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From measuring time to understanding
our position in the universe,

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from mapping the Earth
to navigating the seas,

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from man's earliest inventions
to today's advanced technologies,

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mathematics has been the pivot
on which human life depends.

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The first steps of man's
mathematical journey

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were taken by the ancient cultures
of Egypt, Mesopotamia and Greece -

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cultures which created the basic
language of number and calculation.

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But when ancient Greece
fell into decline,

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mathematical progress
juddered to a halt.

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But that was in the West.

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In the East, mathematics
would reach dynamic new heights.

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But in the West,
much of this mathematical heritage

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has been conveniently forgotten
or shaded from view.

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Due credit has not been given to
the great mathematical breakthroughs

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that ultimately changed
the world we live in.

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This is the untold story
of the mathematics of the East

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that would transform the West and
give birth to the modern world.

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The Great Wall of China
stretches for thousands of miles.

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Nearly 2,000 years in the making,
this vast, defensive wall

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was begun in 220BC
to protect China's growing empire.

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The Great Wall of China
is an amazing feat of engineering

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built over rough and high
countryside.

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As soon as they started building,

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the ancient Chinese realised
they had to make calculations

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about distances, angles of elevation
and amounts of material.

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So it isn't surprising
that this inspired

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some very clever mathematics
to help build Imperial China.

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At the heart of ancient
Chinese mathematics

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was an incredibly simple
number system

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which laid the foundations for the
way we count in the West today.

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When a mathematician wanted to do a
sum, he would use small bamboo rods.

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These rods were arranged to
represent the numbers one to nine.

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They were then placed in columns,

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each column representing units,
tens,

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hundreds, thousands and so on.

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So the number 924
was represented by putting

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the symbol 4 in the units column,
the symbol 2 in the tens column

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and the symbol 9
in the hundreds column.

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This is what we call a decimal
place-value system,

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and it's very similar
to the one we use today.

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We too use numbers from one to nine,
and we use their position

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to indicate whether it's units,
tens, hundreds or thousands.

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But the power of these rods is that
it makes calculations very quick.

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In fact, the way the ancient
Chinese did their calculations

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is very similar to the way
we learn today in school.

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Not only were the ancient Chinese

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the first to use a decimal
place-value system,

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but they did so over 1,000 years
before we adopted it in the West.

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But they only used it
when calculating with the rods.

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When writing the numbers down,

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the ancient Chinese
didn't use the place-value system.

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Instead, they used a far
more laborious method,

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in which special symbols stood for
tens, hundreds, thousands and so on.

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So the number 924
would be written out

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as nine hundreds,
two tens and four.

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Not quite so efficient.

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The problem was

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that the ancient Chinese didn't
have a concept of zero.

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They didn't have a symbol for zero.
It just didn't exist as a number.

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Using the counting rods,

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they would use a blank space
where today we would write a zero.

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The problem came with trying to
write down this number, which is why

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they had to create these new symbols
for tens, hundreds and thousands.

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Without a zero, the written
number was extremely limited.

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But the absence of zero
didn't stop

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the ancient Chinese from
making giant mathematical steps.

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In fact,
there was a widespread fascination

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with number in ancient China.

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According to legend,
the first sovereign of China,

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the Yellow Emperor,
had one of his deities

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create mathematics in 2800BC,

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believing that number held cosmic
significance. And to this day,

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the Chinese still believe in
the mystical power of numbers.

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Odd numbers are seen as male,
even numbers, female.

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The number four
is to be avoided at all costs.

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The number eight
brings good fortune.

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And the ancient Chinese were
drawn to patterns in numbers,

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developing their own
rather early version of sudoku.

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It was called the magic square.

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Legend has it that thousands of
years ago, Emperor Yu was visited

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by a sacred turtle that came out
of the depths of the Yellow River.

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On its back were numbers

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arranged into a magic square,
a little like this.

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In this square,

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which was regarded as having
great religious significance,

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all the numbers in each line -
horizontal, vertical and diagonal -

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all add up to the same number - 15.

