The History of Mathematics. Documentary
MIK2023-03-15
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💫 Short Summary
Ancient Egyptians and Babylonians made significant mathematical advancements in areas such as land surveying, number systems, geometry, and algebra. Greeks introduced deductive proof systems and explored geometric arguments, while the Chinese developed decimal systems and mathematical methods for solving equations. Indian mathematicians revolutionized the concept of zero and negative numbers, making fundamental discoveries in trigonometry and infinite series. Islamic scholars played a crucial role in preserving and advancing mathematical knowledge, introducing algebra and problem-solving methods. Europeans were influenced by Fibonacci and Hindu-Arabic numerals, leading to the spread of a new number system and advancements in solving cubic equations.
✨ Highlights
📊 Transcript
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Ancient Egyptians used mathematics for land management, predicting crop yields, and administering taxes.
03:44Body measurements like palms and cubits were used for land surveying.
Bureaucratic needs led to the evolution of the number system.
Egyptians used a decimal system represented by hieroglyphs but lacked place value understanding.
Early mathematical innovations were important for societal organization and record-keeping.
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The Egyptians were skilled problem solvers in mathematics, as demonstrated through the Rhind Mathematical Papyrus.
06:44They utilized papyrus to document methods for multiplication and division, showcasing their advanced mathematical knowledge.
The papyrus also included practical problems involving fractions, highlighting the Egyptians' use of fractions in trade.
The Eye of Horus hieroglyph was used to symbolize fractions, with each part representing a different fraction.
This symbol hinted at geometric and infinite series, concepts further explored in the Rhind Papyrus.
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Egyptian Advancements in Geometry
11:48Egyptians excelled in calculating the area of a circle and the value of pi by using smaller shapes to approximate larger shapes.
The pyramids, symbolizing Egyptian mathematics, displayed symmetry and potentially the golden ratio.
Egyptians employed a simple form of Pythagoras' theorem to achieve perfect right angles.
Their problem-solving method focused on concrete numbers rather than abstract proofs, demonstrating their distinct approach to mathematics.
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Contributions of Egyptians and Babylonians to Mathematics.
17:40Egyptians showed early calculus concepts in pyramid volume formulas.
Babylonians excelled in numbers and used math for practical problem-solving.
They utilized written instructions for tasks like measuring and weighing.
Despite the absence of algebraic language, they effectively manipulated quantities, laying the foundation for future mathematical advancements.
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The Babylonians developed a base-60 number system that included place value and zero, crucial for large number calculations.
24:20This system was utilized for applications in astronomy and engineering, such as irrigation projects.
Babylonian mathematics also involved quadratic equations, particularly in land measurement problems.
An example of their problem-solving approach was finding the length of a field side using quadratic equations.
The legacy of Babylonian mathematics includes the use of quadratic equations for calculating areas, showcasing their innovative solutions and practical applications in various fields.
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The Babylonians were early mathematicians who made significant advancements in geometry and number theory.
31:12They potentially understood right-angled triangles and Pythagoras' theorem before the Greeks.
The Plimpton 322 tablet is a famous example of their advanced mathematical knowledge, showcasing Pythagorean triples.
Babylonians also made significant advancements in calculating the square root of two, an irrational number.
Their mathematical prowess suggests they had a deep understanding of various mathematical concepts, possibly predating famous mathematicians like Pythagoras.
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Pythagoras' contributions to Greek mathematics and music theory.
39:01Pythagoras introduced deductive proof systems and his famous theorem on right-angled triangles.
Despite some controversy, his impact on mathematics and music theory is significant.
He discovered the harmonic series through whole-number ratios in musical intervals, showcasing the rational explanations provided by mathematics.
Greek mathematicians emphasized beautiful geometric arguments over numerical reliance, influencing mathematical thinking for centuries.
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Development of Mathematics in Ancient Greece.
43:36The discovery of irrational numbers led to the growth of schools like the Academy.
Plato highlighted the significance of mathematics in comprehending reality and introduced Platonic solids.
Alexandria, under the Ptolemies, became a hub for arts, culture, and mathematics.
The library of Alexandria, destroyed in the 7th century, is still a symbol of academic achievement.
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Contributions of Greek mathematicians in ancient times.
50:46Euclid's "The Elements" written around 300 BC contained enduring mathematical assumptions and theorems.
Archimedes specialized in pure mathematics, calculating areas and volumes of shapes, and used mirrors to set Roman ships on fire.
Archimedes' dedication to mathematics ultimately led to his tragic death.
The Roman's practical approach to mathematics contributed to the decline of the library of Alexandria, but mathematician Hypatia worked to safeguard the Greek legacy.
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Hypatia, a prominent figure in Alexandria, was tragically murdered by a Christian mob during Lent, leading to her cult status overshadowing her mathematical achievements.
54:18The ancient Chinese had a decimal place-value system using counting rods for calculations over 1,000 years before the West, but they lacked the concept of zero.
Despite the absence of zero, the Chinese made significant mathematical advancements and viewed numbers as mystical, with odd numbers symbolizing male and even numbers symbolizing female.
Chinese culture explored patterns like the magic square in mathematics, showcasing their unique approach to numerical concepts and calculations.
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Mathematics played a crucial role in Ancient China's governance, including managing the court, calendar, and harem.
59:53Mathematicians used geometric progression to schedule the emperor's interactions with his female companions efficiently.
Math was essential for taxation, legal codes, and standardized systems in the state.
The Nine Chapters, a mathematical textbook from 200BC, trained civil servants in practical problem-solving.
Equations in Ancient China were compared to cryptic crosswords, requiring deduction of unknown numbers through practical examples.
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Development of Mathematical Methods in Ancient China
01:05:26Ancient Chinese developed mathematical methods for solving equations centuries before the West, with Carl Friedrich Gauss rediscovering these methods in the 19th century.
Chinese Remainder Theorem
The Chinese remainder theorem involved solving equations with unknown numbers divided by given numbers, with practical examples like calculating the number of eggs in a tray using remainders.
Practical Applications and Contributions
The mathematical approach using small numbers to represent larger ones became dominant over the centuries, with practical applications in Chinese astronomy and modern internet cryptography. Renowned Chinese mathematician Qin Jiushao made significant contributions to solving cubic equations for three-dimensional shapes.
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The influence of Qin's mathematical breakthroughs on Western discoveries.
01:10:30Qin's innovative method for solving equations was advanced for his time and paved the way for future developments.
The Indian mathematical tradition introduced the decimal place-value system and the number zero, revolutionizing mathematics.
The concept of zero evolved from a mere placeholder to a significant number for calculations, allowing for efficient representation of large numbers.
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Indian mathematicians' contributions to the development of zero, negative numbers, and abstract mathematical concepts.
01:19:10Brahmagupta's understanding of negative numbers allowed for the solving of quadratic equations with unknowns before Western discoveries.
Indian mathematicians introduced new notations and made fundamental discoveries in trigonometry, advancing the field's understanding and application.
Their approach to mathematics as abstract entities, rather than just for counting, led to innovative breakthroughs and an explosion of mathematical ideas.
These contributions shaped the course of mathematical development globally.
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Trigonometry and the sine function play a crucial role in calculating distances in various fields like architecture, engineering, and astronomy.
01:22:20Indian mathematicians utilized trigonometry to make calculations about the solar system, such as the distance between the sun and Earth, and the moon.
Madhava from Kerala, India, discovered the concept of the infinite and used infinite sums to calculate the sine function of any angle.
He also connected infinite series with trigonometry, applying the method to find the value of pi.
Indian mathematicians like Aryabhata have made accurate approximations of pi, a fundamental constant in mathematics and engineering.
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Contributions of Indian and Islamic mathematicians in history.
01:28:10Madhava discovered an exact formula for pi using infinity in the 15th century in Kerala.
Indian mathematicians like Madhava made significant discoveries before Western mathematicians but were not given credit.
The Islamic empire in the Middle East established a vibrant intellectual culture and emphasized the importance of knowledge and mathematical skill.
The House of Wisdom in Baghdad played a crucial role in preserving and advancing mathematical knowledge.
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Introduction of Hindu-Arabic numerals and development of algebra by Al-Khwarizmi revolutionized mathematics and science.
01:32:47Algebra became a new language for analyzing numbers and led to systematic problem-solving methods.
Al-Khwarizmi's algebraic language enabled the solution of quadratic equations and laid the foundation for solving cubic equations.
Persian mathematician Omar Khayyam developed a general method to solve all cubic equations, continuing Al-Khwarizmi's legacy.
Khayyam's mathematical work, aligned with Al-Khwarizmi's algebraic spirit, was significant despite his fame as a poet.
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Development of Mathematics in Europe
01:40:33Fibonacci's promotion of Hindu-Arabic numerals revolutionized calculations, replacing Roman numerals.
Initial suspicion and resistance to new numbers due to fear of fraud and empowerment of the masses.
Fibonacci discovered the Fibonacci sequence while solving a rabbit mating riddle.
The Fibonacci sequence is prevalent in nature beyond just rabbits.
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Tartaglia's breakthrough in solving cubic equations marked the beginning of the Western world's mathematical revolution.
01:46:16Despite facing challenges, Tartaglia developed a formula for all types of cubic equations.
Cardano convinced Tartaglia to reveal his formula with the promise of secrecy, but later shared it with his student Ferrari.
Ferrari used Tartaglia's work to solve quartic equations, leading to the publication of Cardano's formula.
Tartaglia died penniless, leaving behind a significant contribution to the field of mathematics.
00:00Our world is made up of patterns and sequences.
00:04They're all around us.
00:06Day becomes night.
00:09Animals travel across the earth in ever-changing formations.
00:14Landscapes are constantly altering.
00:18One of the reasons mathematics began was because we needed to find a way
00:21of making sense of these natural patterns.
00:27The most basic concepts of maths - space and quantity -
00:32are hard-wired into our brains.
00:36Even animals have a sense of distance and number,
00:39assessing when their pack is outnumbered, and whether to fight or fly,
00:45calculating whether their prey is within striking distance.
00:50Understanding maths is the difference between life and death.
00:56But it was man who took these basic concepts
00:59and started to build upon these foundations.
01:01At some point, humans started to spot patterns,
01:05to make connections, to count and to order the world around them.
01:08With this, a whole new mathematical universe began to emerge.
01:20This is the River Nile.
01:22It's been the lifeline of Egypt for millennia.
01:26I've come here because it's where some of the first signs
01:29of mathematics as we know it today emerged.
01:34People abandoned nomadic life and began settling here as early as 6000BC.
01:39The conditions were perfect for farming.
01:47The most important event for Egyptian agriculture each year was the flooding of the Nile.
01:53So this was used as a marker to start each new year.
01:58Egyptians did record what was going on over periods of time,
02:03so in order to establish a calendar like this,
02:05you need to count how many days, for example,
02:09happened in-between lunar phases,
02:11or how many days happened in-between two floodings of the Nile.
02:19Recording the patterns for the seasons was essential,
02:23not only to their management of the land, but also their religion.
02:27The ancient Egyptians who settled on the Nile banks
02:30believed it was the river god, Hapy, who flooded the river each year.
