 --> 00:22:30,760
However, many of the tablets we have today aren't official documents, but children's exercises.

312
00:22:30,800 --> 00:22:37,400
It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.

313
00:22:37,440 --> 00:22:42,200
So, this is a geometrical textbook from about the 18th century BC.

314
00:22:42,240 --> 00:22:44,680
I hope you can see that there are lots of pictures on it.

315
00:22:44,720 --> 00:22:48,920
And underneath each picture is a text that sets a problem about the picture.

316
00:22:48,960 --> 00:22:55,560
So for instance this one here says, I drew a square, 60 units long,

317
00:22:55,600 --> 00:23:00,680
and inside it, I drew four circles - what are their areas?

318
00:23:00,720 --> 00:23:07,000
This little tablet here was written 1,000 years at least later than the tablet here,

319
00:23:07,040 --> 00:23:09,880
but has a very interesting relationship.

320
00:23:09,920 --> 00:23:12,280
It also has four circles on,

321
00:23:12,320 --> 00:23:17,040
in a square, roughly drawn, but this isn't a textbook, it's a school exercise.

322
00:23:17,080 --> 00:23:21,160
The adult scribe who's teaching the student is being given this

323
00:23:21,200 --> 00:23:25,080
as an example of completed homework or something like that.

324
00:23:26,240 --> 00:23:29,320
Like the Egyptians, the Babylonians appeared interested

325
00:23:29,360 --> 00:23:32,680
in solving practical problems to do with measuring and weighing.

326
00:23:32,720 --> 00:23:37,160
The Babylonian solutions to these problems are written like mathematical recipes.

327
00:23:37,200 --> 00:23:42,760
A scribe would simply follow and record a set of instructions to get a result.

328
00:23:42,800 --> 00:23:47,520
Here's an example of the kind of problem they'd solve.

329
00:23:47,560 --> 00:23:51,520
I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them.

330
00:23:51,560 --> 00:23:56,200
Instead, I'm gonna take four times their weight and add them to the scales.

331
00:23:57,840 --> 00:24:04,400
Now I'm gonna add 20 gin. Gin was the ancient Babylonian measure of weight.

332
00:24:04,440 --> 00:24:07,720
I'm gonna take half of everything here and then add it again...

333
00:24:07,760 --> 00:24:10,040
That's two bundles, and ten gin.

334
00:24:10,080 --> 00:24:16,080
Everything on this side is equal to one mana. One mana was 60 gin.

335
00:24:16,120 --> 00:24:20,040
And here, we have one of the first mathematical equations in history,

336
00:24:20,080 --> 00:24:22,920
everything on this side is equal to one mana.

337
00:24:22,960 --> 00:24:25,960
But how much does the bundle of cinnamon sticks weigh?

338
00:24:26,000 --> 00:24:29,240
Without any algebraic language, they were able to manipulate

339
00:24:29,280 --> 00:24:34,960
the quantities to be able to prove that the cinnamon sticks weighed five gin.

340
00:24:35,000 --> 00:24:40,320
In my mind, it's this kind of problem which gives mathematics a bit of a bad name.

341
00:24:40,360 --> 00:24:44,800
You can blame those ancient Babylonians for all those tortuous problems you had at school.

342
00:24:44,840 --> 00:24:49,680
But the ancient Babylonian scribes excelled at this kind of problem.

343
00:24:49,720 --> 00:24:57,200
Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60.

344
00:24:59,920 --> 00:25:05,080
The Babylonians invented their number system, like the Egyptians, by using their fingers.

345
00:25:05,120 --> 00:25:08,280
But instead of counting through the 10 fingers on their hand,

346
00:25:08,320 --> 00:25:11,240
Babylonians found a moreintriguing way to count body parts.

347
00:25:11,280 --> 00:25:13,760
They used the 12 knuckles on one hand,

348
00:25:13,800 --> 00:25:16,160
and the five fingers on the other to be able to count

349
00:25:16,200 --> 00:25:20,280
12 times 5, ie 60 different numbers.

350
00:25:20,320 --> 00:25:24,760
So for example, this number would have been 2 lots of 12, 24,

351
00:25:24,800 --> 00:25:28,880
and then, 1, 2, 3, 4, 5, to make 29.

352
00:25:32,000 --> 00:25:35,680
The number 60 had another powerful property.

353
00:25:35,720 --> 00:25:39,120
It can be perfectly divided in a multitude of ways.

354
00:25:39,160 --> 00:25:41,120
Here are 60 beans.

355
00:25:41,160 --> 00:25:44,560
I can arrange them in 2 rows of 30.

356
00:25:48,560 --> 00:25:51,280
3 rows of 20.

357
00:25:51,320 --> 00:25:53,680
4 rows of 15.

358
00:25:53,720 --> 00:25:55,920
5 rows of 12.

359
00:25:55,960 --> 00:25:59,080
Or 6 rows of 10.

360
00:25:59,120 --> 00:26:05,320
The divisibility of 60 makes it a perfect base in which to do arithmetic.

361
00:26:05,360 --> 00:26:10,760
The base 60 system was so successful, we still use elements of it today.

362
00:26:10,800 --> 00:26:14,840
Every time we want to tell the time, we recognise units of 60 -

363
00:26:14,880 --> 00:26:18,800
60 seconds in a minute, 60 minutes in an hour.

364
00:26:18,840 --> 00:26:24,560
But the most important feature of the Babylonians' number system was that it recognised place value.

365
00:26:24,600 --> 00:26:29,960
Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording,

366
00:26:30,000 --> 00:26:34,080
the position of each Babylonian number records the power of 60.