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Now, the magic square may be
no more than a fun puzzle,

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but it shows
the ancient Chinese fascination

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with mathematical patterns,
and it wasn't too long

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before they were creating
even bigger magic squares

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with even greater magical
and mathematical powers.

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But mathematics also played

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a vital role in the running
of the emperor's court.

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The calendar and the movement
of the planets

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were of the utmost
importance to the emperor,

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influencing all his decisions, even
down to the way his day was planned,

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so astronomers became prized
members of the imperial court,

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and astronomers were
always mathematicians.

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Everything in the emperor's life
was governed by the calendar,

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and he ran his affairs
with mathematical precision.

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The emperor even got
his mathematical advisors

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to come up with a system
to help him sleep his way

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through the vast number of women
he had in his harem.

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Never one to miss a trick,
the mathematical advisors decided

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to base the harem on a mathematical
idea called a geometric progression.

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Maths has never had
such a fun purpose!

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Legend has it that
in the space of 15 nights,

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the emperor had to sleep
with 121 women...

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..the empress,

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three senior consorts,

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nine wives,

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27 concubines

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and 81 slaves.

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The mathematicians
would soon have realised

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that this was a geometric
progression - a series of numbers

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in which you get
from one number to the next

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by multiplying the same number
each time - in this case, three.

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Each group of women is three times
as large as the previous group,

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so the mathematicians could quickly
draw up a rota to ensure that,

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in the space of 15 nights,

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the emperor slept
with every woman in the harem.

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The first night
was reserved for the empress.

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The next was for the three
senior consorts.

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The nine wives came next,

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and then the 27 concubines were
chosen in rotation, nine each night.

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And then finally,
over a period of nine nights,

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the 81 slaves were dealt with
in groups of nine.

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Being the emperor certainly
required stamina,

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a bit like being a mathematician,

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but the object is clear -

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to procure the best
possible imperial succession.

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The rota ensured that the emperor

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slept with the ladies of highest
rank closest to the full moon,

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when their yin, their female force,

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would be at its highest and be able
to match his yang, or male force.

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The emperor's court wasn't alone
in its dependence on mathematics.

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It was central to the running
of the state.

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Ancient China was a vast and growing
empire with a strict legal code,

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widespread taxation

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and a standardised system
of weights, measures and money.

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The empire needed

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a highly trained civil service,
competent in mathematics.

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And to educate these civil servants
was a mathematical textbook,

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probably written in around 200BC -
the Nine Chapters.

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The book is a compilation
of 246 problems

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in practical areas such as trade,
payment of wages and taxes.

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And at the heart
of these problems lies

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one of the central themes of
mathematics, how to solve equations.

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Equations are a little bit
like cryptic crosswords.

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You're given a certain amount
of information

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about some unknown numbers,
and from that information

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you've got to deduce what
the unknown numbers are.

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For example,
with my weights and scales,

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I've found out that one plum...

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..together with three peaches

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weighs a total of 15 grams.

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But...

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..two plums

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together with one peach

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weighs a total of 10g.

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From this information, I can
deduce what a single plum weighs

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and a single peach weighs,
and this is how I do it.

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If I take the first set of scales,

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one plum and three peaches
weighing 15g,

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and double it, I get two plums
and six peaches weighing 30g.

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If I take this and subtract from it
the second set of scales -

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that's two plums
and a peach weighing 10g -

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I'm left with
an interesting result -

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no plums.

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Having eliminated the plums,

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I've discovered that
five peaches weighs 20g,

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so a single peach weighs 4g,

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and from this I can deduce
that the plum weighs 3g.

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The ancient Chinese went on
to apply similar methods

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to larger and larger numbers
of unknowns,

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using it to solve increasingly
complicated equations.

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What's extraordinary is

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that this particular
system of solving equations

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didn't appear in the West until
the beginning of the 19th century.

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In 1809, while analysing a rock
called Pallas in the asteroid belt,

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00:13:03,860 --> 00:13:05,700
Carl Friedrich Gauss,

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who would become known
as the prince of mathematics,

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00:13:08,500 --> 00:13:10,180
rediscovered this method

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which had been formulated
in ancient China centuries earlier.