02:34And in return for the life-giving water,
02:37the citizens offered a portion of the yield as a thanksgiving.
02:43As settlements grew larger, it became necessary to find ways to administer them.
02:48Areas of land needed to be calculated, crop yields predicted,
02:52taxes charged and collated.
02:54In short, people needed to count and measure.
02:59The Egyptians used their bodies to measure the world,
03:02and it's how their units of measurements evolved.
03:06A palm was the width of a hand,
03:08a cubit an arm length from elbow to fingertips.
03:13Land cubits, strips of land measuring a cubit by 100,
03:16were used by the pharaoh's surveyors to calculate areas.
03:23There's a very strong link between bureaucracy
03:26and the development of mathematics in ancient Egypt.
03:29And we can see this link right from the beginning,
03:32from the invention of the number system,
03:34throughout Egyptian history, really.
03:37For the Old Kingdom, the only evidence we have
03:40are metrological systems, that is measurements for areas, for length.
03:44This points to a bureaucratic need to develop such things.
03:50It was vital to know the area of a farmer's land so he could be taxed accordingly.
03:55Or if the Nile robbed him of part of his land, so he could request a rebate.
04:00It meant that the pharaoh's surveyors were often calculating
04:03the area of irregular parcels of land.
04:07It was the need to solve such practical problems
04:10that made them the earliest mathematical innovators.
04:18The Egyptians needed some way to record the results of their calculations.
04:25Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo,
04:29I was on the hunt for those that recorded some of the first numbers in history.
04:34They were difficult to track down.
04:39But I did find them in the end.
04:45The Egyptians were using a decimal system, motivated by the 10 fingers on our hands.
04:51The sign for one was a stroke,
04:5310, a heel bone, 100, a coil of rope, and 1,000, a Lotus plant.
04:59How much is this T-shirt?
05:01Er, 25.
05:0325!Yes!So that would be 2 knee bones and 5 strokes.
05:09So you're not gonna charge me anything up here?Here, one million!
05:12One million?My God!
05:14This one million.
05:17One million, yeah, that's pretty big!
05:20The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed.
05:27They had no concept of a place value,
05:31so one stroke could only represent one unit,
05:33not 100 or 1,000.
05:35Although you can write a million with just one character,
05:38rather than the seven that we use, if you want to write a million minus one,
05:42then the poor old Egyptian scribe has got to write nine strokes,
05:46nine heel bones, nine coils of rope, and so on,
05:49a total of 54 characters.
05:54Despite the drawback of this number system, the Egyptians were brilliant problem solvers.
06:01We know this because of the few records that have survived.
06:05The Egyptian scribes used sheets of papyrus
06:08to record their mathematical discoveries.
06:11This delicate material made from reeds decayed over time
06:15and many secrets perished with it.
06:18But there's one revealing document that has survived.
06:22The Rhind Mathematical Papyrus is the most important document
06:26we have today for Egyptian mathematics.
06:29We get a good overview of what types of problems
06:33the Egyptians would have dealt with in their mathematics.
06:37We also get explicitly stated how multiplications and divisions were carried out.
06:44The papyri show how to multiply two large numbers together.
06:49But to illustrate the method, let's take two smaller numbers.
06:53Let's do three times six.
06:56The scribe would take the first number, three, and put it in one column.
07:02In the second column, he would place the number one.
07:05Then he would double the numbers in each column, so three becomes six...
07:13..and six would become 12.
07:20And then in the second column, one would become two,
07:23and two becomes four.
07:28Now, here's the really clever bit.
07:30The scribe wants to multiply three by six.
07:33So he takes the powers of two in the second column,
07:37which add up to six. That's two plus four.
07:40Then he moves back to the first column, and just takes
07:43those rows corresponding to the two and the four.
07:46So that's six and the 12.
07:48He adds those together to get the answer of 18.
07:53But for me, the most striking thing about this method
07:56is that the scribe has effectively written that second number in binary.
08:00Six is one lot of four, one lot of two, and no units.
08:05Which is 1-1-0.
08:08The Egyptians have understood the power of binary over 3,000 years
08:12before the mathematician and philosopher Leibniz would reveal their potential.
08:16Today, the whole technological world depends on the same principles
08:21that were used in ancient Egypt.
08:25The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC.
08:31Its problems are concerned with finding solutions to everyday situations.
08:36Several of the problems mention bread and beer,
08:39which isn't surprising as Egyptian workers were paid in food and drink.
08:43One is concerned with how to divide nine loaves
08:46equally between 10 people, without a fight breaking out.
08:51I've got nine loaves of bread here.
08:53I'm gonna take five of them and cut them into halves.
08:58Of course, nine people could shave a 10th off their loaf
09:00and give the pile of crumbs to the 10th person.
09:04But the Egyptians developed a far more elegant solution -
09:07take the next four and divide those into thirds.
09:13But two of the thirds I am now going to cut into fifths,
09:16so each piece will be one fifteenth.
09:21Each person then gets one half
09:26and one third
09:28and one fifteenth.
09:32It is through such seemingly practical problems
09:35that we start to see a more abstract mathematics developing.
09:38Suddenly, new numbers are on the scene - fractions -
09:41and it isn't too long before the Egyptians are exploring the mathematics of these numbers.
09:48Fractions are clearly of practical importance to anyone dividing quantities for trade in the market.
09:54To log these transactions, the Egyptians developed notation which recorded these new numbers.
10:02One of the earliest representations of these fractions
10:05came from a hieroglyph which had great mystical significance.
10:09It's called the Eye of Horus.
10:13Horus was an Old Kingdom god, depicted as half man, half falcon.
10:20According to legend, Horus' father was killed by his other son, Seth.
10:24Horus was determined to avenge the murder.
10:28During one particularly fierce battle,
10:30Seth ripped out Horus' eye, tore it up and scattered it over Egypt.
10:35But the gods were looking favourably on Horus.
10:38They gathered up the scattered pieces and reassembled the eye.
10:45Each part of the eye represented a different fraction.
10:49Each one, half the fraction before.
10:52Although the original eye represented a whole unit,
10:56the reassembled eye is 1/64 short.
10:59Although the Egyptians stopped at 1/64,
11:03implicit in this picture
11:05is the possibility of adding more fractions,
11:08halving them each time, the sum getting closer and closer to one,
11:13but never quite reaching it.
11:16This is the first hint of something called a geometric series,
11:20and it appears at a number of points in the Rhind Papyrus.
11:23But the concept of infinite series would remain hidden
11:26until the mathematicians of Asia discovered it centuries later.
11:34Having worked out a system of numbers, including these new fractions,
11:38it was time for the Egyptians to apply their knowledge
11:41to understanding shapes that they encountered day to day.
11:44These shapes were rarely regular squares or rectangles,
11:48and in the Rhind Papyrus, we find the area of a more organic form, the circle.
11:53What is astounding in the calculation
11:57of the area of the circle is its exactness, really.
12:00How they would have found their method is open to speculation,
12:04because the texts we have
12:06do not show us the methods how they were found.
12:10This calculation is particularly striking because it depends
12:14on seeing how the shape of the circle
12:16can be approximated by shapes that the Egyptians already understood.
12:21The Rhind Papyrus states that a circular field
12:24with a diameter of nine units
12:26is close in area to a square with sides of eight.
12:30But how would this relationship have been discovered?
12:34My favourite theory sees the answer in the ancient game of mancala.
12:39Mancala boards were found carved on the roofs of temples.
12:43Each player starts with an equal number of stones,
12:47and the object of the game is to move them round the board,
12:50capturing your opponent's counters on the way.
12:54As the players sat around waiting to make their next move,
12:58perhaps one of them realised that sometimes the balls fill the circular holes
13:01of the mancala board in a rather nice way.
13:03He might have gone on to experiment with trying to make larger circles.
13:09Perhaps he noticed that 64 stones, the square of 8,
13:13can be used to make a circle with diameter nine stones.
13:17By rearranging the stones, the circle has been approximated by a square.
13:22And because the area of a circle ispi times the radius squared,
13:26the Egyptian calculation gives us the first accurate value for pi.
13:31The area of the circle is 64. Divide this by the radius squared,
13:36in this case 4.5 squared, and you get a value for pi.
13:40So 64 divided by 4.5 squared is 3.16,
13:44just a little under two hundredths away from its true value.
13:48But the really brilliant thing is, the Egyptians
13:51are using these smaller shapes to capture the larger shape.
13:59But there's one imposing andmajestic symbol of Egyptian
14:02mathematics we haven't attempted to unravel yet -
14:05the pyramid.
14:07I've seen so many pictures that I couldn't believe I'd be impressed by them.
14:12But meeting them face to face, you understand why they're called
14:15one of the Seven Wonders of the Ancient World.
14:18They're simply breathtaking.
14:20And how much more impressive they must have been in their day,
14:23when the sides were as smooth as glass, reflecting the desert sun.
14:28To me it looks like there might be mirror pyramidshiding underneath the desert,
14:34which would complete the shapes to make perfectlysymmetrical octahedrons.
14:38Sometimes, in the shimmer of the desert heat, you can almost see these shapes.
14:45It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician.
14:52The pyramids are just a little short to create these perfect shapes,
14:57but some have suggested another important mathematical concept
15:00might be hidden inside the proportions of the Great Pyramid - the golden ratio.
15:06Two lengths are in the golden ratio, if the relationship of the longest
15:11to the shortest is the same as the sum of the two to the longest side.
15:16Such a ratio has been associated with the perfect proportions one finds
15:21all over the natural world, as well as in the work of artists,
15:25architects and designers for millennia.
15:31Whether the architects of the pyramids were conscious of this important mathematical idea,
15:36or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know.
15:41For me, the most impressive thing about the pyramidsis the mathematical brilliance
15:46that went into making them, including the first inkling
15:49of one of the great theorems of the ancient world, Pythagoras' theorem.
15:55In order to get perfect right-angled corners on their buildings
15:58and pyramids, the Egyptians would haveused a rope with knots tied in it.
16:03At some point, the Egyptians realised that if they took a triangle with sides
16:07marked with three knots, four knots and five knots, it guaranteed them aperfect right-angle.
16:14This is because three squared, plus four squared, is equal to five squared.
16:19So we've got a perfect Pythagorean triangle.
16:24In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle.
16:30But I'm pretty sure that the Egyptians hadn't got
16:32this sweeping generalisation of their 3, 4, 5 triangle.
16:37We would not expect to find the general proof
16:41because this is not the style of Egyptian mathematics.
16:44Every problem was solved using concrete numbers and then
16:48if a verification would be carried out at the end, it would use the result
16:52and these concrete, given numbers,
16:54there's no general proof within the Egyptian mathematical texts.
17:00It would be some 2,000 years before the Greeks and Pythagoras
17:03would prove that all right-angled triangles shared certain properties.
17:08This wasn't the only mathematical idea that theEgyptians would anticipate.
17:12In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume
17:19of a pyramid with its peak sliced off,which shows the first hint ofcalculus at work.