367
00:26:41,160 --> 00:26:44,200
Instead of inventing new symbols for bigger and bigger numbers,

368
00:26:44,240 --> 00:26:50,200
they would write 1-1-1, so this number would be 3,661.

369
00:26:53,800 --> 00:26:59,440
The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky.

370
00:27:07,200 --> 00:27:10,600
The Babylonians' calendar was based on the cycles of the moon.

371
00:27:10,640 --> 00:27:14,960
They needed a way of recording astronomically large numbers.

372
00:27:15,000 --> 00:27:19,320
Month by month, year by year, these cycles were recorded.

373
00:27:19,360 --> 00:27:25,480
From about 800BC, there were complete lists of lunar eclipses.

374
00:27:25,520 --> 00:27:31,160
The Babylonian system of measurement was quite sophisticated at that time.

375
00:27:31,200 --> 00:27:33,520
They had a system of angular measurement,

376
00:27:33,560 --> 00:27:37,240
360 degrees in a full circle, each degree was divided

377
00:27:37,280 --> 00:27:41,600
into 60 minutes, a minute was further divided into 60 seconds.

378
00:27:41,640 --> 00:27:48,320
So they had a regular system for measurement, and it was in perfect harmony with their number system,

379
00:27:48,360 --> 00:27:51,960
so it's well suited not only for observation but also for calculation.

380
00:27:52,000 --> 00:27:56,120
But in order to calculate and cope with these large numbers,

381
00:27:56,160 --> 00:28:00,480
the Babylonians needed to invent a new symbol.

382
00:28:00,520 --> 00:28:03,760
And in so doing, they prepared the ground for one of the great

383
00:28:03,800 --> 00:28:06,640
breakthroughs in the history of mathematics - zero.

384
00:28:06,680 --> 00:28:11,000
In the early days, the Babylonians, in order to mark an empty place in

385
00:28:11,040 --> 00:28:14,400
the middle of a number, would simply leave a blank space.

386
00:28:14,440 --> 00:28:19,720
So they needed a way of representing nothing in the middle of a number.

387
00:28:19,760 --> 00:28:25,120
So they used a sign, as a sort of breathing marker, a punctuation mark,

388
00:28:25,160 --> 00:28:29,040
and it comes to mean zero in the middle of a number.

389
00:28:29,080 --> 00:28:34,960
This was the first time zero, in any form, had appeared in the mathematical universe.

390
00:28:35,000 --> 00:28:41,760
But it would be over a 1,000 years before this little place holder would become a number in its own right.

391
00:28:50,400 --> 00:28:54,120
Having established such a sophisticated system of numbers,

392
00:28:54,160 --> 00:28:59,480
they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.

393
00:29:01,880 --> 00:29:06,160
Babylonian engineers and surveyors found ingenious ways of

394
00:29:06,200 --> 00:29:10,160
accessing water, and channelling it to the crop fields.

395
00:29:10,200 --> 00:29:15,520
Yet again, they used mathematics to come up with solutions.

396
00:29:15,560 --> 00:29:18,960
The Orontes valley in Syria is still an agricultural hub,

397
00:29:19,000 --> 00:29:26,080
and the old methods of irrigation are being exploited today, just as they were thousands of years ago.

398
00:29:26,120 --> 00:29:28,920
Many of the problems in Babylonian mathematics

399
00:29:28,960 --> 00:29:34,120
are concerned with measuring land, and it's here we see for the first time

400
00:29:34,160 --> 00:29:39,680
the use of quadratic equations, one of the greatest legacies ofBabylonian mathematics.

401
00:29:39,720 --> 00:29:43,320
Quadratic equations involve things where the unknown quantity

402
00:29:43,360 --> 00:29:46,680
you're trying to identify is multiplied by itself.

403
00:29:46,720 --> 00:29:49,640
We call this squaring because it gives the area of a square,

404
00:29:49,680 --> 00:29:52,800
and it's in the context of calculating the area of land

405
00:29:52,840 --> 00:29:55,720
that thesequadratic equations naturally arise.

406
00:30:01,120 --> 00:30:03,040
Here's a typical problem.

407
00:30:03,080 --> 00:30:05,920
If a field has an area of 55 units

408
00:30:05,960 --> 00:30:10,400
and one side is six units longer than the other,

409
00:30:10,440 --> 00:30:12,320
how long is the shorter side?

410
00:30:14,000 --> 00:30:18,400
The Babylonian solution was to reconfigure the field as a square.

411
00:30:18,440 --> 00:30:21,680
Cut three units off the end

412
00:30:21,720 --> 00:30:24,520
and move this round.

413
00:30:24,560 --> 00:30:29,680
Now, there's a three-by-three piece missing, so let's add this in.

414
00:30:29,720 --> 00:30:34,400
The area of the field has increased by nine units.

415
00:30:34,440 --> 00:30:37,800
This makes the new area 64.

416
00:30:37,840 --> 00:30:41,640
So the sides of the square are eight units.

417
00:30:41,680 --> 00:30:45,080
The problem-solver knows that they've added three to this side.

418
00:30:45,120 --> 00:30:49,280
So, the original length must be five.

419
00:30:50,320 --> 00:30:55,360
It may not look like it, but this is one of the first quadratic equations in history.

420
00:30:57,200 --> 00:31:02,160
In modern mathematics, I would use the symbolic language of algebra to solve this problem.