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00:13:13,780 --> 00:13:17,740
Once again, ancient China
streets ahead of Europe.

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But the Chinese
were to go on to solve

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even more complicated equations
involving far larger numbers.

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In what's become known as
the Chinese remainder theorem,

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00:13:31,380 --> 00:13:35,540
the Chinese came up
with a new kind of problem.

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00:13:35,540 --> 00:13:38,780
In this, we know the number
that's left

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when the equation's unknown number
is divided by a given number -

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say, three, five or seven.

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00:13:46,500 --> 00:13:50,700
Of course, this is a fairly
abstract mathematical problem,

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00:13:50,700 --> 00:13:54,860
but the ancient Chinese still
couched it in practical terms.

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So a woman in the market has
a tray of eggs,

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00:13:59,940 --> 00:14:02,780
but she doesn't know
how many eggs she's got.

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00:14:02,780 --> 00:14:06,140
What she does know is that
if she arranges them in threes,

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00:14:06,140 --> 00:14:09,300
she has one egg left over.

200
00:14:09,300 --> 00:14:13,220
If she arranges them in fives,
she gets two eggs left over.

201
00:14:13,220 --> 00:14:16,180
But if she arranged them
in rows of seven,

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00:14:16,180 --> 00:14:18,980
she found she had
three eggs left over.

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00:14:18,980 --> 00:14:22,660
The ancient Chinese found a
systematic way to calculate

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00:14:22,660 --> 00:14:26,660
that the smallest number of eggs she
could have had in the tray is 52.

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00:14:26,660 --> 00:14:29,740
But the more amazing thing is
that you can capture

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00:14:29,740 --> 00:14:31,580
such a large number, like 52,

207
00:14:31,580 --> 00:14:35,100
by using these small numbers
like three, five and seven.

208
00:14:35,100 --> 00:14:37,100
This way of looking at numbers

209
00:14:37,100 --> 00:14:40,860
would become a dominant theme
over the last two centuries.

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00:14:49,500 --> 00:14:54,220
By the 6th century AD, the Chinese
remainder theorem was being used

211
00:14:54,220 --> 00:14:57,700
in ancient Chinese astronomy
to measure planetary movement.

212
00:14:57,700 --> 00:15:01,100
But today it still
has practical uses.

213
00:15:01,100 --> 00:15:05,940
Internet cryptography
encodes numbers using mathematics

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00:15:05,940 --> 00:15:10,140
that has its origins
in the Chinese remainder theorem.

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00:15:17,940 --> 00:15:19,740
By the 13th century,

216
00:15:19,740 --> 00:15:22,820
mathematics was long established
on the curriculum,

217
00:15:22,820 --> 00:15:26,820
with over 30 mathematics schools
scattered across the country.

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00:15:26,820 --> 00:15:30,780
The golden age of
Chinese maths had arrived.

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00:15:33,140 --> 00:15:36,700
And its most important mathematician
was called Qin Jiushao.

220
00:15:38,900 --> 00:15:43,620
Legend has it that Qin Jiushao
was something of a scoundrel.

221
00:15:43,620 --> 00:15:47,620
He was a fantastically
corrupt imperial administrator

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00:15:47,620 --> 00:15:51,180
who crisscrossed China,
lurching from one post to another.

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00:15:51,180 --> 00:15:54,700
Repeatedly sacked for embezzling
government money,

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00:15:54,700 --> 00:15:57,620
he poisoned anyone
who got in his way.

225
00:16:00,060 --> 00:16:02,580
Qin Jiushao
was reputedly described as

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00:16:02,580 --> 00:16:05,020
as violent as a tiger or a wolf

227
00:16:05,020 --> 00:16:08,100
and as poisonous
as a scorpion or a viper

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00:16:08,100 --> 00:16:11,220
so, not surprisingly,
he made a fierce warrior.

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00:16:11,220 --> 00:16:14,140
For ten years, he fought
against the invading Mongols,

230
00:16:14,140 --> 00:16:17,820
but for much of that time he was
complaining that his military life

231
00:16:17,820 --> 00:16:19,860
took him away
from his true passion.