17:25For a culture like Egypt that is famous for its pyramids, you would expect problems like this
17:32to have been a regular feature within the mathematical texts.
17:35The calculation of the volume of a truncated pyramid is one of the most
17:40advanced bits, according to our modern standards of mathematics,
17:45that we have from ancient Egypt.
17:48The architects and engineers would certainly have wanted such a formula
17:52to calculate the amount of materials required to build it.
17:55But it's a mark of the sophistication
17:58of Egyptian mathematics that they were ableto produce such a beautiful method.
18:08To understand how they derived their formula, start with a pyramid
18:12built such that the highest point sits directly over one corner.
18:17Three of these can be put together to make a rectangular box,
18:22so the volume of the skewed pyramid is a third the volume of the box.
18:27That is, the height, times the length, times the width, divided by three.
18:33Now comes an argument which shows the very first hints of the calculus at work,
18:38thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.
18:44Suppose you could cut the pyramid into slices, you could then slide
18:48the layers across to make the more symmetrical pyramid you see in Giza.
18:54However, the volume of the pyramid has not changed, despite the rearrangement of the layers.
18:58So the same formula works.
19:04The Egyptians were amazing innovators,
19:08and their ability to generate new mathematics was staggering.
19:11For me, they revealed the power of geometry and numbers, and made the first moves
19:16towards some of the exciting mathematical discoveries to come.
19:20But there was another civilisation that had mathematics to rival that of Egypt.
19:25And we know much more about their achievements.
19:33This is Damascus, over 5,000 years old,
19:37and still vibrant and bustling today.
19:40It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.
19:46The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.
19:52In order to expand and run their empire, they became masters of managing and manipulating numbers.
20:00We have law codes for instance that tell us
20:03about the way society is ordered.
20:05The people we know most about are the scribes, the professionally literate
20:09and numerate people who kept the records for the wealthy families and for the temples and palaces.
20:14Scribe schools existed from around 2500BC.
20:19Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.
20:26Scribe records were kept on clay tablets,
20:29which allowed the Babylonians to manage and advance their empire.
20:33However, many of the tablets we have today aren't official documents, but children's exercises.
20:40It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.
20:46So, this is a geometrical textbook from about the 18th century BC.
20:51I hope you can see that there are lots of pictures on it.
20:54And underneath each picture is a text that sets a problem about the picture.
20:58So for instance this one here says, I drew a square, 60 units long,
21:04and inside it, I drew four circles - what are their areas?
21:10This little tablet here was written 1,000 years at least later than the tablet here,
21:16but has a very interesting relationship.
21:19It also has four circles on,
21:21in a square, roughly drawn, but this isn't a textbook, it's a school exercise.
21:26The adult scribe who's teaching the student is being given this
21:30as an example of completed homework or something like that.
21:35Like the Egyptians, the Babylonians appeared interested
21:38in solving practical problems to do with measuring and weighing.
21:42The Babylonian solutions to these problems are written like mathematical recipes.
21:46A scribe would simply follow and record a set of instructions to get a result.
21:52Here's an example of the kind of problem they'd solve.
21:56I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them.
22:00Instead, I'm gonna take four times their weight and add them to the scales.
22:07Now I'm gonna add 20 gin. Gin was the ancient Babylonian measure of weight.
22:13I'm gonna take half of everything here and then add it again...
22:17That's two bundles, and ten gin.
22:19Everything on this side is equal to one mana. One mana was 60 gin.
22:25And here, we have one of the first mathematical equations in history,
22:29everything on this side is equal to one mana.
22:32But how much does the bundle of cinnamon sticks weigh?
22:35Without any algebraic language, they were able to manipulate
22:38the quantities to be able to prove that the cinnamon sticks weighed five gin.
22:44In my mind, it's this kind of problem which gives mathematics a bit of a bad name.
22:49You can blame those ancient Babylonians for all those tortuous problems you had at school.
22:54But the ancient Babylonian scribes excelled at this kind of problem.
22:59Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60.
23:09The Babylonians invented their number system, like the Egyptians, by using their fingers.
23:14But instead of counting through the 10 fingers on their hand,
23:17Babylonians found a moreintriguing way to count body parts.
23:20They used the 12 knuckles on one hand,
23:23and the five fingers on the other to be able to count
23:2512 times 5, ie 60 different numbers.
23:29So for example, this number would have been 2 lots of 12, 24,
23:34and then, 1, 2, 3, 4, 5, to make 29.
23:41The number 60 had another powerful property.
23:45It can be perfectly divided in a multitude of ways.
23:48Here are 60 beans.
23:50I can arrange them in 2 rows of 30.
23:573 rows of 20.
24:004 rows of 15.
24:035 rows of 12.
24:05Or 6 rows of 10.
24:08The divisibility of 60 makes it a perfect base in which to do arithmetic.
24:14The base 60 system was so successful, we still use elements of it today.
24:20Every time we want to tell the time, we recognise units of 60 -
24:2460 seconds in a minute, 60 minutes in an hour.
24:28But the most important feature of the Babylonians' number system was that it recognised place value.
24:33Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording,
24:39the position of each Babylonian number records the power of 60.
24:50Instead of inventing new symbols for bigger and bigger numbers,
24:53they would write 1-1-1, so this number would be 3,661.
25:03The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky.
25:16The Babylonians' calendar was based on the cycles of the moon.
25:20They needed a way of recording astronomically large numbers.
25:24Month by month, year by year, these cycles were recorded.
25:28From about 800BC, there were complete lists of lunar eclipses.
25:34The Babylonian system of measurement was quite sophisticated at that time.
25:40They had a system of angular measurement,
25:42360 degrees in a full circle, each degree was divided
25:46into 60 minutes, a minute was further divided into 60 seconds.
25:51So they had a regular system for measurement, and it was in perfect harmony with their number system,
25:57so it's well suited not only for observation but also for calculation.
26:01But in order to calculate and cope with these large numbers,
26:05the Babylonians needed to invent a new symbol.
26:09And in so doing, they prepared the ground for one of the great
26:13breakthroughs in the history of mathematics - zero.
26:16In the early days, the Babylonians, in order to mark an empty place in
26:20the middle of a number, would simply leave a blank space.
26:23So they needed a way of representing nothing in the middle of a number.
26:29So they used a sign, as a sort of breathing marker, a punctuation mark,
26:34and it comes to mean zero in the middle of a number.
26:38This was the first time zero, in any form, had appeared in the mathematical universe.
26:44But it would be over a 1,000 years before this little place holder would become a number in its own right.
26:59Having established such a sophisticated system of numbers,
27:03they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.
27:11Babylonian engineers and surveyors found ingenious ways of
27:15accessing water, and channelling it to the crop fields.
27:19Yet again, they used mathematics to come up with solutions.
27:24The Orontes valley in Syria is still an agricultural hub,
27:28and the old methods of irrigation are being exploited today, just as they were thousands of years ago.
27:35Many of the problems in Babylonian mathematics
27:38are concerned with measuring land, and it's here we see for the first time
27:43the use of quadratic equations, one of the greatest legacies of Babylonian mathematics.
27:49Quadratic equations involve things where the unknown quantity
27:52you're trying to identify is multiplied by itself.
27:56We call this squaring because it gives the area of a square,
27:59and it's in the context of calculating the area of land
28:02that thesequadratic equations naturally arise.
28:10Here's a typical problem.
28:12If a field has an area of 55 units
28:15and one side is six units longer than the other,
28:19how long is the shorter side?
28:23The Babylonian solution was to reconfigure the field as a square.
28:27Cut three units off the end
28:31and move this round.
28:33Now, there's a three-by-three piece missing, so let's add this in.
28:39The area of the field has increased by nine units.
28:43This makes the new area 64.
28:47So the sides of the square are eight units.
28:51The problem-solver knows that they've added three to this side.
28:54So, the original length must be five.
28:59It may not look like it, but this is one of the first quadratic equations in history.
29:06In modern mathematics, I would use the symbolic language of algebra to solve this problem.
29:11The amazing feat of the Babylonians is that they were using these geometric games to find the value,
29:16without any recourse to symbols or formulas.
29:19The Babylonians were enjoying problem-solving for its own sake.
29:23They were falling in love with mathematics.
29:38The Babylonians' fascination with numbers soon found a place in their leisure time, too.
29:43They were avid game-players.
29:45The Babylonians and their descendants have been playing
29:47a version of backgammon for over 5,000 years.
29:52The Babylonians played board games,
29:55from very posh board games in royal tombs to little bits of board games found in schools,
30:01to board games scratched on the entrances of palaces,
30:05so that the guardsman must have played when they were bored,
30:09and they used dice to move their counters round.
30:13People who played games were using numbers in their leisure time to try and outwit their opponent,
30:18doing mental arithmetic very fast,
30:21and so they were calculating in their leisure time,
30:26without even thinking about it as being mathematical hard work.
30:32Now's my chance.
30:33'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.'
30:39It's up to you.Six... I need to move something.
30:42'But it wasn't as easy as I thought.'
30:45Ah! What the hell was that?
30:47Yeah.This is one, this is two.
30:51Now you're in trouble.
30:53So I can't move anything. You cannot move these.
30:56Oh, gosh.
30:59There you go.
31:02Three and four.
31:04'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.'
31:09Yeah.
31:12Put it there. Good game.
31:16The Babylonians are recognised as one of the first cultures
31:19to use symmetrical mathematical shapes to make dice,
31:23but there is more heated debates about whether they might also
31:26have been the first to discover the secrets of another important shape.
31:30The right-angled triangle.
31:36We've already seen how the Egyptians use a 3-4-5 right-angled triangle.
31:41But what the Babylonians knew about this shape and others like it is much more sophisticated.
31:46This is the most famous and controversial ancient tablet we have.
31:51It's called Plimpton 322.
31:54Many mathematicians are convinced it shows the Babylonians
31:58could well have known the principle regarding right-angled triangles,
32:02that the square on the diagonal is the sum of the squares on the sides,
32:06and known it centuries before the Greeks claimed it.
32:11This is a copy of arguably the most famous Babylonian tablet,
32:15which is Plimpton 322,
32:17and these numbers here reflect the width or height of a triangle,
32:21this being the diagonal, the other side would be over here,
32:26and the square of this column plus the square of the number in this column
32:32equals the square of the diagonal.
32:35They are arranged in an order of steadily decreasing angle,
32:40on a very uniform basis, showing that somebody
32:43had a lot of understanding of how the numbers fit together.
32:53Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.
32:59It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem,
33:05and it's a conclusion that generations of historians have been seduced by.
33:10But there could be a much simpler explanation
33:13for the sets of three numbers which fulfil Pythagoras' theorem.
33:16It's not a systematic explanation of Pythagorean triples, it's simply
33:21a mathematics teacher doing some quite complicated calculations,
33:26but in order to produce some very simple numbers,
33:30in order to set his students problems about right-angled triangles,
33:35and in that sense it's about Pythagorean triples only incidentally.
33:42The most valuable clues to what they understood could lie elsewhere.
33:48This small school exercise tablet is nearly 4,000 years old
33:52and reveals just what the Babylonians did know about right-angled triangles.
