The Language of the Universe The Story of Maths


The Language of the Universe

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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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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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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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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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But I'm pretty sure that the Egyptians hadn't got

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this sweeping generalisation of their 3, 4, 5 triangle.

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We would not expect to find the general proof

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because this is not the style of Egyptian mathematics.

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Every problem was solved using concrete numbers and then

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if a verification would be carried out at the end, it would use the result

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and these concrete, given numbers,

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there's no general proof within the Egyptian mathematical texts.

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It would be some 2,000 years before the Greeks and Pythagoras

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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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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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For a culture like Egypt that is famous for its pyramids, you would expect problems like this

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to have been a regular feature within the mathematical texts.

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The calculation of the volume of a truncated pyramid is one of the most

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advanced bits, according to our modern standards of mathematics,

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that we have from ancient Egypt.

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The architects and engineers would certainly have wanted such a formula

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to calculate the amount of materials required to build it.

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But it's a mark of the sophistication

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of Egyptian mathematics that they were able to produce such a beautiful method.

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To understand how they derived their formula, start with a pyramid

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built such that the highest point sits directly over one corner.

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Three of these can be put together to make a rectangular box,

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so the volume of the skewed pyramid is a third the volume of the box.

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That is, the height, times the length, times the width, divided by three.

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Now comes an argument which shows the very first hints of the calculus at work,

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thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.

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Suppose you could cut the pyramid into slices, you could then slide

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the layers across to make the more symmetrical pyramid you see in Giza.

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However, the volume of the pyramid has not changed, despite the rearrangement of the layers.

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So the same formula works.

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The Egyptians were amazing innovators,

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and their ability to generate new mathematics was staggering.

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For me, they revealed the power of geometry and numbers, and made the first moves

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towards some of the exciting mathematical discoveries to come.

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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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This is Damascus, over 5,000 years old,

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and still vibrant and bustling today.

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It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.

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The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.

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In order to expand and run their empire, they became masters of managing and manipulating numbers.

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We have law codes for instance that tell us

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about the way society is ordered.

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The people we know most about are the scribes, the professionally literate

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and numerate people who kept the records for the wealthy families and for the temples and palaces.

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Scribe schools existed from around 2500BC.

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Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.

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Scribe records were kept on clay tablets,

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which allowed the Babylonians to manage and advance their empire.

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However, many of the tablets we have today aren't official documents, but children's exercises.

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It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.

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So, this is a geometrical textbook from about the 18th century BC.

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I hope you can see that there are lots of pictures on it.

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And underneath each picture is a text that sets a problem about the picture.

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So for instance this one here says, I drew a square, 60 units long,

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and inside it, I drew four circles - what are their areas?

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This little tablet here was written 1,000 years at least later than the tablet here,

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but has a very interesting relationship.

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It also has four circles on,

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in a square, roughly drawn, but this isn't a textbook, it's a school exercise.

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The adult scribe who's teaching the student is being given this

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as an example of completed homework or something like that.

0:23:210:23:25

Like the Egyptians, the Babylonians appeared interested

0:23:260:23:29

in solving practical problems to do with measuring and weighing.

0:23:290:23:32

The Babylonian solutions to these problems are written like mathematical recipes.

0:23:320:23:37

A scribe would simply follow and record a set of instructions to get a result.

0:23:370:23:43

Here's an example of the kind of problem they'd solve.

0:23:430:23:47

I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them.

0:23:470:23:51

Instead, I'm gonna take four times their weight and add them to the scales.

0:23:510:23:56

Now I'm gonna add 20 gin. Gin was the ancient Babylonian measure of weight.

0:23:580:24:04

I'm gonna take half of everything here and then add it again...

0:24:040:24:07

That's two bundles, and ten gin.

0:24:070:24:10

Everything on this side is equal to one mana. One mana was 60 gin.

0:24:100:24:16

And here, we have one of the first mathematical equations in history,

0:24:160:24:20

everything on this side is equal to one mana.

0:24:200:24:23

But how much does the bundle of cinnamon sticks weigh?

0:24:230:24:26

Without any algebraic language, they were able to manipulate

0:24:260:24:29

the quantities to be able to prove that the cinnamon sticks weighed five gin.

0:24:290:24:35

In my mind, it's this kind of problem which gives mathematics a bit of a bad name.