421
00:31:02,200 --> 00:31:07,160
The amazing feat of the Babylonians is that they were using these geometric games to find the value,

422
00:31:07,200 --> 00:31:09,960
without any recourse to symbols or formulas.

423
00:31:10,000 --> 00:31:13,680
The Babylonians were enjoying problem-solving for its own sake.

424
00:31:13,720 --> 00:31:17,720
They were falling in love with mathematics.

425
00:31:28,880 --> 00:31:33,840
The Babylonians' fascination with numbers soon found a place in their leisure time, too.

426
00:31:33,880 --> 00:31:35,720
They were avid game-players.

427
00:31:35,760 --> 00:31:38,520
The Babylonians and their descendants have been playing

428
00:31:38,560 --> 00:31:42,920
a version of backgammon for over 5,000 years.

429
00:31:42,960 --> 00:31:45,600
The Babylonians played board games,

430
00:31:45,640 --> 00:31:51,960
from very posh board games in royal tombs to little bits of board games found in schools,

431
00:31:52,000 --> 00:31:56,040
to board games scratched on the entrances of palaces,

432
00:31:56,080 --> 00:32:00,280
so that the guardsman must have played when they were bored,

433
00:32:00,320 --> 00:32:03,520
and they used dice to move their counters round.

434
00:32:04,480 --> 00:32:09,560
People who played games were using numbers in their leisure time to try and outwit their opponent,

435
00:32:09,600 --> 00:32:12,440
doing mental arithmetic very fast,

436
00:32:12,480 --> 00:32:17,040
and so they were calculating in their leisure time,

437
00:32:17,080 --> 00:32:20,760
without even thinking about it as being mathematical hard work.

438
00:32:23,120 --> 00:32:24,360
Now's my chance.

439
00:32:24,400 --> 00:32:29,760
'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.'

440
00:32:29,800 --> 00:32:33,320
It's up to you.Six... I need to move something.

441
00:32:33,360 --> 00:32:36,320
'But it wasn't as easy as I thought.'

442
00:32:36,360 --> 00:32:38,440
Ah! What the hell was that?

443
00:32:38,480 --> 00:32:42,200
Yeah.This is one, this is two.

444
00:32:42,240 --> 00:32:43,960
Now you're in trouble.

445
00:32:44,000 --> 00:32:47,560
So I can't move anything. You cannot move these.

446
00:32:47,600 --> 00:32:48,960
Oh, gosh.

447
00:32:50,320 --> 00:32:52,080
There you go.

448
00:32:53,120 --> 00:32:54,720
Three and four.

449
00:32:54,760 --> 00:33:00,480
'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.'

450
00:33:00,520 --> 00:33:01,880
Yeah.

451
00:33:02,920 --> 00:33:05,600
Put it there. Good game.

452
00:33:06,920 --> 00:33:10,400
The Babylonians are recognised as one of the first cultures

453
00:33:10,440 --> 00:33:13,600
to use symmetrical mathematical shapes to make dice,

454
00:33:13,640 --> 00:33:17,200
but there is more heated debates about whether they might also

455
00:33:17,240 --> 00:33:20,680
have been the first to discover the secrets of another important shape.

456
00:33:20,720 --> 00:33:23,800
The right-angled triangle.

457
00:33:26,800 --> 00:33:32,120
We've already seen how the Egyptians use a 3-4-5 right-angled triangle.

458
00:33:32,160 --> 00:33:37,360
But what the Babylonians knew about this shape and others like it is much more sophisticated.

459
00:33:37,400 --> 00:33:41,880
This is the most famous and controversial ancient tablet we have.

460
00:33:41,920 --> 00:33:44,240
It's called Plimpton 322.

461
00:33:45,280 --> 00:33:48,840
Many mathematicians are convinced it shows the Babylonians

462
00:33:48,880 --> 00:33:53,120
could well have known the principle regarding right-angled triangles,

463
00:33:53,160 --> 00:33:57,160
that the square on the diagonal is the sum of the squares on the sides,

464
00:33:57,200 --> 00:34:00,040
and known it centuries before the Greeks claimed it.

465
00:34:01,680 --> 00:34:06,080
This is a copy of arguably the most famous Babylonian tablet,

466
00:34:06,120 --> 00:34:07,800
which is Plimpton 322,

467
00:34:07,840 --> 00:34:12,440
and these numbers here reflect the width or height of a triangle,

468
00:34:12,480 --> 00:34:17,280
this being the diagonal, the other side would be over here,

469
00:34:17,320 --> 00:34:23,240
and the square of this column plus the square of the number in this column

470
00:34:23,280 --> 00:34:26,120
equals the square of the diagonal.

471
00:34:26,160 --> 00:34:30,880
They are arranged in an order of steadily decreasing angle,

472
00:34:30,920 --> 00:34:33,760
on a very uniform basis, showing that somebody

473
00:34:33,800 --> 00:34:38,360
had a lot of understanding of how the numbers fit together.

474
00:34:44,480 --> 00:34:50,560
Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.

475
00:34:50,600 --> 00:34:55,920
It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem,

476
00:34:55,960 --> 00:35:00,960
and it's a conclusion that generations of historians have been seduced by.

477
00:35:01,000 --> 00:35:03,720
But there could be a much simpler explanation

478
00:35:03,760 --> 00:35:07,520
for the sets of three numbers which fulfil Pythagoras' theorem.