33:57It uses a principle of Pythagoras' theorem to find the value of an astounding new number.
34:07Drawn along the diagonal is a really very good approximation to the square root of two,
34:14and so that shows us that it was known and used in school environments.
34:20Why's this important?
34:22Because the square root of two is what we now call an irrational number,
34:27that is, if we write it out in decimals, or even in sexigesimal places,
34:33it doesn't end, the numbers go on forever after the decimal point.
34:38The implications of this calculation are far-reaching.
34:42Firstly, it means the Babylonians knew something of Pythagoras' theorem
34:471,000 years before Pythagoras.
34:48Secondly, the fact that they can calculate this number to an accuracy of four decimal places
34:54shows an amazing arithmetic facility, as well as a passion for mathematical detail.
35:01The Babylonians' mathematical dexterity was astounding,
35:05and for nearly 2,000 years they spearheaded intellectual progress in the ancient world.
35:12But when their imperial power began to wane, so did their intellectual vigour.
35:25By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.
35:34This is Palmyra in central Syria, a once-great city built by the Greeks.
35:42The mathematical expertise needed to build structures with such geometric perfection is impressive.
35:51Just like the Babylonians before them, the Greeks were passionate about mathematics.
35:59The Greeks were clever colonists.
36:02They took the best from the civilisations they invaded
36:05to advance their own power and influence,
36:07but they were soon making contributions themselves.
36:11In my opinion, their greatest innovation was to do with a shift in the mind.
36:16What they initiated would influence humanity for centuries.
36:20They gave us the power of proof.
36:23Somehow they decided that they had to have a deductive system for their mathematics
36:28and the typical deductive system was to begin with certain axioms, which you assume are true.
36:34It's as if you assume a certain theorem is true without proving it.
36:38And then, using logical methods and very careful steps,
36:43from these axioms you prove theorems
36:46and from those theorems you prove more theorems, and it just snowballs.
36:52Proof is what gives mathematics its strength.
36:56It's the power or proof which means that the discoveries of the Greeks
37:00are as true today as they were 2,000 years ago.
37:04I needed to head west into the heart of the old Greek empire to learn more.
37:17For me, Greek mathematics has always been heroic and romantic.
37:24I'm on my way to Samos, less than a mile from the Turkish coast.
37:29This place has become synonymous with the birth of Greek mathematics,
37:34and it's down to the legend of one man.
37:40His name is Pythagoras.
37:42The legends that surround his life and work have contributed
37:45to the celebrity status he has gained over the last 2,000 years.
37:49He's credited, rightly or wrongly, with beginning the transformation
37:54from mathematics as a tool for accounting to the analytic subject we recognise today.
38:03Pythagoras is a controversial figure.
38:06Because he left no mathematical writings, many have questioned
38:09whether he indeed solved any of the theorems attributed to him.
38:14He founded a school in Samos in the sixth century BC,
38:17but his teachings were considered suspect and the Pythagoreans a bizarre sect.
38:24There is good evidence that there were schools of Pythagoreans,
38:28and they may have looked more like sects than what we associate with philosophical schools,
38:35because they didn't just share knowledge, they also shared a way of life.
38:40There may have been communal living and they all seemed to have been
38:45involved in the politics of their cities.
38:49One feature that makes them unusual in the ancient world is that they included women.
38:55But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians -
39:01the properties of right-angled triangles.
39:05What's known as Pythagoras' theorem
39:07states that if you take any right-angled triangle,
39:10build squares on all the sides, then the area of the largest square
39:14is equal to the sum of the squares on the two smaller sides.
39:22It's at this point for me that mathematics is born
39:25and a gulf opens up between the other sciences,
39:29and the proof is as simple as it is devastating in its implications.
39:33Place four copies of the right-angled triangle
39:37on top of this surface.
39:39The square that you now see
39:40has sides equal to the hypotenuse of the triangle.
39:44By sliding these triangles around,
39:46we see how we can break the area of the large square up
39:49into the sum of two smaller squares,
39:52whose sides are given by the two short sides of the triangle.
39:56In other words, the square on the hypotenuse is equal to the sum
40:01of the squares on the other sides. Pythagoras' theorem.
40:07It illustrates one of the characteristic themes of Greek mathematics -
40:11the appeal to beautiful arguments in geometry rather than a reliance on number.
40:20Pythagoras may have fallen out of favour and many of the discoveries accredited to him
40:25have been contested recently, but there's one mathematical theory that I'm loath to take away from him.
40:31It's to do with music and the discoveryof the harmonic series.
40:36The story goes that, walking past a blacksmith's one day,
40:40Pythagoras heard anvils being struck,
40:42and noticed how the notes being produced sounded in perfect harmony.
40:47He believed that there must be some rational explanation
40:51to make sense of why the notes sounded so appealing.
40:55The answer was mathematics.
41:02Experimenting with a stringed instrument, Pythagoras discovered that the intervals between
41:07harmonious musical notes were always represented as whole-number ratios.
41:14And here's how he might have constructed his theory.
41:19First, play a note on the open string.
41:22MAN PLAYS NOTE
41:24Next, take half the length.
41:28The note almost sounds the same as the first note.
41:31In fact it's an octave higher, but the relationship is so strong, we give these notes the same name.
41:36Now take a third the length.
41:40We get another note which sounds harmonious next to the first two,
41:44but take a length of string which is not in a whole-number ratio and all we get is dissonance.
41:55According to legend, Pythagoras was so excited by this discovery
42:00that he concluded the whole universe was built from numbers.
42:03But he and his followers were in for a rather unsettling challenge to their world view
42:09and it came about as a result of the theorem which bears Pythagoras' name.
42:16Legend has it, one of his followers, a mathematician called Hippasus,
42:21set out to find the length of the diagonal
42:24for a right-angled triangle with two sides measuring one unit.
42:28Pythagoras' theorem implied that the length of the diagonal was a number whose square was two.
42:34The Pythagoreans assumed that the answer would be a fraction,
42:38but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it.
42:45Eventually he realised his mistake.
42:47It was the assumption that the value was a fraction at all which was wrong.
42:52The value of the square root of two was the number that the Babylonians etched into the Yale tablet.
42:58However, they didn't recognise the special character of this number.
43:02But Hippasus did.
43:04It was an irrational number.
43:10The discovery of this new number, and others like it, is akin to an explorer
43:13discovering a new continent, or a naturalist finding a new species.
43:18But these irrational numbers didn't fit the Pythagorean world view.
43:22Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy,
43:28but Hippasus let slip the discovery
43:31and was promptly drowned for his attempts to broadcast their research.
43:36But these mathematical discoveries could not be easily suppressed.
43:41Schools of philosophy and science started to flourish all over Greece, building on these foundations.
43:47The most famous of these was the Academy.
43:51Plato founded this school in Athens in 387 BC.
43:56Although we think of him today as a philosopher, he was one of mathematics' most important patrons.
44:03Plato was enraptured by the Pythagorean world view
44:06and considered mathematics the bedrock of knowledge.
44:11Some people would say that Plato is the most influential figure
44:16for our perception of Greek mathematics.
44:19He argued that mathematics is an important form of knowledge
44:24and does have a connection with reality.
44:26So by knowing mathematics, we know more about reality.
44:32In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking
44:38the secrets of the universe, a view still held by scientists today.
44:42Indeed, the importance Plato attached to geometry is encapsulated
44:46in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here."
44:56Plato proposed that the universe could be crystallised into five regular symmetrical shapes.
45:02These shapes, which we now call the Platonic solids,
45:05were composed of regular polygons, assembled to create
45:08three-dimensional symmetrical objects.
45:12The tetrahedron represented fire.
45:14The icosahedron, made from 20 triangles, represented water.
45:19The stable cube was Earth.
45:21The eight-faced octahedron was air.
45:25And the fifth Platonic solid, the dodecahedron,
45:28made out of 12 pentagons, was reserved for the shape
45:31that captured Plato's view of the universe.
45:38Plato's theory would have a seismic influence and continued to inspire
45:42mathematicians and astronomers for over 1,500 years.
45:47In addition to the breakthroughs made in the Academy,
45:50mathematical triumphs were also emerging from the edge of the Greek empire,
45:55and owed as much to the mathematical heritage of the Egyptians as the Greeks.
46:00Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,
46:07and its famous library soon gained a reputation to rival Plato's academy.
46:13The kings of Alexandria were prepared to invest in the arts and culture,
46:20in technology, mathematics, grammar,
46:24because patronage for cultural pursuits
46:28was one way of showing that you were a more prestigious ruler,
46:36and had a better entitlement to greatness.
46:41The old library and its precious contents were destroyed
46:44when the Muslims conquered Egypt in the 7th Century.
46:48But its spirit is alive in a new building.
46:52Today, the library remains a place of discovery and scholarship.
47:01Mathematicians and philosophers flocked to Alexandria,
47:04driven by their thirst for knowledge and the pursuit of excellence.
47:07The patrons of the library were the first professional scientists,
47:11individuals who were paid for their devotion to research.
47:15But of all those early pioneers,
47:17my hero is the enigmatic Greek mathematician Euclid.
47:25We know very little about Euclid's life,
47:27but his greatest achievements were as a chronicler of mathematics.
47:31Around 300 BC, he wrote the most important text book of all time -
47:37The Elements. In The Elements,
47:39we find the culmination of the mathematical revolution
47:43which had taken place in Greece.
47:47It's built on a series of mathematical assumptions, called axioms.
47:51For example, a line can be drawn between any two points.
47:56From these axioms, logical deductions are made and mathematical theorems established.
48:04The Elements contains formulas for calculating the volumes of cones
48:08and cylinders, proofs about geometric series,
48:12perfect numbers and primes.
48:14The climax of The Elements is a proof that there are only five Platonic solids.
48:22For me, this last theorem captures the power of mathematics.
48:26It's one thing to build five symmetrical solids,
48:29quite another to come up with a watertight, logical argument for why there can't be a sixth.
48:35The Elements unfolds like a wonderful, logical mystery novel.
48:39But this is a story which transcends time.
48:42Scientific theories get knocked down, from one generation to the next,
48:46but the theorems in The Elements are as true today as they were 2,000 years ago.
48:52When you stop and think about it, it's really amazing.
48:56It's the same theorems that we teach.
48:57We may teach them in a slightly different way, we may organise them differently,
49:02but it's Euclidean geometry that is still valid,
49:06and even in higher mathematics, when you go to higher dimensional spaces,
49:10you're still using Euclidean geometry.
49:14Alexandria must have been an inspiring place for the ancient scholars,
49:18and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port.
49:24One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.
49:32He would become a mathematical visionary.
49:35The best Greek mathematicians, they were always pushing the limits,
49:40pushing the envelope.
49:42So, Archimedes...
49:44did what he could with polygons,
49:47with solids.
49:50He then moved on to centres of gravity.
49:52He then moved on to the spiral.
49:57This instinct to try and mathematise everything
50:03is something that I see as a legacy.
50:08One of Archimedes' specialities was weapons of mass destruction.
50:12They were used against the Romans when they invaded his home of Syracuse in 212 BC.