0:24:350:24:40

You can blame those ancient Babylonians for all those tortuous problems you had at school.

0:24:400:24:45

But the ancient Babylonian scribes excelled at this kind of problem.

0:24:450:24:50

Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60.

0:24:500:24:57

The Babylonians invented their number system, like the Egyptians, by using their fingers.

0:25:000:25:05

But instead of counting through the 10 fingers on their hand,

0:25:050:25:08

Babylonians found a more intriguing way to count body parts.

0:25:080:25:11

They used the 12 knuckles on one hand,

0:25:110:25:14

and the five fingers on the other to be able to count

0:25:140:25:16

12 times 5, ie 60 different numbers.

0:25:160:25:20

So for example, this number would have been 2 lots of 12, 24,

0:25:200:25:25

and then, 1, 2, 3, 4, 5, to make 29.

0:25:250:25:29

The number 60 had another powerful property.

0:25:320:25:35

It can be perfectly divided in a multitude of ways.

0:25:350:25:39

Here are 60 beans.

0:25:390:25:41

I can arrange them in 2 rows of 30.

0:25:410:25:44

3 rows of 20.

0:25:480:25:51

4 rows of 15.

0:25:510:25:53

5 rows of 12.

0:25:530:25:56

Or 6 rows of 10.

0:25:560:25:59

The divisibility of 60 makes it a perfect base in which to do arithmetic.

0:25:590:26:04

The base 60 system was so successful, we still use elements of it today.

0:26:040:26:11

Every time we want to tell the time, we recognise units of 60 -

0:26:110:26:15

60 seconds in a minute, 60 minutes in an hour.

0:26:150:26:19

But the most important feature of the Babylonians' number system was that it recognised place value.

0:26:190:26:24

Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording,

0:26:240:26:30

the position of each Babylonian number records the power of 60.

0:26:300:26:34

Instead of inventing new symbols for bigger and bigger numbers,

0:26:410:26:44

they would write 1-1-1, so this number would be 3,661.

0:26:440:26:50

The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky.

0:26:540:26:59

The Babylonians' calendar was based on the cycles of the moon.

0:27:070:27:10

They needed a way of recording astronomically large numbers.

0:27:100:27:15

Month by month, year by year, these cycles were recorded.

0:27:150:27:19

From about 800BC, there were complete lists of lunar eclipses.

0:27:190:27:25

The Babylonian system of measurement was quite sophisticated at that time.

0:27:250:27:30

They had a system of angular measurement,

0:27:300:27:32

360 degrees in a full circle, each degree was divided

0:27:320:27:36

into 60 minutes, a minute was further divided into 60 seconds.

0:27:360:27:41

So they had a regular system for measurement, and it was in perfect harmony with their number system,

0:27:410:27:48

so it's well suited not only for observation but also for calculation.

0:27:480:27:52

But in order to calculate and cope with these large numbers,

0:27:520:27:56

the Babylonians needed to invent a new symbol.

0:27:560:28:00

And in so doing, they prepared the ground for one of the great

0:28:000:28:03

breakthroughs in the history of mathematics - zero.

0:28:030:28:06

In the early days, the Babylonians, in order to mark an empty place in

0:28:060:28:11

the middle of a number, would simply leave a blank space.

0:28:110:28:14

So they needed a way of representing nothing in the middle of a number.

0:28:140:28:19

So they used a sign, as a sort of breathing marker, a punctuation mark,

0:28:190:28:25

and it comes to mean zero in the middle of a number.

0:28:250:28:28

This was the first time zero, in any form,

0:28:280:28:31

had appeared in the mathematical universe.

0:28:310:28:35

But it would be over a 1,000 years before this little place holder would become a number in its own right.

0:28:350:28:42

Having established such a sophisticated system of numbers,

0:28:500:28:53

they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.

0:28:530:28:59

Babylonian engineers and surveyors found ingenious ways of

0:29:020:29:06

accessing water, and channelling it to the crop fields.

0:29:060:29:10

Yet again, they used mathematics to come up with solutions.

0:29:100:29:15

The Orontes valley in Syria is still an agricultural hub,

0:29:150:29:19

and the old methods of irrigation are being exploited today, just as they were thousands of years ago.