479
00:35:07,560 --> 00:35:12,560
It's not a systematic explanation of Pythagorean triples, it's simply

480
00:35:12,600 --> 00:35:17,400
a mathematics teacher doing some quite complicated calculations,

481
00:35:17,440 --> 00:35:20,920
but in order to produce some very simple numbers,

482
00:35:20,960 --> 00:35:25,880
in order to set his students problems about right-angled triangles,

483
00:35:25,920 --> 00:35:30,760
and in that sense it's about Pythagorean triples only incidentally.

484
00:35:33,280 --> 00:35:38,800
The most valuable clues to what they understood could lie elsewhere.

485
00:35:38,840 --> 00:35:43,120
This small school exercise tablet is nearly 4,000 years old

486
00:35:43,160 --> 00:35:48,560
and reveals just what the Babylonians did know about right-angled triangles.

487
00:35:48,600 --> 00:35:54,120
It uses a principle of Pythagoras' theorem to find the value of an astounding new number.

488
00:35:57,720 --> 00:36:05,520
Drawn along the diagonal is a really very good approximation to the square root of two,

489
00:36:05,560 --> 00:36:10,640
and so that shows us that it was known and used in school environments.

490
00:36:10,680 --> 00:36:12,640
Why's this important?

491
00:36:12,680 --> 00:36:18,200
Because the square root of two is what we now call an irrational number,

492
00:36:18,240 --> 00:36:23,720
that is, if we write it out in decimals, or even in sexigesimal places,

493
00:36:23,760 --> 00:36:28,120
it doesn't end, the numbers go on forever after the decimal point.

494
00:36:29,040 --> 00:36:33,400
The implications of this calculation are far-reaching.

495
00:36:33,440 --> 00:36:37,680
Firstly, it means the Babylonians knew something of Pythagoras' theorem

496
00:36:37,720 --> 00:36:39,560
1,000 years before Pythagoras.

497
00:36:39,600 --> 00:36:45,320
Secondly, the fact that they can calculate this number to an accuracy of four decimal places

498
00:36:45,360 --> 00:36:50,360
shows an amazing arithmetic facility, as well as a passion for mathematical detail.

499
00:36:52,000 --> 00:36:56,200
The Babylonians' mathematical dexterity was astounding,

500
00:36:56,240 --> 00:37:02,840
and for nearly 2,000 years they spearheaded intellectual progress in the ancient world.

501
00:37:02,880 --> 00:37:08,040
But when their imperial power began to wane, so did their intellectual vigour.

502
00:37:16,200 --> 00:37:23,040
By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.

503
00:37:25,000 --> 00:37:30,760
This is Palmyra in central Syria, a once-great city built by the Greeks.

504
00:37:33,600 --> 00:37:40,760
The mathematical expertise needed to build structures with such geometric perfection is impressive.

505
00:37:41,920 --> 00:37:48,080
Just like the Babylonians before them, the Greeks were passionate about mathematics.

506
00:37:50,320 --> 00:37:52,840
The Greeks were clever colonists.

507
00:37:52,880 --> 00:37:56,040
They took the best from the civilisations they invaded

508
00:37:56,080 --> 00:37:58,480
to advance their own power and influence,

509
00:37:58,520 --> 00:38:01,640
but they were soon making contributions themselves.

510
00:38:01,680 --> 00:38:06,840
In my opinion, their greatest innovation was to do with a shift in the mind.

511
00:38:06,880 --> 00:38:11,320
What they initiated would influence humanity for centuries.

512
00:38:11,360 --> 00:38:14,280
They gave us the power of proof.

513
00:38:14,320 --> 00:38:19,400
Somehow they decided that they had to have a deductive system for their mathematics

514
00:38:19,440 --> 00:38:25,480
and the typical deductive system was to begin with certain axioms, which you assume are true.

515
00:38:25,520 --> 00:38:28,840
It's as if you assume a certain theorem is true without proving it.

516
00:38:28,880 --> 00:38:34,360
And then, using logical methods and very careful steps,

517
00:38:34,400 --> 00:38:37,240
from these axioms you prove theorems

518
00:38:37,280 --> 00:38:42,160
and from those theorems you prove more theorems, and it just snowballs.

519
00:38:43,320 --> 00:38:46,760
Proof is what gives mathematics its strength.

520
00:38:46,800 --> 00:38:51,120
It's the power or proof which means that the discoveries of the Greeks

521
00:38:51,160 --> 00:38:55,240
are as true today as they were 2,000 years ago.

522
00:38:55,280 --> 00:39:00,880
I needed to head west into the heart of the old Greek empire to learn more.

523
00:39:08,520 --> 00:39:13,760
For me, Greek mathematics has always been heroic and romantic.

524
00:39:15,080 --> 00:39:20,000
I'm on my way to Samos, less than a mile from the Turkish coast.

525
00:39:20,040 --> 00:39:24,760
This place has become synonymous with the birth of Greek mathematics,

526
00:39:24,800 --> 00:39:27,680
and it's down to the legend of one man.

527
00:39:30,800 --> 00:39:32,880
His name is Pythagoras.

528
00:39:32,920 --> 00:39:36,280
The legends that surround his life and work have contributed

529
00:39:36,320 --> 00:39:40,080
to the celebrity status he has gained over the last 2,000 years.

530
00:39:40,120 --> 00:39:44,720
He's credited, rightly or wrongly, with beginning the transformation

531
00:39:44,760 --> 00:39:50,000
from mathematics as a tool for accounting to the analytic subject we recognise today.

532
00:39:53,960 --> 00:39:56,920
Pythagoras is a controversial figure.

533
00:39:56,960 --> 00:40:00,120
Because he left no mathematical writings, many have questioned

534
00:40:00,160 --> 00:40:04,680
whether he indeed solved any of the theorems attributed to him.