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00:16:19,860 --> 00:16:22,860
No, not corruption, but mathematics.

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00:16:34,420 --> 00:16:36,980
Qin started trying
to solve equations

234
00:16:36,980 --> 00:16:40,260
that grew out of trying
to measure the world around us.

235
00:16:40,260 --> 00:16:42,140
Quadratic equations involve numbers

236
00:16:42,140 --> 00:16:46,940
that are squared, or to the power
of two - say, five times five.

237
00:16:47,620 --> 00:16:49,980
The ancient Mesopotamians

238
00:16:49,980 --> 00:16:52,380
had already realised
that these equations

239
00:16:52,380 --> 00:16:55,740
were perfect for measuring flat,
two-dimensional shapes,

240
00:16:55,740 --> 00:16:57,500
like Tiananmen Square.

241
00:17:00,460 --> 00:17:02,660
But Qin was interested

242
00:17:02,660 --> 00:17:06,620
in more complicated equations -
cubic equations.

243
00:17:08,340 --> 00:17:11,100
These involve numbers
which are cubed,

244
00:17:11,100 --> 00:17:15,860
or to the power of three -
say, five times five times five,

245
00:17:15,860 --> 00:17:19,660
and they were perfect for capturing
three-dimensional shapes,

246
00:17:19,660 --> 00:17:22,060
like Chairman Mao's mausoleum.

247
00:17:23,580 --> 00:17:26,340
Qin found a way
of solving cubic equations,

248
00:17:26,340 --> 00:17:28,900
and this is how it worked.

249
00:17:32,100 --> 00:17:34,460
Say Qin wants to know

250
00:17:34,460 --> 00:17:37,940
the exact dimensions
of Chairman Mao's mausoleum.

251
00:17:40,140 --> 00:17:42,420
He knows the volume of the building

252
00:17:42,420 --> 00:17:45,660
and the relationships
between the dimensions.

253
00:17:47,340 --> 00:17:49,660
In order to get his answer,

254
00:17:49,660 --> 00:17:54,220
Qin uses what he knows
to produce a cubic equation.

255
00:17:54,220 --> 00:17:57,740
He then makes
an educated guess at the dimensions.

256
00:17:57,740 --> 00:18:01,660
Although he's captured a good
proportion of the mausoleum,

257
00:18:01,660 --> 00:18:03,940
there are still bits left over.

258
00:18:05,460 --> 00:18:09,500
Qin takes these bits
and creates a new cubic equation.

259
00:18:09,500 --> 00:18:11,460
He can now refine his first guess

260
00:18:11,460 --> 00:18:15,540
by trying to find a solution to
this new cubic equation, and so on.

261
00:18:18,660 --> 00:18:21,980
Each time he does this,
the pieces he's left with

262
00:18:21,980 --> 00:18:26,780
get smaller and smaller and his
guesses get better and better.

263
00:18:28,460 --> 00:18:31,980
What's striking is that Qin's
method for solving equations

264
00:18:31,980 --> 00:18:35,220
wasn't discovered in the West
until the 17th century,

265
00:18:35,220 --> 00:18:39,700
when Isaac Newton came up with a
very similar approximation method.

266
00:18:39,700 --> 00:18:42,180
The power of this technique is

267
00:18:42,180 --> 00:18:46,340
that it can be applied
to even more complicated equations.

268
00:18:46,340 --> 00:18:50,060
Qin even used his techniques
to solve an equation

269
00:18:50,060 --> 00:18:52,300
involving numbers
up to the power of ten.

270
00:18:52,300 --> 00:18:56,340
This was extraordinary stuff -
highly complex mathematics.

271
00:18:58,740 --> 00:19:01,140
Qin may have been years
ahead of his time,

272
00:19:01,140 --> 00:19:03,460
but there was a problem
with his technique.

273
00:19:03,460 --> 00:19:06,300
It only gave him
an approximate solution.

274
00:19:06,300 --> 00:19:10,220
That might be good enough for an
engineer - not for a mathematician.