0:29:190:29:26

Many of the problems in Babylonian mathematics

0:29:260:29:29

are concerned with measuring land, and it's here we see for the first time

0:29:290:29:34

the use of quadratic equations, one of the greatest legacies of Babylonian mathematics.

0:29:340:29:39

Quadratic equations involve things where the unknown quantity

0:29:390:29:43

you're trying to identify is multiplied by itself.

0:29:430:29:46

We call this squaring because it gives the area of a square,

0:29:460:29:49

and it's in the context of calculating the area of land

0:29:490:29:53

that these quadratic equations naturally arise.

0:29:530:29:55

Here's a typical problem.

0:30:010:30:03

If a field has an area of 55 units

0:30:030:30:06

and one side is six units longer than the other,

0:30:060:30:10

how long is the shorter side?

0:30:100:30:12

The Babylonian solution was to reconfigure the field as a square.

0:30:140:30:18

Cut three units off the end

0:30:180:30:21

and move this round.

0:30:210:30:24

Now, there's a three-by-three piece missing, so let's add this in.

0:30:240:30:29

The area of the field has increased by nine units.

0:30:290:30:34

This makes the new area 64.

0:30:340:30:38

So the sides of the square are eight units.

0:30:380:30:41

The problem-solver knows that they've added three to this side.

0:30:410:30:45

So, the original length must be five.

0:30:450:30:49

It may not look like it, but this is one of the first quadratic equations in history.

0:30:500:30:55

In modern mathematics, I would use the symbolic language of algebra to solve this problem.

0:30:570:31:02

The amazing feat of the Babylonians is that they were using these geometric games to find the value,

0:31:020:31:07

without any recourse to symbols or formulas.

0:31:070:31:10

The Babylonians were enjoying problem-solving for its own sake.

0:31:100:31:13

They were falling in love with mathematics.

0:31:130:31:17

The Babylonians' fascination with numbers soon found a place in their leisure time, too.

0:31:290:31:34

They were avid game-players.

0:31:340:31:35

The Babylonians and their descendants have been playing

0:31:350:31:38

a version of backgammon for over 5,000 years.

0:31:380:31:43

The Babylonians played board games,

0:31:430:31:45

from very posh board games in royal tombs to little bits of board games found in schools,

0:31:450:31:52

to board games scratched on the entrances of palaces,

0:31:520:31:56

so that the guardsmen must have played when they were bored,

0:31:560:32:00

and they used dice to move their counters round.

0:32:000:32:03

People who played games were using numbers in their leisure time to try and outwit their opponent,

0:32:040:32:09

doing mental arithmetic very fast,

0:32:090:32:12

and so they were calculating in their leisure time,

0:32:120:32:17

without even thinking about it as being mathematical hard work.

0:32:170:32:21

Now's my chance.

0:32:230:32:24

'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.'

0:32:240:32:30

-It's up to you.

-Six... I need to move something.

0:32:300:32:33

'But it wasn't as easy as I thought.'

0:32:330:32:36

Ah! What the hell was that?

0:32:360:32:38

-Yeah.

-This is one, this is two.

0:32:380:32:42

Now you're in trouble.

0:32:420:32:44

-So I can't move anything.

-You cannot move these.

0:32:440:32:47

Oh, gosh.

0:32:470:32:49

There you go.

0:32:500:32:52

Three and four.

0:32:530:32:54

'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.'

0:32:540:33:00

Yeah.

0:33:000:33:02

Put it there. Good game.

0:33:030:33:05

The Babylonians are recognised as one of the first cultures

0:33:070:33:10

to use symmetrical mathematical shapes to make dice,

0:33:100:33:13

but there is more heated debate about whether they might also

0:33:130:33:17

have been the first to discover the secrets of another important shape.

0:33:170:33:20

The right-angled triangle.

0:33:200:33:24

We've already seen how the Egyptians use a 3-4-5 right-angled triangle.

0:33:270:33:32

But what the Babylonians knew about this shape and others like it is much more sophisticated.

0:33:320:33:37

This is the most famous and controversial ancient tablet we have.

0:33:370:33:42

It's called Plimpton 322.

0:33:420:33:44

Many mathematicians are convinced it shows the Babylonians

0:33:450:33:49

could well have known the principle regarding right-angled triangles,

0:33:490:33:53

that the square on the diagonal is the sum of the squares on the sides,

0:33:530:33:57

and known it centuries before the Greeks claimed it.