50:18He also designed mirrors, which harnessed the power of the sun,
50:22to set the Roman ships on fire.
50:25But to Archimedes, these endeavours were mere amusements in geometry.
50:30He had loftier ambitions.
50:35Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake
50:42and not for the ignoble trade of engineering or the sordid quest for profit.
50:46One of his finest investigations into pure mathematics
50:50was to produce formulas to calculate the areas of regular shapes.
50:56Archimedes' method was to capture new shapes by using shapes he already understood.
51:02So, for example, to calculate the area of a circle,
51:05he would enclose it inside a triangle, and then by doubling the number of sides on the triangle,
51:10the enclosing shape would get closer and closer to the circle.
51:14Indeed, we sometimes call a circle
51:16a polygon with an infinite number of sides.
51:19But by estimating the area of the circle, Archimedes is, in fact,
51:23getting a value for pi, the most important number in mathematics.
51:29However, it was calculating the volumes of solid objects where Archimedes excelled.
51:35He found a way to calculate the volume of a sphere
51:38by slicing it up and approximating each slice as a cylinder.
51:42He then added up the volumes of the slices
51:45to get an approximate value for the sphere.
51:49But his act of genius was to see what happens
51:52if you make the slices thinner and thinner.
51:54In the limit, the approximation becomes an exact calculation.
52:03But it was Archimedes' commitment to mathematics that would be his undoing.
52:10Archimedes was contemplating a problem about circles traced in the sand.
52:15When a Roman soldier accosted him,
52:18Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.
52:24But the Roman soldier was not interested in Archimedes' problem and killed him on the spot.
52:29Even in death, Archimedes' devotion to mathematics was unwavering.
52:55By the middle of the 1st Century BC,
52:59the Romans had tightened their grip on the old Greek empire.
53:03They were less smitten with the beauty of mathematics
53:05and were more concerned with its practical applications.
53:09This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.
53:15But one mathematician was determined to keep the legacy of the Greeks alive.
53:19Hypatia was exceptional, a female mathematician,
53:24and a pagan in the piously Christian Roman empire.
53:29Hypatia was very prestigious and very influential in her time.
53:34She was a teacher with a lot of students, a lot of followers.
53:40She was politically influential in Alexandria.
53:44So it's this combination of...
53:47high knowledge and high prestige that may have made her
53:53a figure of hatred for...
53:57the Christian mob.
54:04One morning during Lent, Hypatia was dragged off her chariot
54:08by a zealous Christian mob and taken to a church.
54:12There, she was tortured and brutally murdered.
54:18The dramatic circumstances of her life and death
54:21fascinated later generations.
54:24Sadly, her cult status eclipsed her mathematical achievements.
54:30She was, in fact, a brilliant teacher and theorist,
54:33and her death dealt a final blow to the Greek mathematical heritage of Alexandria.
54:46My travels have taken me on a fascinating journey to uncover
54:50the passion and innovation of the world's earliest mathematicians.
54:55It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece
55:00that are the foundations on which my subject is built today.
55:04But this is just the beginning of my mathematical odyssey.
55:08The next leg of my journey lies east, in the depths of Asia,
55:12where mathematicians scaled even greater heights
55:15in pursuit of knowledge.
55:17With this new era came a new language of algebra and numbers,
55:21better suited to telling the next chapter in the story of maths.
55:26You can learn more about the story of maths
55:29with the Open University at...
55:42This is what we call a decimal
place-value system,
55:45and it's very similar
to the one we use today.
55:48We too use numbers from one to nine,
and we use their position
55:51to indicate whether it's units,
tens, hundreds or thousands.
55:55But the power of these rods is that
it makes calculations very quick.
55:59In fact, the way the ancient
Chinese did their calculations
56:03is very similar to the way
we learn today in school.
56:11Not only were the ancient Chinese
56:12the first to use a decimal
place-value system,
56:16but they did so over 1,000 years
before we adopted it in the West.
56:20But they only used it
when calculating with the rods.
56:24When writing the numbers down,
56:27the ancient Chinese
didn't use the place-value system.
56:32Instead, they used a far
more laborious method,
56:36in which special symbols stood for
tens, hundreds, thousands and so on.
56:42So the number 924
would be written out
56:45as nine hundreds,
two tens and four.
56:50Not quite so efficient.
56:53The problem was
56:55that the ancient Chinese didn't
have a concept of zero.
56:58They didn't have a symbol for zero.
It just didn't exist as a number.
57:01Using the counting rods,
57:03they would use a blank space
where today we would write a zero.
57:07The problem came with trying to
write down this number, which is why
57:11they had to create these new symbols
for tens, hundreds and thousands.
57:14Without a zero, the written
number was extremely limited.
57:21But the absence of zero
didn't stop
57:24the ancient Chinese from
making giant mathematical steps.
57:28In fact,
there was a widespread fascination
57:31with number in ancient China.
57:33According to legend,
the first sovereign of China,
57:37the Yellow Emperor,
had one of his deities
57:40create mathematics in 2800BC,
57:43believing that number held cosmic
significance. And to this day,
57:48the Chinese still believe in
the mystical power of numbers.
57:55Odd numbers are seen as male,
even numbers, female.
58:00The number four
is to be avoided at all costs.
58:03The number eight
brings good fortune.
58:07And the ancient Chinese were
drawn to patterns in numbers,
58:10developing their own
rather early version of sudoku.
58:16It was called the magic square.
58:23Legend has it that thousands of
years ago, Emperor Yu was visited
58:27by a sacred turtle that came out
of the depths of the Yellow River.
58:31On its back were numbers
58:33arranged into a magic square,
a little like this.
58:45In this square,
58:46which was regarded as having
great religious significance,
58:50all the numbers in each line -
horizontal, vertical and diagonal -
58:54all add up to the same number - 15.
59:01Now, the magic square may be
no more than a fun puzzle,
59:04but it shows
the ancient Chinese fascination
59:06with mathematical patterns,
and it wasn't too long
59:09before they were creating
even bigger magic squares
59:12with even greater magical
and mathematical powers.
59:23But mathematics also played
59:26a vital role in the running
of the emperor's court.
59:30The calendar and the movement
of the planets
59:34were of the utmost
importance to the emperor,
59:37influencing all his decisions, even
down to the way his day was planned,
59:42so astronomers became prized
members of the imperial court,
59:46and astronomers were
always mathematicians.
59:53Everything in the emperor's life
was governed by the calendar,
59:57and he ran his affairs
with mathematical precision.
01:00:01The emperor even got
his mathematical advisors
01:00:04to come up with a system
to help him sleep his way
01:00:07through the vast number of women
he had in his harem.
01:00:10Never one to miss a trick,
the mathematical advisors decided
01:00:13to base the harem on a mathematical
idea called a geometric progression.
01:00:18Maths has never had
such a fun purpose!
01:00:21Legend has it that
in the space of 15 nights,
01:00:25the emperor had to sleep
with 121 women...
01:00:35..the empress,
01:00:37three senior consorts,
01:00:39nine wives,
01:00:4227 concubines
01:00:44and 81 slaves.
01:00:48The mathematicians
would soon have realised
01:00:51that this was a geometric
progression - a series of numbers
01:00:54in which you get
from one number to the next
01:00:57by multiplying the same number
each time - in this case, three.
01:01:03Each group of women is three times
as large as the previous group,
01:01:07so the mathematicians could quickly
draw up a rota to ensure that,
01:01:11in the space of 15 nights,
01:01:13the emperor slept
with every woman in the harem.
01:01:18The first night
was reserved for the empress.
01:01:22The next was for the three
senior consorts.
01:01:25The nine wives came next,
01:01:27and then the 27 concubines were
chosen in rotation, nine each night.
01:01:34And then finally,
over a period of nine nights,
01:01:37the 81 slaves were dealt with
in groups of nine.
01:01:46Being the emperor certainly
required stamina,
01:01:49a bit like being a mathematician,
01:01:51but the object is clear -
01:01:53to procure the best
possible imperial succession.
01:01:57The rota ensured that the emperor
01:01:59slept with the ladies of highest
rank closest to the full moon,
01:02:03when their yin, their female force,
01:02:05would be at its highest and be able
to match his yang, or male force.
01:02:15The emperor's court wasn't alone
in its dependence on mathematics.
01:02:19It was central to the running
of the state.
01:02:22Ancient China was a vast and growing
empire with a strict legal code,
01:02:27widespread taxation
01:02:28and a standardised system
of weights, measures and money.
01:02:34The empire needed
01:02:36a highly trained civil service,
competent in mathematics.
01:02:42And to educate these civil servants
was a mathematical textbook,
01:02:46probably written in around 200BC -
the Nine Chapters.
01:02:53The book is a compilation
of 246 problems
01:02:57in practical areas such as trade,
payment of wages and taxes.
01:03:04And at the heart
of these problems lies
01:03:07one of the central themes of
mathematics, how to solve equations.
01:03:15Equations are a little bit
like cryptic crosswords.
01:03:18You're given a certain amount
of information
01:03:20about some unknown numbers,
and from that information
01:03:23you've got to deduce what
the unknown numbers are.
01:03:26For example,
with my weights and scales,
01:03:28I've found out that one plum...
01:03:31..together with three peaches
01:03:34weighs a total of 15 grams.
01:03:40But...
01:03:42..two plums
01:03:44together with one peach
01:03:47weighs a total of 10g.
01:03:49From this information, I can
deduce what a single plum weighs
01:03:53and a single peach weighs,
and this is how I do it.
01:03:59If I take the first set of scales,
01:04:01one plum and three peaches
weighing 15g,
01:04:04and double it, I get two plums
and six peaches weighing 30g.
01:04:13If I take this and subtract from it
the second set of scales -
01:04:17that's two plums
and a peach weighing 10g -
01:04:19I'm left with
an interesting result -
01:04:22no plums.
01:04:24Having eliminated the plums,
01:04:27I've discovered that
five peaches weighs 20g,
01:04:30so a single peach weighs 4g,
01:04:33and from this I can deduce
that the plum weighs 3g.
01:04:37The ancient Chinese went on
to apply similar methods
01:04:41to larger and larger numbers
of unknowns,
01:04:44using it to solve increasingly
complicated equations.
01:04:49What's extraordinary is
01:04:51that this particular
system of solving equations
01:04:54didn't appear in the West until
the beginning of the 19th century.
01:04:58In 1809, while analysing a rock
called Pallas in the asteroid belt,
01:05:02Carl Friedrich Gauss,
01:05:04who would become known
as the prince of mathematics,
01:05:07rediscovered this method
01:05:08which had been formulated
in ancient China centuries earlier.
01:05:12Once again, ancient China
streets ahead of Europe.
01:05:20But the Chinese
were to go on to solve
01:05:22even more complicated equations
involving far larger numbers.
01:05:26In what's become known as
the Chinese remainder theorem,
01:05:29the Chinese came up
with a new kind of problem.
01:05:34In this, we know the number
that's left
01:05:37when the equation's unknown number
is divided by a given number -
01:05:41say, three, five or seven.