535
00:40:04,720 --> 00:40:07,720
He founded a school in Samos in the sixth century BC,

536
00:40:07,760 --> 00:40:13,200
but his teachings were considered suspect and the Pythagoreans a bizarre sect.

537
00:40:14,760 --> 00:40:19,480
There is good evidence that there were schools of Pythagoreans,

538
00:40:19,520 --> 00:40:25,680
and they may have looked more like sects than what we associate with philosophical schools,

539
00:40:25,720 --> 00:40:30,680
because they didn't just share knowledge, they also shared a way of life.

540
00:40:30,720 --> 00:40:35,840
There may have been communal living and they all seemed to have been

541
00:40:35,880 --> 00:40:39,760
involved in the politics of their cities.

542
00:40:39,800 --> 00:40:45,200
One feature that makes them unusual in the ancient world is that they included women.

543
00:40:46,360 --> 00:40:52,040
But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians -

544
00:40:52,080 --> 00:40:55,800
the properties of right-angled triangles.

545
00:40:55,840 --> 00:40:58,000
What's known as Pythagoras' theorem

546
00:40:58,040 --> 00:41:01,120
states that if you take any right-angled triangle,

547
00:41:01,160 --> 00:41:05,080
build squares on all the sides, then the area of the largest square

548
00:41:05,120 --> 00:41:09,080
is equal to the sum of the squares on the two smaller sides.

549
00:41:13,040 --> 00:41:16,440
It's at this point for me that mathematics is born

550
00:41:16,480 --> 00:41:19,640
and a gulf opens up between the other sciences,

551
00:41:19,680 --> 00:41:24,360
and the proof is as simple as it is devastating in its implications.

552
00:41:24,400 --> 00:41:27,840
Place four copies of the right-angled triangle

553
00:41:27,880 --> 00:41:29,600
on top of this surface.

554
00:41:29,640 --> 00:41:31,480
The square that you now see

555
00:41:31,520 --> 00:41:35,200
has sides equal to the hypotenuse of the triangle.

556
00:41:35,240 --> 00:41:37,360
By sliding these triangles around,

557
00:41:37,400 --> 00:41:40,480
we see how we can break the area of the large square up

558
00:41:40,520 --> 00:41:42,920
into the sum of two smaller squares,

559
00:41:42,960 --> 00:41:47,040
whose sides are given by the two short sides of the triangle.

560
00:41:47,080 --> 00:41:51,800
In other words, the square on the hypotenuse is equal to the sum

561
00:41:51,840 --> 00:41:55,600
of the squares on the other sides. Pythagoras' theorem.

562
00:41:57,840 --> 00:42:02,160
It illustrates one of the characteristic themes of Greek mathematics -

563
00:42:02,200 --> 00:42:07,360
the appeal to beautiful arguments in geometry rather than a reliance on number.

564
00:42:11,200 --> 00:42:15,760
Pythagoras may have fallen out of favour and many of the discoveries accredited to him

565
00:42:15,800 --> 00:42:21,600
have been contested recently, but there's one mathematical theory that I'm loath to take away from him.

566
00:42:21,640 --> 00:42:25,600
It's to do with music and the discoveryof the harmonic series.

567
00:42:27,480 --> 00:42:31,240
The story goes that, walking past a blacksmith's one day,

568
00:42:31,280 --> 00:42:33,560
Pythagoras heard anvils being struck,

569
00:42:33,600 --> 00:42:38,560
and noticed how the notes being produced sounded in perfect harmony.

570
00:42:38,600 --> 00:42:42,000
He believed that there must be some rational explanation

571
00:42:42,040 --> 00:42:45,840
to make sense of why the notes sounded so appealing.

572
00:42:45,880 --> 00:42:48,320
The answer was mathematics.

573
00:42:53,280 --> 00:42:57,880
Experimenting with a stringed instrument, Pythagoras discovered that the intervals between

574
00:42:57,920 --> 00:43:03,200
harmonious musical notes were always represented as whole-number ratios.

575
00:43:05,000 --> 00:43:07,920
And here's how he might have constructed his theory.

576
00:43:10,520 --> 00:43:13,360
First, play a note on the open string.

577
00:43:13,400 --> 00:43:14,880
MAN PLAYS NOTE

578
00:43:14,920 --> 00:43:16,800
Next, take half the length.

579
00:43:18,760 --> 00:43:21,920
The note almost sounds the same as the first note.

580
00:43:21,960 --> 00:43:26,880
In fact it's an octave higher, but the relationship is so strong, we give these notes the same name.

581
00:43:26,920 --> 00:43:28,720
Now take a third the length.

582
00:43:31,400 --> 00:43:35,400
We get another note which sounds harmonious next to the first two,

583
00:43:35,440 --> 00:43:41,000
but take a length of string which is not in a whole-number ratio and all we get is dissonance.

584
00:43:46,400 --> 00:43:50,760
According to legend, Pythagoras was so excited by this discovery

585
00:43:50,800 --> 00:43:54,200
that he concluded the whole universe was built from numbers.

586
00:43:54,240 --> 00:43:59,800
But he and his followers were in for a rather unsettling challenge to their world view

587
00:43:59,840 --> 00:44:04,880
and it came about as a result of the theorem which bears Pythagoras' name.

588
00:44:06,920 --> 00:44:12,160
Legend has it, one of his followers, a mathematician called Hippasus,

589
00:44:12,200 --> 00:44:15,240
set out to find the length of the diagonal

590
00:44:15,280 --> 00:44:19,520
for a right-angled triangle with two sides measuring one unit.