0:33:570:34:00

This is a copy of arguably the most famous Babylonian tablet,

0:34:010:34:06

which is Plimpton 322,

0:34:060:34:08

and these numbers here reflect the width or height of a triangle,

0:34:080:34:12

this being the diagonal, the other side would be over here,

0:34:120:34:17

and the square of this column

0:34:170:34:19

plus the square of the number in this column

0:34:190:34:23

equals the square of the diagonal.

0:34:230:34:26

They are arranged in an order of steadily decreasing angle,

0:34:260:34:31

on a very uniform basis, showing that somebody

0:34:310:34:34

had a lot of understanding of how the numbers fit together.

0:34:340:34:38

Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.

0:34:440:34:50

It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem,

0:34:500:34:56

and it's a conclusion that generations of historians have been seduced by.

0:34:560:35:01

But there could be a much simpler explanation

0:35:010:35:03

for the sets of three numbers which fulfil Pythagoras' theorem.

0:35:030:35:07

It's not a systematic explanation of Pythagorean triples, it's simply

0:35:070:35:12

a mathematics teacher doing some quite complicated calculations,

0:35:120:35:17

but in order to produce some very simple numbers,

0:35:170:35:21

in order to set his students problems about right-angled triangles,

0:35:210:35:26

and in that sense it's about Pythagorean triples only incidentally.

0:35:260:35:31

The most valuable clues to what they understood could lie elsewhere.

0:35:330:35:39

This small school exercise tablet is nearly 4,000 years old

0:35:390:35:43

and reveals just what the Babylonians did know about right-angled triangles.

0:35:430:35:48

It uses a principle of Pythagoras' theorem to find the value of an astounding new number.

0:35:480:35:54

Drawn along the diagonal is a really very good approximation to the square root of two,

0:35:570:36:05

and so that shows us that it was known and used in school environments.

0:36:050:36:10

Why's this important?

0:36:100:36:12

Because the square root of two is what we now call an irrational number,

0:36:120:36:18

that is, if we write it out in decimals, or even in sexigesimal places,

0:36:180:36:23

it doesn't end, the numbers go on forever after the decimal point.

0:36:230:36:28

The implications of this calculation are far-reaching.

0:36:290:36:33

Firstly, it means the Babylonians knew something of Pythagoras' theorem

0:36:330:36:37

1,000 years before Pythagoras.

0:36:370:36:39

Secondly, the fact that they can calculate this number to an accuracy of four decimal places

0:36:390:36:45

shows an amazing arithmetic facility, as well as a passion for mathematical detail.

0:36:450:36:50

The Babylonians' mathematical dexterity was astounding,

0:36:520:36:56

and for nearly 2,000 years they spearheaded intellectual progress in the ancient world.

0:36:560:37:03

But when their imperial power began to wane, so did their intellectual vigour.

0:37:030:37:08

By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.

0:37:160:37:23

This is Palmyra in central Syria, a once-great city built by the Greeks.

0:37:250:37:31

The mathematical expertise needed to build structures with such geometric perfection is impressive.

0:37:330:37:41

Just like the Babylonians before them, the Greeks were passionate about mathematics.

0:37:420:37:48

The Greeks were clever colonists.

0:37:500:37:53

They took the best from the civilisations they invaded

0:37:530:37:56

to advance their own power and influence,

0:37:560:37:58

but they were soon making contributions themselves.

0:37:580:38:01

In my opinion, their greatest innovation was to do with a shift in the mind.

0:38:010:38:07

What they initiated would influence humanity for centuries.

0:38:070:38:11

They gave us the power of proof.

0:38:110:38:14

Somehow they decided that they had to have a deductive system

0:38:140:38:18

for their mathematics

0:38:180:38:19

and the typical deductive system

0:38:190:38:21

was to begin with certain axioms, which you assume are true.

0:38:210:38:25

It's as if you assume a certain theorem is true without proving it.

0:38:250:38:29

And then, using logical methods and very careful steps,

0:38:290:38:34

from these axioms you prove theorems

0:38:340:38:37

and from those theorems you prove more theorems, and it just snowballs.