01:05:45Of course, this is a fairly
abstract mathematical problem,
01:05:49but the ancient Chinese still
couched it in practical terms.
01:05:55So a woman in the market has
a tray of eggs,
01:05:58but she doesn't know
how many eggs she's got.
01:06:01What she does know is that
if she arranges them in threes,
01:06:04she has one egg left over.
01:06:07If she arranges them in fives,
she gets two eggs left over.
01:06:11But if she arranged them
in rows of seven,
01:06:14she found she had
three eggs left over.
01:06:17The ancient Chinese found a
systematic way to calculate
01:06:21that the smallest number of eggs she
could have had in the tray is 52.
01:06:25But the more amazing thing is
that you can capture
01:06:28such a large number, like 52,
01:06:30by using these small numbers
like three, five and seven.
01:06:33This way of looking at numbers
01:06:35would become a dominant theme
over the last two centuries.
01:06:48By the 6th century AD, the Chinese
remainder theorem was being used
01:06:52in ancient Chinese astronomy
to measure planetary movement.
01:06:56But today it still
has practical uses.
01:06:59Internet cryptography
encodes numbers using mathematics
01:07:04that has its origins
in the Chinese remainder theorem.
01:07:16By the 13th century,
01:07:18mathematics was long established
on the curriculum,
01:07:21with over 30 mathematics schools
scattered across the country.
01:07:25The golden age of
Chinese maths had arrived.
01:07:31And its most important mathematician
was called Qin Jiushao.
01:07:37Legend has it that Qin Jiushao
was something of a scoundrel.
01:07:42He was a fantastically
corrupt imperial administrator
01:07:46who crisscrossed China,
lurching from one post to another.
01:07:49Repeatedly sacked for embezzling
government money,
01:07:53he poisoned anyone
who got in his way.
01:07:58Qin Jiushao
was reputedly described as
01:08:01as violent as a tiger or a wolf
01:08:03and as poisonous
as a scorpion or a viper
01:08:06so, not surprisingly,
he made a fierce warrior.
01:08:09For ten years, he fought
against the invading Mongols,
01:08:12but for much of that time he was
complaining that his military life
01:08:16took him away
from his true passion.
01:08:18No, not corruption, but mathematics.
01:08:33Qin started trying
to solve equations
01:08:35that grew out of trying
to measure the world around us.
01:08:38Quadratic equations involve numbers
01:08:40that are squared, or to the power
of two - say, five times five.
01:08:46The ancient Mesopotamians
01:08:48had already realised
that these equations
01:08:50were perfect for measuring flat,
two-dimensional shapes,
01:08:54like Tiananmen Square.
01:08:59But Qin was interested
01:09:01in more complicated equations -
cubic equations.
01:09:06These involve numbers
which are cubed,
01:09:09or to the power of three -
say, five times five times five,
01:09:14and they were perfect for capturing
three-dimensional shapes,
01:09:18like Chairman Mao's mausoleum.
01:09:22Qin found a way
of solving cubic equations,
01:09:24and this is how it worked.
01:09:30Say Qin wants to know
01:09:33the exact dimensions
of Chairman Mao's mausoleum.
01:09:38He knows the volume of the building
01:09:41and the relationships
between the dimensions.
01:09:45In order to get his answer,
01:09:48Qin uses what he knows
to produce a cubic equation.
01:09:52He then makes
an educated guess at the dimensions.
01:09:56Although he's captured a good
proportion of the mausoleum,
01:10:00there are still bits left over.
01:10:04Qin takes these bits
and creates a new cubic equation.
01:10:08He can now refine his first guess
01:10:10by trying to find a solution to
this new cubic equation, and so on.
01:10:17Each time he does this,
the pieces he's left with
01:10:20get smaller and smaller and his
guesses get better and better.
01:10:27What's striking is that Qin's
method for solving equations
01:10:30wasn't discovered in the West
until the 17th century,
01:10:33when Isaac Newton came up with a
very similar approximation method.
01:10:38The power of this technique is
01:10:40that it can be applied
to even more complicated equations.
01:10:44Qin even used his techniques
to solve an equation
01:10:48involving numbers
up to the power of ten.
01:10:50This was extraordinary stuff -
highly complex mathematics.
01:10:57Qin may have been years
ahead of his time,
01:10:59but there was a problem
with his technique.
01:11:02It only gave him
an approximate solution.
01:11:04That might be good enough for an
engineer - not for a mathematician.
01:11:08Mathematics is an exact science.
We like things to be precise,
01:11:12and Qin just couldn't
come up with a formula
01:11:15to give him an exact solution
to these complicated equations.
01:11:26China had made
great mathematical leaps,
01:11:29but the next great mathematical
breakthroughs were to happen
01:11:33in a country lying
to the southwest of China -
01:11:35a country that had a rich
mathematical tradition
01:11:39that would change
the face of maths for ever.
01:12:12India's first great mathematical
gift lay in the world of number.
01:12:17Like the Chinese, the Indians had
discovered the mathematical benefits
01:12:21of the decimal place-value system
01:12:23and were using it by the middle
of the 3rd century AD.
01:12:29It's been suggested that
the Indians learned the system
01:12:33from Chinese merchants travelling
in India with their counting rods,
01:12:37or they may well just have
stumbled across it themselves.
01:12:41It's all such a long time ago
that it's shrouded in mystery.
01:12:47We may never know how the Indians
came up with their number system,
01:12:50but we do know that they refined
and perfected it,
01:12:53creating the ancestors for the nine
numerals used across the world now.
01:12:57Many rank the Indian
system of counting
01:13:00as one of the greatest intellectual
innovations of all time,
01:13:03developing into the closest thing
we could call a universal language.
01:13:26But there was one number missing,
01:13:28and it was the Indians who
would introduce it to the world.
01:13:38The earliest known recording of this
number dates from the 9th century,
01:13:43though it was probably in
practical use for centuries before.
01:13:48This strange new numeral
is engraved on the wall
01:13:52of small temple in the fort
of Gwalior in central India.
01:14:00So here we are in one of the holy
sites of the mathematical world,
01:14:04and what I'm looking for
is in this inscription on the wall.
01:14:08Up here are some numbers, and...
01:14:11here's the new number.
01:14:13It's zero.
01:14:20It's astonishing to think
that before the Indians invented it,
01:14:24there was no number zero.
01:14:26To the ancient Greeks,
it simply hadn't existed.
01:14:30To the Egyptians, the Mesopotamians
and, as we've seen, the Chinese,
01:14:34zero had been in use but as
a placeholder, an empty space
01:14:38to show a zero inside a number.
01:14:44The Indians transformed zero
from a mere placeholder
01:14:47into a number
that made sense in its own right -
01:14:49a number for calculation,
for investigation.
01:14:53This brilliant conceptual leap
would revolutionise mathematics.
01:15:01Now, with just ten digits - zero
to nine - it was suddenly possible
01:15:05to capture astronomically large
numbers
01:15:08in an incredibly efficient way.
01:15:13But why did the Indians
make this imaginative leap?
01:15:17Well, we'll never know for sure,
01:15:19but it's possible that the idea and
symbol that the Indians use for zero
01:15:23came from calculations they did
with stones in the sand.
01:15:26When stones were removed
from the calculation,
01:15:29a small, round hole was left
in its place,
01:15:32representing the movement
from something to nothing.
01:15:38But perhaps there is also a cultural
reason for the invention of zero.
01:15:43HORNS BLOW AND DRUMS BANG
01:15:46METALLIC BEATING
01:15:51For the ancient Indians, the
concepts of nothingness and eternity
01:15:56lay at the very heart
of their belief system.
01:16:03BELL CLANGS AND SILENCE FALLS
01:16:08In the religions of India, the
universe was born from nothingness,
01:16:12and nothingness is
the ultimate goal of humanity.
01:16:15So it's perhaps not surprising
01:16:17that a culture that so
enthusiastically embraced the void
01:16:21should be happy
with the notion of zero.
01:16:24The Indians even used the word for
the philosophical idea of the void,
01:16:29shunya, to represent
the new mathematical term "zero".
01:16:46In the 7th century, the brilliant
Indian mathematician Brahmagupta
01:16:51proved some of the essential
properties of zero.
01:17:00Brahmagupta's rules
about calculating with zero
01:17:03are taught in schools
all over the world to this day.
01:17:08One plus zero equals one.
01:17:12One minus zero equals one.
01:17:15One times zero is equal to zero.
01:17:23But Brahmagupta came a cropper when
he tried to do one divided by zero.
01:17:27After all, what number
times zero equals one?
01:17:30It would require a new mathematical
concept, that of infinity,
01:17:34to make sense of dividing by zero,
01:17:36and the breakthrough was made by a
12th-century Indian mathematician
01:17:40called Bhaskara II,
and it works like this.
01:17:43If I take a fruit and I divide
it into halves, I get two pieces,
01:17:50so one divided by a half is two.
01:17:53If I divide it into thirds,
I get three pieces.
01:17:56So when I divide it into smaller
and smaller fractions,
01:17:59I get more and more pieces,
so ultimately,
01:18:03when I divide by a piece
01:18:05which is of zero size,
I'll have infinitely many pieces.
01:18:09So for Bhaskara,
one divided by zero is infinity.
01:18:21But the Indians would go further
in their calculations with zero.
01:18:26For example, if you take three
from three and get zero,
01:18:31what happens when you take
four from three?
01:18:34It looks like you have nothing,
01:18:36but the Indians recognised
that this
01:18:39was a new sort of nothing -
negative numbers.
01:18:42The Indians called them "debts",
because they solved equations like,
01:18:46"If I have three batches
of material and take four away,
01:18:49"how many have I left?"
01:18:55This may seem odd and impractical,
01:18:57but that was the beauty
of Indian mathematics.
01:19:00Their ability to come up
with negative numbers and zero
01:19:03was because they thought of
numbers as abstract entities.
01:19:07They weren't just for counting
and measuring pieces of cloth.
01:19:10They had a life of their own,
floating free of the real world.
01:19:13This led to an explosion
of mathematical ideas.
01:19:29The Indians' abstract approach
to mathematics soon revealed
01:19:33a new side to the problem of
how to solve quadratic equations.
01:19:37That is equations including
numbers to the power of two.
01:19:42Brahmagupta's understanding of
negative numbers allowed him to see
01:19:46that quadratic equations
always have two solutions,
01:19:49one of which could be negative.
01:19:54Brahmagupta went even further,
01:19:55solving quadratic equations
with two unknowns,
01:19:58a question which wouldn't be
considered in the West until 1657,
01:20:02when French mathematician Fermat
01:20:04challenged his colleagues
with the same problem.
01:20:07Little did he know that they'd
been beaten to a solution
01:20:10by Brahmagupta
1,000 years earlier.
01:20:18Brahmagupta was beginning to find
abstract ways of solving equations,
01:20:23but astonishingly,
he was also developing
01:20:26a new mathematical language
to express that abstraction.
01:20:31Brahmagupta was experimenting with
ways of writing his equations down,
01:20:35using the initials
of the names of different colours
01:20:39to represent unknowns
in his equations.