591
00:44:19,560 --> 00:44:25,280
Pythagoras' theorem implied that the length of the diagonal was a number whose square was two.

592
00:44:25,320 --> 00:44:29,320
The Pythagoreans assumed that the answer would be a fraction,

593
00:44:29,360 --> 00:44:35,760
but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it.

594
00:44:35,800 --> 00:44:38,360
Eventually he realised his mistake.

595
00:44:38,400 --> 00:44:43,080
It was the assumption that the value was a fraction at all which was wrong.

596
00:44:43,120 --> 00:44:49,200
The value of the square root of two was the number that the Babylonians etched into the Yale tablet.

597
00:44:49,240 --> 00:44:53,080
However, they didn't recognise the special character of this number.

598
00:44:53,120 --> 00:44:54,800
But Hippasus did.

599
00:44:54,840 --> 00:44:57,320
It was an irrational number.

600
00:45:00,680 --> 00:45:04,560
The discovery of this new number, and others like it, is akin to an explorer

601
00:45:04,600 --> 00:45:09,000
discovering a new continent, or a naturalist finding a new species.

602
00:45:09,040 --> 00:45:13,280
But these irrational numbers didn't fit the Pythagorean world view.

603
00:45:13,320 --> 00:45:18,880
Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy,

604
00:45:18,920 --> 00:45:21,600
but Hippasus let slip the discovery

605
00:45:21,640 --> 00:45:25,360
and was promptly drowned for his attempts to broadcast their research.

606
00:45:26,880 --> 00:45:32,200
But these mathematical discoveries could not be easily suppressed.

607
00:45:32,240 --> 00:45:37,680
Schools of philosophy and science started to flourish all over Greece, building on these foundations.

608
00:45:37,720 --> 00:45:42,120
The most famous of these was the Academy.

609
00:45:42,160 --> 00:45:47,320
Plato founded this school in Athens in 387 BC.

610
00:45:47,360 --> 00:45:53,800
Although we think of him today as a philosopher, he was one of mathematics' most important patrons.

611
00:45:53,840 --> 00:45:57,480
Plato was enraptured by the Pythagorean world view

612
00:45:57,520 --> 00:46:01,800
and considered mathematics the bedrock of knowledge.

613
00:46:01,840 --> 00:46:06,840
Some people would say that Plato is the most influential figure

614
00:46:06,880 --> 00:46:09,840
for our perception of Greek mathematics.

615
00:46:09,880 --> 00:46:14,880
He argued that mathematics is an important form of knowledge

616
00:46:14,920 --> 00:46:17,360
and does have a connection with reality.

617
00:46:17,400 --> 00:46:23,240
So by knowing mathematics, we know more about reality.

618
00:46:23,280 --> 00:46:29,000
In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking

619
00:46:29,040 --> 00:46:33,240
the secrets of the universe, a view still held by scientists today.

620
00:46:33,280 --> 00:46:37,240
Indeed, the importance Plato attached to geometry is encapsulated

621
00:46:37,280 --> 00:46:43,720
in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here."

622
00:46:47,320 --> 00:46:53,480
Plato proposed that the universe could be crystallised into five regular symmetrical shapes.

623
00:46:53,520 --> 00:46:56,400
These shapes, which we now call the Platonic solids,

624
00:46:56,440 --> 00:46:59,360
were composed of regular polygons, assembled to create

625
00:46:59,400 --> 00:47:02,840
three-dimensional symmetrical objects.

626
00:47:02,880 --> 00:47:05,480
The tetrahedron represented fire.

627
00:47:05,520 --> 00:47:09,720
The icosahedron, made from 20 triangles, represented water.

628
00:47:09,760 --> 00:47:11,920
The stable cube was Earth.

629
00:47:11,960 --> 00:47:15,640
The eight-faced octahedron was air.

630
00:47:15,680 --> 00:47:19,200
And the fifth Platonic solid, the dodecahedron,

631
00:47:19,240 --> 00:47:22,040
made out of 12 pentagons, was reserved for the shape

632
00:47:22,080 --> 00:47:25,760
that captured Plato's view of the universe.

633
00:47:29,400 --> 00:47:33,400
Plato's theory would have a seismic influence and continued to inspire

634
00:47:33,440 --> 00:47:37,160
mathematicians and astronomers for over 1,500 years.

635
00:47:38,160 --> 00:47:40,880
In addition to the breakthroughs made in the Academy,

636
00:47:40,920 --> 00:47:45,640
mathematical triumphs were also emerging from the edge of the Greek empire,

637
00:47:45,680 --> 00:47:51,280
and owed as much to the mathematical heritage of the Egyptians as the Greeks.

638
00:47:51,320 --> 00:47:57,760
Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,

639
00:47:57,800 --> 00:48:04,080
and its famous library soon gained a reputation to rival Plato's academy.

640
00:48:04,120 --> 00:48:11,520
The kings of Alexandria were prepared to invest in the arts and culture,

641
00:48:11,560 --> 00:48:14,720
in technology, mathematics, grammar,

642
00:48:14,760 --> 00:48:19,440
because patronage for cultural pursuits

643
00:48:19,480 --> 00:48:26,760
was one way of showing that you were a more prestigious ruler,

644
00:48:26,800 --> 00:48:30,080
and had a better entitlement to greatness.

645
00:48:31,840 --> 00:48:34,840
The old library and its precious contents were destroyed

646
00:48:34,880 --> 00:48:38,600
when the Muslims conquered Egypt in the 7th Century.