0:38:370:38:42

Proof is what gives mathematics its strength.

0:38:430:38:47

It's the power of proof which means that the discoveries of the Greeks

0:38:470:38:51

are as true today as they were 2,000 years ago.

0:38:510:38:55

I needed to head west into the heart of the old Greek empire to learn more.

0:38:550:39:01

For me, Greek mathematics has always been heroic and romantic.

0:39:080:39:14

I'm on my way to Samos, less than a mile from the Turkish coast.

0:39:150:39:20

This place has become synonymous with the birth of Greek mathematics,

0:39:200:39:25

and it's down to the legend of one man.

0:39:250:39:27

His name is Pythagoras.

0:39:310:39:33

The legends that surround his life and work have contributed

0:39:330:39:36

to the celebrity status he has gained over the last 2,000 years.

0:39:360:39:40

He's credited, rightly or wrongly, with beginning the transformation

0:39:400:39:44

from mathematics as a tool for accounting to the analytic subject we recognise today.

0:39:440:39:50

Pythagoras is a controversial figure.

0:39:540:39:57

Because he left no mathematical writings, many have questioned

0:39:570:40:00

whether he indeed solved any of the theorems attributed to him.

0:40:000:40:04

He founded a school in Samos in the sixth century BC,

0:40:040:40:07

but his teachings were considered suspect and the Pythagoreans a bizarre sect.

0:40:070:40:13

There is good evidence that there were schools of Pythagoreans,

0:40:140:40:19

and they may have looked more like sects

0:40:190:40:22

than what we associate with philosophical schools,

0:40:220:40:25

because they didn't just share knowledge, they also shared a way of life.

0:40:250:40:30

There may have been communal living and they all seemed to have been

0:40:300:40:36

involved in the politics of their cities.

0:40:360:40:40

One feature that makes them unusual in the ancient world is that they included women.

0:40:400:40:45

But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians -

0:40:460:40:52

the properties of right-angled triangles.

0:40:520:40:56

What's known as Pythagoras' theorem

0:40:560:40:58

states that if you take any right-angled triangle,

0:40:580:41:01

build squares on all the sides, then the area of the largest square

0:41:010:41:05

is equal to the sum of the squares on the two smaller sides.

0:41:050:41:09

It's at this point for me that mathematics is born

0:41:130:41:16

and a gulf opens up between the other sciences,

0:41:160:41:19

and the proof is as simple as it is devastating in its implications.

0:41:190:41:24

Place four copies of the right-angled triangle

0:41:240:41:28

on top of this surface.

0:41:280:41:29

The square that you now see

0:41:290:41:31

has sides equal to the hypotenuse of the triangle.

0:41:310:41:35

By sliding these triangles around,

0:41:350:41:37

we see how we can break the area of the large square up

0:41:370:41:40

into the sum of two smaller squares,

0:41:400:41:43

whose sides are given by the two short sides of the triangle.

0:41:430:41:47

In other words, the square on the hypotenuse is equal to the sum

0:41:470:41:52

of the squares on the other sides. Pythagoras' theorem.

0:41:520:41:55

It illustrates one of the characteristic themes of Greek mathematics -

0:41:580:42:02

the appeal to beautiful arguments in geometry rather than a reliance on number.

0:42:020:42:07

Pythagoras may have fallen out of favour and many of the discoveries accredited to him

0:42:110:42:16

have been contested recently, but there's one mathematical theory that I'm loath to take away from him.

0:42:160:42:21

It's to do with music and the discovery of the harmonic series.

0:42:210:42:25

The story goes that, walking past a blacksmith's one day,

0:42:270:42:31

Pythagoras heard anvils being struck,

0:42:310:42:33

and noticed how the notes being produced sounded in perfect harmony.

0:42:330:42:38

He believed that there must be some rational explanation

0:42:380:42:42

to make sense of why the notes sounded so appealing.

0:42:420:42:46

The answer was mathematics.

0:42:460:42:48

Experimenting with a stringed instrument, Pythagoras discovered that the intervals between

0:42:530:42:58

harmonious musical notes were always represented as whole-number ratios.

0:42:580:43:02

And here's how he might have constructed his theory.

0:43:050:43:08

First, play a note on the open string.

0:43:100:43:13

MAN PLAYS NOTE

0:43:130:43:15

Next, take half the length.