01:20:43A new mathematical language
was coming to life,
01:20:46which would ultimately lead
to the x's and y's
01:20:48which fill today's
mathematical journals.
01:21:05But it wasn't just new
notation that was being developed.
01:21:12Indian mathematicians
were responsible for making
01:21:14fundamental new discoveries
in the theory of trigonometry.
01:21:21The power of trigonometry
is that it acts like a dictionary,
01:21:25translating geometry into numbers
and back.
01:21:28Although first developed by the
ancient Greeks,
01:21:32it was in the hands
of the Indian mathematicians
01:21:34that the subject truly flourished.
01:21:36At its heart lies the study
of right-angled triangles.
01:21:42In trigonometry,
you can use this angle here
01:21:46to find the ratios of the opposite
side to the longest side.
01:21:51There's a function
called the sine function
01:21:53which, when you input the angle,
outputs the ratio.
01:21:56So for example in this triangle,
the angle is about 30 degrees,
01:22:00so the output of the sine function
is a ratio of one to two,
01:22:04telling me that this side is half
the length of the longest side.
01:22:11The sine function
enables you to calculate distances
01:22:15when you're not able to make
an accurate measurement.
01:22:20To this day, it's used
in architecture and engineering.
01:22:24The Indians used it
to survey the land around them,
01:22:26navigate the seas and, ultimately,
chart the depths of space itself.
01:22:33It was central to the work
of observatories,
01:22:36like this one in Delhi,
01:22:38where astronomers
would study the stars.
01:22:41The Indian astronomers
could use trigonometry
01:22:43to work out the relative distance
between Earth and the moon
01:22:47and Earth and the sun.
01:22:48You can only make the calculation
when the moon is half full,
01:22:52because that's when it's
directly opposite the sun,
01:22:55so that the sun, moon and Earth
create a right-angled triangle.
01:23:01Now, the Indians could measure
01:23:03that the angle between the sun
and the observatory
01:23:06was one-seventh of a degree.
01:23:09The sine function of
one-seventh of a degree
01:23:13gives me the ratio of 400:1.
01:23:16This means the sun is 400 times
further from Earth than the moon is.
01:23:22So using trigonometry,
01:23:24the Indian mathematicians
could explore the solar system
01:23:27without ever having
to leave the surface of the Earth.
01:23:37The ancient Greeks had been the
first to explore the sine function,
01:23:41listing precise values
for some angles,
01:23:45but they couldn't calculate
the sines of every angle.
01:23:49The Indians were to go much further,
setting themselves a mammoth task.
01:23:54The search was on to find a way
01:23:56to calculate the sine function
of any angle you might be given.
01:24:16The breakthrough in the search for
the sine function of every angle
01:24:20would be made here in
Kerala in south India.
01:24:23In the 15th century,
this part of the country
01:24:26became home to one of the most
brilliant schools of mathematicians
01:24:30to have ever worked.
01:24:33Their leader was called Madhava,
and he was to make
01:24:37some extraordinary
mathematical discoveries.
01:24:44The key to Madhava's success
was the concept of the infinite.
01:24:48Madhava discovered that you could
add up infinitely many things
01:24:51with dramatic effects.
01:24:53Previous cultures had been nervous
of these infinite sums,
01:24:56but Madhava
was happy to play with them.
01:24:59For example,
here's how one can be made up
01:25:01by adding
infinitely many fractions.
01:25:05I'm heading from zero
to one on my boat,
01:25:10but I can split my journey up
into infinitely many fractions.
01:25:14So I can get to a half,
01:25:17then I can sail on a quarter,
01:25:20then an eighth, then a sixteenth,
and so on.
01:25:23The smaller the fractions I move,
the nearer to one I get,
01:25:28but I'll only get there once I've
added up infinitely many fractions.
01:25:34Physically and philosophically,
01:25:37it seems rather a challenge
to add up infinitely many things,
01:25:40but the power of mathematics is
to make sense of the impossible.
01:25:44By producing a language
01:25:46to articulate and manipulate
the infinite,
01:25:48you can prove
that after infinitely many steps
01:25:51you'll reach your destination.
01:25:56Such infinite sums are called
infinite series, and Madhava
01:26:00was doing a lot of research
into the connections
01:26:03between these series
and trigonometry.
01:26:07First, he realised that
he could use the same principle
01:26:11of adding up infinitely many
fractions to capture
01:26:13one of the most important
numbers in mathematics - pi.
01:26:19Pi is the ratio of the circle's
circumference to its diameter.
01:26:24It's a number that appears
in all sorts of mathematics,
01:26:28but is especially useful
for engineers,
01:26:31because any measurements
involving curves soon require pi.
01:26:37So for centuries, mathematicians
searched for a precise value for pi.
01:26:47It was in 6th-century India
that the mathematician Aryabhata
01:26:51gave a very accurate approximation
for pi - namely 3.1416.
01:26:56He went on to use this
01:26:57to make a measurement
of the circumference of the Earth,
01:27:00and he got it as 24,835 miles,
01:27:04which amazingly is only 70 miles
away from its true value.
01:27:08But it was in Kerala
in the 15th century
01:27:11that Madhava realised
he could use infinity
01:27:14to get an exact formula for pi.
01:27:19By successively adding
and subtracting different fractions,
01:27:23Madhava could hone in
on an exact formula for pi.
01:27:28First, he moved four steps
up the number line.
01:27:33That took him way past pi.
01:27:36So next he took
four-thirds of a step,
01:27:40or one-and-one-third
steps, back.
01:27:43Now he'd come too far
the other way.
01:27:46So he headed forward
four-fifths of a step.
01:27:50Each time, he alternated between
four divided by the next odd number.
01:28:01He zigzagged up and down
the number line,
01:28:05getting closer and closer to pi.
01:28:07He discovered that if you went
through all the odd numbers,
01:28:10infinitely many of them,
you would hit pi exactly.
01:28:18I was taught at university
that this formula for pi
01:28:21was discovered by the 17th-century
German mathematician Leibniz,
01:28:25but amazingly, it was actually
discovered here in Kerala
01:28:28two centuries earlier by Madhava.
01:28:30He went on to use
the same sort of mathematics
01:28:33to get infinite-series expressions
01:28:35for the sine formula
in trigonometry.
01:28:37And the wonderful thing is that
you can use these formulas now
01:28:41to calculate the sine of any angle
to any degree of accuracy.
01:28:55It seems incredible that
the Indians made these discoveries
01:28:59centuries before
Western mathematicians.
01:29:05And it says a lot about our attitude
in the West to non-Western cultures
01:29:09that we nearly always
claim their discoveries as our own.
01:29:13What is clear is the West has
been very slow to give due credit
01:29:17to the major breakthroughs
made in non-Western mathematics.
01:29:21Madhava wasn't the only
mathematician to suffer this way.
01:29:24As the West came into contact
more and more with the East
01:29:27during the 18th and 19th centuries,
01:29:29there was a widespread dismissal
and denigration
01:29:32of the cultures
they were colonising.
01:29:34The natives, it was assumed,
couldn't have anything
01:29:36of intellectual worth
to offer the West.
01:29:39It's only now, at the beginning
of the 21st century,
01:29:42that history is being rewritten.
01:29:43But Eastern mathematics was to have
a major impact in Europe,
01:29:48thanks to the development
of one of the major powers
01:29:51of the medieval world.
01:30:16In the 7th century,
a new empire began to spread
01:30:19across the Middle East.
01:30:21The teachings
of the Prophet Mohammed
01:30:24inspired a vast
and powerful Islamic empire
01:30:27which soon stretched
from India in the east
01:30:29to here in Morocco
in the west.
01:30:40And at the heart of this empire
lay a vibrant intellectual culture.
01:30:50A great library and centre of
learning was established in Baghdad.
01:30:55Called the House of Wisdom,
its teaching spread
01:30:58throughout the Islamic empire,
01:31:00reaching schools
like this one here in Fez.
01:31:04Subjects studied included astronomy,
medicine,
01:31:07chemistry, zoology
01:31:09and mathematics.
01:31:12The Muslim scholars collected
and translated many ancient texts,
01:31:17effectively saving
them for posterity.
01:31:19In fact, without their intervention,
we may never have known
01:31:22about the ancient cultures of
Egypt, Babylon, Greece and India.
01:31:26But the scholars at the
House of Wisdom weren't content
01:31:29simply with translating
other people's mathematics.
01:31:32They wanted to create
a mathematics of their own,
01:31:35to push the subject forward.
01:31:41Such intellectual curiosity
was actively encouraged
01:31:45in the early centuries
of the Islamic empire.
01:31:50The Koran asserted
the importance of knowledge.
01:31:53Learning was nothing less
than a requirement of God.
01:32:00In fact, the needs of Islam
demanded mathematical skill.
01:32:04The devout needed to calculate
the time of prayer
01:32:06and the direction of Mecca
to pray towards,
01:32:09and the prohibition
of depicting the human form
01:32:12meant that they had to use
01:32:14much more geometric patterns
to cover their buildings.
01:32:17The Muslim artists discovered all
the different sorts of symmetry
01:32:21that you can depict
on a two-dimensional wall.
01:32:32The director of the House of Wisdom
in Baghdad
01:32:35was a Persian scholar
called Muhammad Al-Khwarizmi.
01:32:42Al-Khwarizmi was an exceptional
mathematician who was responsible
01:32:47for introducing two key
mathematical concepts to the West.
01:32:51Al-Khwarizmi recognised
the incredible potential
01:32:54that the Hindu numerals had
01:32:56to revolutionise
mathematics and science.
01:32:59His work explaining
the power of these numbers
01:33:01to speed up calculations
and do things effectively
01:33:04was so influential that it wasn't
long before they were adopted
01:33:08as the numbers of choice amongst the
mathematicians of the Islamic world.
01:33:12In fact, these numbers
have now become known
01:33:14as the Hindu-Arabic numerals.
01:33:17These numbers -
one to nine and zero -
01:33:20are the ones we use today
all over the world.
01:33:28But Al-Khwarizmi was to create
a whole new mathematical language.
01:33:35It was called algebra
01:33:37and was named after the title of
his book Al-jabr W'al-muqabala,
01:33:41or Calculation By Restoration
Or Reduction.
01:33:49Algebra is the grammar that
underlies the way that numbers work.
01:33:55It's a language
that explains the patterns
01:33:57that lie behind
the behaviour of numbers.
01:34:00It's a bit like a code
for running a computer program.
01:34:04The code will work whatever the
numbers you feed in to the program.
01:34:09For example, mathematicians
might have discovered
01:34:13that if you take a number
and square it,
01:34:15that's always one more
than if you'd taken
01:34:18the numbers either side
and multiplied those together.
01:34:21For example, five times five is 25,
01:34:24which is one more
than four times six - 24.
01:34:28Six times six is always one more
than five times seven and so on.
01:34:32But how can you be sure
01:34:33that this is going to work
whatever numbers you take?
01:34:37To explain the pattern underlying
these calculations,
01:34:39let's use the dyeing holes
in this tannery.
01:34:50If we take a square of
25 holes, running five by five,
01:34:55and take one row of five away
and add it to the bottom,
01:34:59we get six by four
with one left over.