275
00:19:10,220 --> 00:19:13,780
Mathematics is an exact science.
We like things to be precise,

276
00:19:13,780 --> 00:19:16,660
and Qin just couldn't
come up with a formula

277
00:19:16,660 --> 00:19:20,180
to give him an exact solution
to these complicated equations.

278
00:19:28,180 --> 00:19:30,620
China had made
great mathematical leaps,

279
00:19:30,620 --> 00:19:34,580
but the next great mathematical
breakthroughs were to happen

280
00:19:34,580 --> 00:19:37,380
in a country lying
to the southwest of China -

281
00:19:37,380 --> 00:19:40,700
a country that had a rich
mathematical tradition

282
00:19:40,700 --> 00:19:43,700
that would change
the face of maths for ever.

283
00:20:14,180 --> 00:20:18,900
India's first great mathematical
gift lay in the world of number.

284
00:20:18,900 --> 00:20:22,980
Like the Chinese, the Indians had
discovered the mathematical benefits

285
00:20:22,980 --> 00:20:24,900
of the decimal place-value system

286
00:20:24,900 --> 00:20:28,860
and were using it by the middle
of the 3rd century AD.

287
00:20:30,580 --> 00:20:34,540
It's been suggested that
the Indians learned the system

288
00:20:34,540 --> 00:20:38,740
from Chinese merchants travelling
in India with their counting rods,

289
00:20:38,740 --> 00:20:42,980
or they may well just have
stumbled across it themselves.

290
00:20:42,980 --> 00:20:46,460
It's all such a long time ago
that it's shrouded in mystery.

291
00:20:48,660 --> 00:20:52,180
We may never know how the Indians
came up with their number system,

292
00:20:52,180 --> 00:20:55,220
but we do know that they refined
and perfected it,

293
00:20:55,220 --> 00:20:59,140
creating the ancestors for the nine
numerals used across the world now.

294
00:20:59,140 --> 00:21:01,820
Many rank the Indian
system of counting

295
00:21:01,820 --> 00:21:05,380
as one of the greatest intellectual
innovations of all time,

296
00:21:05,380 --> 00:21:09,540
developing into the closest thing
we could call a universal language.

297
00:21:27,980 --> 00:21:29,940
But there was one number missing,

298
00:21:29,940 --> 00:21:33,780
and it was the Indians who
would introduce it to the world.

299
00:21:40,300 --> 00:21:44,740
The earliest known recording of this
number dates from the 9th century,

300
00:21:44,740 --> 00:21:48,420
though it was probably in
practical use for centuries before.

301
00:21:50,060 --> 00:21:53,900
This strange new numeral
is engraved on the wall

302
00:21:53,900 --> 00:21:57,700
of small temple in the fort
of Gwalior in central India.

303
00:22:01,820 --> 00:22:05,740
So here we are in one of the holy
sites of the mathematical world,

304
00:22:05,740 --> 00:22:09,180
and what I'm looking for
is in this inscription on the wall.

305
00:22:10,140 --> 00:22:12,940
Up here are some numbers, and...

306
00:22:12,940 --> 00:22:15,220
here's the new number.

307
00:22:15,220 --> 00:22:17,220
It's zero.

308
00:22:21,940 --> 00:22:25,460
It's astonishing to think
that before the Indians invented it,

309
00:22:25,460 --> 00:22:27,980
there was no number zero.

310
00:22:27,980 --> 00:22:31,620
To the ancient Greeks,
it simply hadn't existed.

311
00:22:31,620 --> 00:22:35,860
To the Egyptians, the Mesopotamians
and, as we've seen, the Chinese,

312
00:22:35,860 --> 00:22:40,060
zero had been in use but as
a placeholder, an empty space

313
00:22:40,060 --> 00:22:42,380
to show a zero inside a number.

314
00:22:45,660 --> 00:22:48,740
The Indians transformed zero
from a mere placeholder

315
00:22:48,740 --> 00:22:51,340
into a number
that made sense in its own right -

316
00:22:51,340 --> 00:22:54,620
a number for calculation,
for investigation.