0:43:150:43:17

The note almost sounds the same as the first note.

0:43:180:43:22

In fact it's an octave higher, but the relationship is so strong, we give these notes the same name.

0:43:220:43:27

Now take a third the length.

0:43:270:43:28

We get another note which sounds harmonious next to the first two,

0:43:310:43:35

but take a length of string which is not in a whole-number ratio and all we get is dissonance.

0:43:350:43:41

According to legend, Pythagoras was so excited by this discovery

0:43:460:43:51

that he concluded the whole universe was built from numbers.

0:43:510:43:54

But he and his followers were in for a rather unsettling challenge to their world view

0:43:540:44:00

and it came about as a result of the theorem which bears Pythagoras' name.

0:44:000:44:05

Legend has it, one of his followers, a mathematician called Hippasus,

0:44:070:44:12

set out to find the length of the diagonal

0:44:120:44:15

for a right-angled triangle with two sides measuring one unit.

0:44:150:44:19

Pythagoras' theorem implied that the length of the diagonal was a number whose square was two.

0:44:190:44:25

The Pythagoreans assumed that the answer would be a fraction,

0:44:250:44:29

but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it.

0:44:290:44:36

Eventually he realised his mistake.

0:44:360:44:38

It was the assumption that the value was a fraction at all which was wrong.

0:44:380:44:43

The value of the square root of two was the number that the Babylonians etched into the Yale tablet.

0:44:430:44:49

However, they didn't recognise the special character of this number.

0:44:490:44:53

But Hippasus did.

0:44:530:44:55

It was an irrational number.

0:44:550:44:57

The discovery of this new number, and others like it, is akin to an explorer

0:45:000:45:04

discovering a new continent, or a naturalist finding a new species.

0:45:040:45:09

But these irrational numbers didn't fit the Pythagorean world view.

0:45:090:45:13

Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy,

0:45:130:45:19

but Hippasus let slip the discovery

0:45:190:45:21

and was promptly drowned for his attempts to broadcast their research.

0:45:210:45:25

But these mathematical discoveries could not be easily suppressed.

0:45:270:45:32

Schools of philosophy and science started to flourish all over Greece, building on these foundations.

0:45:320:45:37

The most famous of these was the Academy.

0:45:370:45:42

Plato founded this school in Athens in 387 BC.

0:45:420:45:47

Although we think of him today as a philosopher, he was one of mathematics' most important patrons.

0:45:470:45:54

Plato was enraptured by the Pythagorean world view

0:45:540:45:57

and considered mathematics the bedrock of knowledge.

0:45:570:46:02

Some people would say that Plato is the most influential figure

0:46:020:46:07

for our perception of Greek mathematics.

0:46:070:46:10

He argued that mathematics is an important form of knowledge

0:46:100:46:15

and does have a connection with reality.

0:46:150:46:17

So by knowing mathematics, we know more about reality.

0:46:170:46:23

In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking

0:46:230:46:29

the secrets of the universe, a view still held by scientists today.

0:46:290:46:33

Indeed, the importance Plato attached to geometry is encapsulated

0:46:330:46:37

in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here."

0:46:370:46:43

Plato proposed that the universe could be crystallised into five regular symmetrical shapes.

0:46:470:46:53

These shapes, which we now call the Platonic solids,

0:46:530:46:56

were composed of regular polygons, assembled to create

0:46:560:46:59

three-dimensional symmetrical objects.

0:46:590:47:03

The tetrahedron represented fire.

0:47:030:47:05

The icosahedron, made from 20 triangles, represented water.

0:47:050:47:09

The stable cube was Earth.

0:47:090:47:12

The eight-faced octahedron was air.

0:47:120:47:15

And the fifth Platonic solid, the dodecahedron,

0:47:150:47:19

made out of 12 pentagons, was reserved for the shape

0:47:190:47:22

that captured Plato's view of the universe.

0:47:220:47:26

Plato's theory would have a seismic influence and continued to inspire

0:47:290:47:33

mathematicians and astronomers for over 1,500 years.

0:47:330:47:37

In addition to the breakthroughs made in the Academy,

0:47:380:47:41

mathematical triumphs were also emerging from the edge of the Greek empire,

0:47:410:47:45

and owed as much to the mathematical heritage of the Egyptians as the Greeks.