01:35:04But however many holes there
are on the side of the square,
01:35:07we can always move one row of holes
down in a similar way
01:35:11to be left with a rectangle
of holes with one left over.
01:35:17Algebra was a huge breakthrough.
01:35:19Here was a new language
01:35:21to be able to analyse
the way that numbers worked.
01:35:24Previously, the Indians
and the Chinese
01:35:26had considered
very specific problems,
01:35:29but Al-Khwarizmi went
from the specific to the general.
01:35:32He developed systematic ways
to be able to analyse problems
01:35:36so that the solutions would work
whatever the numbers that you took.
01:35:39This language is used
across the mathematical world today.
01:35:45Al-Khwarizmi's great breakthrough
came when he applied algebra
01:35:49to quadratic equations -
01:35:51that is equations including
numbers to the power of two.
01:35:54The ancient Mesopotamians
had devised
01:35:57a cunning method to solve
particular quadratic equations,
01:36:01but it was Al-Khwarizmi's
abstract language of algebra
01:36:05that could finally express
why this method always worked.
01:36:10This was a great conceptual leap
01:36:13and would ultimately lead to a
formula that could be used to solve
01:36:16any quadratic equation,
whatever the numbers involved.
01:36:29The next mathematical Holy Grail
01:36:31was to find a general method that
could solve all cubic equations -
01:36:35equations including numbers
to the power of three.
01:36:56It was an 11th-century
Persian mathematician
01:36:59who took up the challenge of
cracking the problem of the cubic.
01:37:07His name was Omar Khayyam,
and he travelled widely
01:37:10across the Middle East,
calculating as he went.
01:37:16But he was famous for another,
very different, reason.
01:37:20Khayyam was a celebrated poet,
01:37:23author of the great
epic poem the Rubaiyat.
01:37:29It may seem a bit odd that a poet
was also a master mathematician.
01:37:34After all, the combination
doesn't immediately spring to mind.
01:37:37But there's quite a lot of
similarity between the disciplines.
01:37:41Poetry, with its rhyming structure
and rhythmic patterns,
01:37:44resonates strongly with constructing
a logical mathematical proof.
01:37:51Khayyam's major mathematical work
01:37:54was devoted to finding the general
method to solve all cubic equations.
01:37:59Rather than looking
at particular examples,
01:38:02Khayyam carried out a systematic
analysis of the problem,
01:38:07true to the algebraic spirit
of Al-Khwarizmi.
01:38:12Khayyam's analysis revealed
for the first time
01:38:15that there were several
different sorts of cubic equation.
01:38:18But he was still very influenced
01:38:20by the geometric heritage
of the Greeks.
01:38:23He couldn't separate the algebra
from the geometry.
01:38:26In fact, he wouldn't even consider
equations in higher degrees,
01:38:29because they described objects
in more than three dimensions,
01:38:32something he saw as impossible.
01:38:34Although the geometry allowed him
01:38:36to analyse these cubic equations
to some extent,
01:38:39he still couldn't come up
with a purely algebraic solution.
01:38:44It would be another 500 years before
mathematicians could make the leap
01:38:50and find a general solution
to the cubic equation.
01:38:55And that leap would finally be made
in the West - in Italy.
01:39:14During the centuries in which China,
India and the Islamic empire
01:39:17had been in the ascendant,
01:39:19Europe had fallen under
the shadow of the Dark Ages.
01:39:25All intellectual life, including the
study of mathematics, had stagnated.
01:39:34But by the 13th century,
things were beginning to change.
01:39:40Led by Italy, Europe was starting
to explore and trade with the East.
01:39:45With that contact came the spread
of Eastern knowledge to the West.
01:39:50It was the son of a customs official
01:39:52that would become Europe's first
great medieval mathematician.
01:39:55As a child, he travelled around
North Africa with his father,
01:39:59where he learnt about the
developments of Arabic mathematics
01:40:02and especially the benefits
of the Hindu-Arabic numerals.
01:40:05When he got home to Italy
he wrote a book
01:40:07that would be hugely influential
01:40:09in the development
of Western mathematics.
01:40:28That mathematician was
Leonardo of Pisa,
01:40:30better known as Fibonacci,
01:40:33and in his Book Of Calculating,
01:40:36Fibonacci promoted
the new number system,
01:40:39demonstrating how simple it was
compared to the Roman numerals
01:40:43that were in use across Europe.
01:40:45Calculations were far easier,
a fact that had huge consequences
01:40:50for anyone dealing with numbers -
01:40:54pretty much everyone,
from mathematicians to merchants.
01:40:57But there was widespread
suspicion of these new numbers.
01:41:01Old habits die hard, and the
authorities just didn't trust them.
01:41:05Some believed that they would
be more open to fraud -
01:41:08that you could tamper with them.
01:41:09Others believed that they'd be
so easy to use for calculations
01:41:13that it would empower the masses,
taking authority away
01:41:16from the intelligentsia who knew
how to use the old sort of numbers.
01:41:26The city of Florence
even banned them in 1299,
01:41:30but over time,
common sense prevailed,
01:41:33the new system spread
throughout Europe,
01:41:36and the old Roman system
slowly became defunct.
01:41:39At last, the Hindu-Arabic numerals,
zero to nine, had triumphed.
01:41:47Today Fibonacci is best known for
the discovery of some numbers,
01:41:50now called the Fibonacci sequence,
that arose when he was trying
01:41:54to solve a riddle
about the mating habits of rabbits.
01:41:57Suppose a farmer
has a pair of rabbits.
01:41:59Rabbits take two months
to reach maturity,
01:42:02and after that they give birth to
another pair of rabbits each month.
01:42:06So the problem was how to determine
01:42:08how many pairs of rabbits there
will be in any given month.
01:42:13Well, during the first month
you have one pair of rabbits,
01:42:18and since they haven't matured,
they can't reproduce.
01:42:23During the second month,
there is still only one pair.
01:42:27But at the beginning of the
third month, the first pair
01:42:30reproduces for the first time,
so there are two pairs of rabbits.
01:42:35At the beginning
of the fourth month,
01:42:37the first pair reproduces again,
01:42:39but the second pair is not mature
enough, so there are three pairs.
01:42:45In the fifth month,
the first pair reproduces
01:42:48and the second pair
reproduces for the first time,
01:42:52but the third pair is still too
young, so there are five pairs.
01:42:57The mating ritual continues,
01:42:59but what you soon realise is
01:43:01the number of pairs of rabbits
you have in any given month
01:43:04is the sum of the pairs of
rabbits that you have had
01:43:08in each of the two previous months,
so the sequence goes...
01:43:121...1...2...3...
01:43:165...8...13...
01:43:2021...34...55...and so on.
01:43:24The Fibonacci numbers are
nature's favourite numbers.
01:43:28It's not just rabbits that use them.
01:43:30The number of petals on a flower
is invariably a Fibonacci number.
01:43:34They run up and down pineapples
if you count the segments.
01:43:38Even snails use them
to grow their shells.
01:43:41Wherever you find growth in nature,
you find the Fibonacci numbers.
01:43:50But the next major breakthrough
in European mathematics
01:43:53wouldn't happen
until the early 16th century.
01:43:56It would involve
01:43:58finding the general method that
would solve all cubic equations,
01:44:03and it would happen here
in the Italian city of Bologna.
01:44:09The University of Bologna
was the crucible
01:44:12of European mathematical thought at
the beginning of the 16th century.
01:44:19Pupils from all over Europe
flocked here and developed
01:44:23a new form of spectator sport -
the mathematical competition.
01:44:30Large audiences would gather to
watch mathematicians
01:44:33challenge each other with numbers, a
kind of intellectual fencing match.
01:44:38But even in this
questioning atmosphere
01:44:41it was believed that some problems
were just unsolvable.
01:44:45It was generally assumed
that finding a general method
01:44:50to solve all cubic equations
was impossible.
01:44:53But one scholar
was to prove everyone wrong.
01:44:59His name was Tartaglia,
01:45:01but he certainly didn't look
01:45:03the heroic architect
of a new mathematics.
01:45:06At the age of 12, he'd been
slashed across the face
01:45:10with a sabre
by a rampaging French army.
01:45:12The result
was a terrible facial scar
01:45:15and a devastating speech
impediment.
01:45:17In fact, Tartaglia was the nickname
he'd been given as a child
01:45:21and means "the stammerer".
01:45:28Shunned by his schoolmates,
01:45:30Tartaglia lost himself
in mathematics, and it wasn't long
01:45:36before he'd found the formula
to solve one type of cubic equation.
01:45:42But Tartaglia soon discovered
01:45:43that he wasn't the only one
to believe he'd cracked the cubic.
01:45:47A young Italian called Fior
was boasting
01:45:50that he too held the secret
formula for solving cubic equations.
01:45:55When news broke
about the discoveries
01:45:58made by the two mathematicians,
01:46:01a competition was arranged to
pit them against each other.
01:46:05The intellectual fencing match
of the century was about to begin.
01:46:16The trouble was that Tartaglia
01:46:18only knew how to solve one sort
of cubic equation,
01:46:21and Fior was ready to challenge him
01:46:23with questions
about a different sort.
01:46:26But just a few days
before the contest,
01:46:28Tartaglia worked out how to
solve this different sort,
01:46:31and with this new weapon in his
arsenal he thrashed his opponent,
01:46:34solving all the questions
in under two hours.
01:46:40Tartaglia went on
01:46:42to find the formula to solve
all types of cubic equations.
01:46:47News soon spread,
and a mathematician in Milan
01:46:49called Cardano became so
desperate to find the solution
01:46:53that he persuaded a reluctant
Tartaglia to reveal the secret,
01:46:58but on one condition -
01:47:00that Cardano keep the secret
and never publish.
01:47:06But Cardano couldn't resist
01:47:08discussing Tartaglia's solution
with his brilliant student, Ferrari.
01:47:12As Ferrari got to grips
with Tartaglia's work,
01:47:15he realised that he could use it
to solve
01:47:18the more complicated quartic
equation, an amazing achievement.
01:47:21Cardano couldn't deny his
student his just rewards,
01:47:24and he broke his vow of secrecy,
publishing Tartaglia's work
01:47:28together with Ferrari's
brilliant solution of the quartic.
01:47:34Poor Tartaglia never recovered
and died penniless,
01:47:38and to this day, the formula
that solves the cubic equation
01:47:41is known as Cardano's formula.
01:47:53Tartaglia may not have won glory
in his lifetime,
01:47:56but his mathematics managed to
solve a problem that had bewildered
01:48:00the great mathematicians
of China, India and the Arab world.
01:48:06It was the first
great mathematical breakthrough
01:48:10to happen in modern Europe.
01:48:15The Europeans now had in their
hands the new language of algebra,
01:48:19the powerful techniques
of the Hindu-Arabic numerals
01:48:23and the beginnings
of the mastery of the infinite.
01:48:26It was time for the Western world
01:48:27to start writing
its own mathematical stories
01:48:30in the language of the East.
01:48:31The mathematical revolution
was about to begin.
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