647
00:48:38,640 --> 00:48:41,440
But its spirit is alive in a new building.

648
00:48:43,480 --> 00:48:47,320
Today, the library remains a place of discovery and scholarship.

649
00:48:51,840 --> 00:48:55,120
Mathematicians and philosophers flocked to Alexandria,

650
00:48:55,160 --> 00:48:58,280
driven by their thirst for knowledge and the pursuit of excellence.

651
00:48:58,320 --> 00:49:02,240
The patrons of the library were the first professional scientists,

652
00:49:02,280 --> 00:49:05,800
individuals who were paid for their devotion to research.

653
00:49:05,840 --> 00:49:07,920
But of all those early pioneers,

654
00:49:07,960 --> 00:49:12,080
my hero is the enigmatic Greek mathematician Euclid.

655
00:49:15,800 --> 00:49:18,320
We know very little about Euclid's life,

656
00:49:18,360 --> 00:49:22,560
but his greatest achievements were as a chronicler of mathematics.

657
00:49:22,600 --> 00:49:27,800
Around 300 BC, he wrote the most important text book of all time -

658
00:49:27,840 --> 00:49:30,280
The Elements. In The Elements,

659
00:49:30,320 --> 00:49:34,040
we find the culmination of the mathematical revolution

660
00:49:34,080 --> 00:49:36,160
which had taken place in Greece.

661
00:49:38,120 --> 00:49:42,440
It's built on a series of mathematical assumptions, called axioms.

662
00:49:42,480 --> 00:49:47,200
For example, a line can be drawn between any two points.

663
00:49:47,240 --> 00:49:51,960
From these axioms, logical deductions are made and mathematical theorems established.

664
00:49:55,120 --> 00:49:59,560
The Elements contains formulas for calculating the volumes of cones

665
00:49:59,600 --> 00:50:02,600
and cylinders, proofs about geometric series,

666
00:50:02,640 --> 00:50:05,360
perfect numbers and primes.

667
00:50:05,400 --> 00:50:09,960
The climax of The Elements is a proof that there are only five Platonic solids.

668
00:50:12,800 --> 00:50:17,480
For me, this last theorem captures the power of mathematics.

669
00:50:17,520 --> 00:50:20,280
It's one thing to build five symmetrical solids,

670
00:50:20,320 --> 00:50:25,800
quite another to come up with a watertight, logical argument for why there can't be a sixth.

671
00:50:25,840 --> 00:50:29,800
The Elements unfolds like a wonderful, logical mystery novel.

672
00:50:29,840 --> 00:50:32,920
But this is a story which transcends time.

673
00:50:32,960 --> 00:50:36,760
Scientific theories get knocked down, from one generation to the next,

674
00:50:36,800 --> 00:50:43,120
but the theorems in The Elements are as true today as they were 2,000 years ago.

675
00:50:43,160 --> 00:50:46,680
When you stop and think about it, it's really amazing.

676
00:50:46,720 --> 00:50:48,360
It's the same theorems that we teach.

677
00:50:48,400 --> 00:50:53,160
We may teach them in a slightly different way, we may organise them differently,

678
00:50:53,200 --> 00:50:57,400
but it's Euclidean geometry that is still valid,

679
00:50:57,440 --> 00:51:01,520
and even in higher mathematics, when you go to higher dimensional spaces,

680
00:51:01,560 --> 00:51:03,760
you're still using Euclidean geometry.

681
00:51:05,320 --> 00:51:09,280
Alexandria must have been an inspiring place for the ancient scholars,

682
00:51:09,320 --> 00:51:15,560
and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port.

683
00:51:15,600 --> 00:51:21,880
One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.

684
00:51:22,880 --> 00:51:26,400
He would become a mathematical visionary.

685
00:51:26,440 --> 00:51:31,280
The best Greek mathematicians, they were always pushing the limits,

686
00:51:31,320 --> 00:51:32,760
pushing the envelope.

687
00:51:32,800 --> 00:51:35,400
So, Archimedes...

688
00:51:35,440 --> 00:51:38,400
did what he could with polygons,

689
00:51:38,440 --> 00:51:40,720
with solids.

690
00:51:40,760 --> 00:51:43,560
He then moved on to centres of gravity.

691
00:51:43,600 --> 00:51:47,880
He then moved on to the spiral.

692
00:51:47,920 --> 00:51:54,000
This instinct to try and mathematise everything

693
00:51:54,040 --> 00:51:57,640
is something that I see as a legacy.

694
00:51:58,760 --> 00:52:03,480
One of Archimedes' specialities was weapons of mass destruction.

695
00:52:03,520 --> 00:52:09,560
They were used against the Romans when they invaded his home of Syracuse in 212 BC.

696
00:52:09,600 --> 00:52:13,400
He also designed mirrors, which harnessed the power of the sun,

697
00:52:13,440 --> 00:52:15,960
to set the Roman ships on fire.

698
00:52:16,000 --> 00:52:20,720
But to Archimedes, these endeavours were mere amusements in geometry.

699
00:52:20,760 --> 00:52:23,480
He had loftier ambitions.

700
00:52:26,280 --> 00:52:32,760
Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake

701
00:52:32,800 --> 00:52:37,000
and not for the ignoble trade of engineering or the sordid quest for profit.

702
00:52:37,040 --> 00:52:41,040
One of his finest investigations into pure mathematics

703
00:52:41,080 --> 00:52:45,040
was to produce formulas to calculate the areas of regular shapes.