0:47:450:47:51

Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,

0:47:510:47:58

and its famous library soon gained a reputation to rival Plato's Academy.

0:47:580:48:04

The kings of Alexandria were prepared to invest in the arts and culture,

0:48:040:48:11

in technology, mathematics, grammar,

0:48:110:48:14

because patronage for cultural pursuits

0:48:140:48:19

was one way of showing that you were a more prestigious ruler,

0:48:190:48:27

and had a better entitlement to greatness.

0:48:270:48:30

The old library and its precious contents were destroyed

0:48:320:48:35

But its spirit is alive in a new building.

0:48:350:48:38

Today, the library remains a place of discovery and scholarship.

0:48:400:48:44

Mathematicians and philosophers flocked to Alexandria,

0:48:480:48:51

driven by their thirst for knowledge and the pursuit of excellence.

0:48:510:48:55

The patrons of the library were the first professional scientists,

0:48:550:48:59

individuals who were paid for their devotion to research.

0:48:590:49:02

But of all those early pioneers,

0:49:020:49:04

my hero is the enigmatic Greek mathematician Euclid.

0:49:040:49:08

We know very little about Euclid's life,

0:49:120:49:15

but his greatest achievements were as a chronicler of mathematics.

0:49:150:49:19

Around 300 BC, he wrote the most important text book of all time -

0:49:190:49:24

The Elements. In The Elements,

0:49:240:49:27

we find the culmination of the mathematical revolution

0:49:270:49:31

which had taken place in Greece.

0:49:310:49:32

It's built on a series of mathematical assumptions, called axioms.

0:49:340:49:39

For example, a line can be drawn between any two points.

0:49:390:49:44

From these axioms, logical deductions are made and mathematical theorems established.

0:49:440:49:48

The Elements contains formulas for calculating the volumes of cones

0:49:510:49:56

and cylinders, proofs about geometric series,

0:49:560:49:59

perfect numbers and primes.

0:49:590:50:02

The climax of The Elements is a proof that there are only five Platonic solids.

0:50:020:50:06

For me, this last theorem captures the power of mathematics.

0:50:090:50:14

It's one thing to build five symmetrical solids,

0:50:140:50:17

quite another to come up with a watertight, logical argument for why there can't be a sixth.

0:50:170:50:22

The Elements unfolds like a wonderful, logical mystery novel.

0:50:220:50:26

But this is a story which transcends time.

0:50:260:50:29

Scientific theories get knocked down, from one generation to the next,

0:50:290:50:33

but the theorems in The Elements are as true today as they were 2,000 years ago.

0:50:330:50:39

When you stop and think about it, it's really amazing.

0:50:390:50:43

It's the same theorems that we teach.

0:50:430:50:45

We may teach them in a slightly different way, we may organise them differently,

0:50:450:50:49

but it's Euclidean geometry that is still valid,

0:50:490:50:54

and even in higher mathematics, when you go to higher dimensional spaces,

0:50:540:50:58

you're still using Euclidean geometry.

0:50:580:51:00

Alexandria must have been an inspiring place for the ancient scholars,

0:51:020:51:06

and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port.

0:51:060:51:12

One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.

0:51:120:51:18

He would become a mathematical visionary.

0:51:190:51:23

The best Greek mathematicians, they were always pushing the limits,

0:51:230:51:28

pushing the envelope.

0:51:280:51:29

So, Archimedes...

0:51:290:51:32

did what he could with polygons,

0:51:320:51:35

with solids.

0:51:350:51:37

He then moved on to centres of gravity.

0:51:370:51:40

He then moved on to the spiral.

0:51:400:51:44

This instinct to try and mathematise everything

0:51:440:51:50

is something that I see as a legacy.

0:51:500:51:54

One of Archimedes' specialities was weapons of mass destruction.

0:51:550:52:00

They were used against the Romans when they invaded his home of Syracuse in 212 BC.

0:52:000:52:06

He also designed mirrors, which harnessed the power of the sun,

0:52:060:52:10

to set the Roman ships on fire.

0:52:100:52:12

But to Archimedes, these endeavours were mere amusements in geometry.

0:52:120:52:17

He had loftier ambitions.