704
00:52:47,000 --> 00:52:52,680
Archimedes' method was to capture new shapes by using shapes he already understood.

705
00:52:52,720 --> 00:52:55,920
So, for example, to calculate the area of a circle,

706
00:52:55,960 --> 00:53:01,120
he would enclose it inside a triangle, and then by doubling the number of sides on the triangle,

707
00:53:01,160 --> 00:53:05,520
the enclosing shape would get closer and closer to the circle.

708
00:53:05,560 --> 00:53:07,560
Indeed, we sometimes call a circle

709
00:53:07,600 --> 00:53:10,560
a polygon with an infinite number of sides.

710
00:53:10,600 --> 00:53:14,400
But by estimating the area of the circle, Archimedes is, in fact,

711
00:53:14,440 --> 00:53:18,680
getting a value for pi, the most important number in mathematics.

712
00:53:19,720 --> 00:53:25,960
However, it was calculating the volumes of solid objects where Archimedes excelled.

713
00:53:26,000 --> 00:53:29,000
He found a way to calculate the volume of a sphere

714
00:53:29,040 --> 00:53:33,480
by slicing it up and approximating each slice as a cylinder.

715
00:53:33,520 --> 00:53:36,320
He then added up the volumes of the slices

716
00:53:36,360 --> 00:53:39,680
to get an approximate value for the sphere.

717
00:53:39,720 --> 00:53:42,640
But his act of genius was to see what happens

718
00:53:42,680 --> 00:53:45,480
if you make the slices thinner and thinner.

719
00:53:45,520 --> 00:53:50,240
In the limit, the approximation becomes an exact calculation.

720
00:53:54,320 --> 00:53:59,240
But it was Archimedes' commitment to mathematics that would be his undoing.

721
00:54:01,360 --> 00:54:06,160
Archimedes was contemplating a problem about circles traced in the sand.

722
00:54:06,200 --> 00:54:08,800
When a Roman soldier accosted him,

723
00:54:08,840 --> 00:54:14,840
Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.

724
00:54:14,880 --> 00:54:20,120
But the Roman soldier was not interested in Archimedes' problem and killed him on the spot.

725
00:54:20,160 --> 00:54:25,000
Even in death, Archimedes' devotion to mathematics was unwavering.

726
00:54:46,600 --> 00:54:49,680
By the middle of the 1st Century BC,

727
00:54:49,720 --> 00:54:53,720
the Romans had tightened their grip on the old Greek empire.

728
00:54:53,760 --> 00:54:56,520
They were less smitten with the beauty of mathematics

729
00:54:56,560 --> 00:54:59,840
and were more concerned with its practical applications.

730
00:54:59,880 --> 00:55:05,720
This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.

731
00:55:05,760 --> 00:55:09,960
But one mathematician was determined to keep the legacy of the Greeks alive.

732
00:55:10,000 --> 00:55:14,840
Hypatia was exceptional, a female mathematician,

733
00:55:14,880 --> 00:55:18,000
and a pagan in the piously Christian Roman empire.

734
00:55:19,920 --> 00:55:24,760
Hypatia was very prestigious and very influential in her time.

735
00:55:24,800 --> 00:55:30,640
She was a teacher with a lot of students, a lot of followers.

736
00:55:30,680 --> 00:55:34,880
She was politically influential in Alexandria.

737
00:55:34,920 --> 00:55:37,760
So it's this combination of...

738
00:55:37,800 --> 00:55:44,040
high knowledge and high prestige that may have made her

739
00:55:44,080 --> 00:55:47,600
a figure of hatred for...

740
00:55:47,640 --> 00:55:49,280
the Christian mob.

741
00:55:55,000 --> 00:55:59,000
One morning during Lent, Hypatia was dragged off her chariot

742
00:55:59,040 --> 00:56:03,040
by a zealous Christian mob and taken to a church.

743
00:56:03,080 --> 00:56:06,760
There, she was tortured and brutally murdered.

744
00:56:09,520 --> 00:56:12,440
The dramatic circumstances of her life and death

745
00:56:12,480 --> 00:56:15,200
fascinated later generations.

746
00:56:15,240 --> 00:56:20,880
Sadly, her cult status eclipsed her mathematical achievements.

747
00:56:20,920 --> 00:56:23,920
She was, in fact, a brilliant teacher and theorist,

748
00:56:23,960 --> 00:56:29,640
and her death dealt a final blow to the Greek mathematical heritage of Alexandria.

749
00:56:37,040 --> 00:56:40,880
My travels have taken me on a fascinating journey to uncover

750
00:56:40,920 --> 00:56:46,080
the passion and innovation of the world's earliest mathematicians.

751
00:56:46,120 --> 00:56:51,120
It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece

752
00:56:51,160 --> 00:56:55,520
that are the foundations on which my subject is built today.

753
00:56:55,560 --> 00:56:58,960
But this is just the beginning of my mathematical odyssey.

754
00:56:59,000 --> 00:57:02,600
The next leg of my journey lies east, in the depths of Asia,

755
00:57:02,640 --> 00:57:05,760
where mathematicians scaled even greater heights

756
00:57:05,800 --> 00:57:08,000
in pursuit of knowledge.

757
00:57:08,040 --> 00:57:11,920
With this new era came a new language of algebra and numbers,

758
00:57:11,960 --> 00:57:16,120
better suited to telling the next chapter in the story of maths.

759
00:57:17,600 --> 00:57:19,800
You can learn more about the story of maths

760
00:57:19,840 --> 00:57:23,040
with the Open University at...

761
00:57:39,280 --> 00:57:42,280
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