0:52:170:52:20

Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake

0:52:230:52:29

and not for the ignoble trade of engineering or the sordid quest for profit.

0:52:290:52:33

One of his finest investigations into pure mathematics

0:52:330:52:37

was to produce formulas to calculate the areas of regular shapes.

0:52:370:52:41

Archimedes' method was to capture new shapes by using shapes he already understood.

0:52:430:52:49

So, for example, to calculate the area of a circle,

0:52:490:52:52

he would enclose it inside a triangle, and then by doubling the number of sides on the triangle,

0:52:520:52:57

the enclosing shape would get closer and closer to the circle.

0:52:570:53:02

Indeed, we sometimes call a circle

0:53:020:53:04

a polygon with an infinite number of sides.

0:53:040:53:07

But by estimating the area of the circle, Archimedes is, in fact,

0:53:070:53:11

getting a value for pi, the most important number in mathematics.

0:53:110:53:15

However, it was calculating the volumes of solid objects where Archimedes excelled.

0:53:160:53:22

He found a way to calculate the volume of a sphere

0:53:220:53:25

by slicing it up and approximating each slice as a cylinder.

0:53:250:53:30

He then added up the volumes of the slices

0:53:300:53:33

to get an approximate value for the sphere.

0:53:330:53:36

But his act of genius was to see what happens

0:53:360:53:39

if you make the slices thinner and thinner.

0:53:390:53:42

In the limit, the approximation becomes an exact calculation.

0:53:420:53:47

But it was Archimedes' commitment to mathematics that would be his undoing.

0:53:510:53:56

Archimedes was contemplating a problem about circles traced in the sand.

0:53:580:54:02

When a Roman soldier accosted him,

0:54:020:54:05

Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.

0:54:050:54:11

But the Roman soldier was not interested in Archimedes' problem and killed him on the spot.

0:54:110:54:16

Even in death, Archimedes' devotion to mathematics was unwavering.

0:54:160:54:21

By the middle of the 1st century BC,

0:54:430:54:46

the Romans had tightened their grip on the old Greek empire.

0:54:460:54:50

They were less smitten with the beauty of mathematics

0:54:500:54:53

and were more concerned with its practical applications.

0:54:530:54:56

This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.

0:54:560:55:02

But one mathematician was determined to keep the legacy of the Greeks alive.

0:55:020:55:06

Hypatia was exceptional, a female mathematician,

0:55:060:55:11

and a pagan in the piously Christian Roman empire.

0:55:110:55:14

Hypatia was very prestigious and very influential in her time.

0:55:160:55:21

She was a teacher with a lot of students, a lot of followers.

0:55:210:55:27

She was politically influential in Alexandria.

0:55:270:55:31

So it's this combination of...

0:55:310:55:34

high knowledge and high prestige that may have made her

0:55:340:55:40

a figure of hatred for...

0:55:400:55:44

the Christian mob.

0:55:440:55:46

One morning during Lent, Hypatia was dragged off her chariot

0:55:510:55:55

by a zealous Christian mob and taken to a church.

0:55:550:55:59

There, she was tortured and brutally murdered.

0:55:590:56:03

The dramatic circumstances of her life and death

0:56:060:56:09

fascinated later generations.

0:56:090:56:12

Sadly, her cult status eclipsed her mathematical achievements.

0:56:120:56:17

She was, in fact, a brilliant teacher and theorist,

0:56:170:56:20

and her death dealt a final blow to the Greek mathematical heritage of Alexandria.

0:56:200:56:26

My travels have taken me on a fascinating journey to uncover

0:56:330:56:37

the passion and innovation of the world's earliest mathematicians.

0:56:370:56:42

It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece

0:56:420:56:47

that are the foundations on which my subject is built today.

0:56:470:56:52

But this is just the beginning of my mathematical odyssey.

0:56:520:56:55

The next leg of my journey lies east, in the depths of Asia,

0:56:550:56:59

where mathematicians scaled even greater heights

0:56:590:57:02

in pursuit of knowledge.

0:57:020:57:04

With this new era came a new language of algebra and numbers,

0:57:040:57:08

better suited to telling the next chapter in the story of maths.

0:57:080:57:12

You can learn more about the story of maths

0:57:120:57:16

with the Open University at...

0:57:160:57:19

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0:57:360:57:39

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