317
00:22:54,620 --> 00:22:58,820
This brilliant conceptual leap
would revolutionise mathematics.

318
00:23:02,740 --> 00:23:07,100
Now, with just ten digits - zero
to nine - it was suddenly possible

319
00:23:07,100 --> 00:23:10,100
to capture astronomically large
numbers

320
00:23:10,100 --> 00:23:12,380
in an incredibly efficient way.

321
00:23:15,380 --> 00:23:18,700
But why did the Indians
make this imaginative leap?

322
00:23:18,700 --> 00:23:20,620
Well, we'll never know for sure,

323
00:23:20,620 --> 00:23:24,860
but it's possible that the idea and
symbol that the Indians use for zero

324
00:23:24,860 --> 00:23:28,060
came from calculations they did
with stones in the sand.

325
00:23:28,060 --> 00:23:31,380
When stones were removed
from the calculation,

326
00:23:31,380 --> 00:23:34,140
a small, round hole was left
in its place,

327
00:23:34,140 --> 00:23:37,500
representing the movement
from something to nothing.

328
00:23:40,140 --> 00:23:44,460
But perhaps there is also a cultural
reason for the invention of zero.

329
00:23:44,460 --> 00:23:48,020
HORNS BLOW AND DRUMS BANG

330
00:23:48,020 --> 00:23:50,940
METALLIC BEATING

331
00:23:53,380 --> 00:23:57,860
For the ancient Indians, the
concepts of nothingness and eternity

332
00:23:57,860 --> 00:24:00,780
lay at the very heart
of their belief system.

333
00:24:05,260 --> 00:24:07,700
BELL CLANGS AND SILENCE FALLS

334
00:24:10,220 --> 00:24:14,220
In the religions of India, the
universe was born from nothingness,

335
00:24:14,220 --> 00:24:17,340
and nothingness is
the ultimate goal of humanity.

336
00:24:17,340 --> 00:24:19,180
So it's perhaps not surprising

337
00:24:19,180 --> 00:24:23,020
that a culture that so
enthusiastically embraced the void

338
00:24:23,020 --> 00:24:26,220
should be happy
with the notion of zero.

339
00:24:26,220 --> 00:24:30,420
The Indians even used the word for
the philosophical idea of the void,

340
00:24:30,420 --> 00:24:34,260
shunya, to represent
the new mathematical term "zero".

341
00:24:47,620 --> 00:24:53,020
In the 7th century, the brilliant
Indian mathematician Brahmagupta

342
00:24:53,020 --> 00:24:56,020
proved some of the essential
properties of zero.

343
00:25:01,820 --> 00:25:04,660
Brahmagupta's rules
about calculating with zero

344
00:25:04,660 --> 00:25:08,620
are taught in schools
all over the world to this day.

345
00:25:09,580 --> 00:25:12,580
One plus zero equals one.

346
00:25:13,620 --> 00:25:16,980
One minus zero equals one.

347
00:25:16,980 --> 00:25:20,260
One times zero is equal to zero.

348
00:25:24,460 --> 00:25:29,020
But Brahmagupta came a cropper when
he tried to do one divided by zero.

349
00:25:29,020 --> 00:25:32,220
After all, what number
times zero equals one?

350
00:25:32,220 --> 00:25:36,100
It would require a new mathematical
concept, that of infinity,

351
00:25:36,100 --> 00:25:38,340
to make sense of dividing by zero,

352
00:25:38,340 --> 00:25:42,260
and the breakthrough was made by a
12th-century Indian mathematician

353
00:25:42,260 --> 00:25:45,380
called Bhaskara II,
and it works like this.

354
00:25:45,380 --> 00:25:51,540
If I take a fruit and I divide
it into halves, I get two pieces,

355
00:25:51,540 --> 00:25:54,420
so one divided by a half is two.

356
00:25:54,420 --> 00:25:57,820
If I divide it into thirds,
I get three pieces.

357
00:25:57,820 --> 00:26:01,260
So when I divide it into smaller
and smaller fractions,

358
00:26:01,260 --> 00:26:04,980
I get more and more pieces,
so ultimately,

359
00:26:04,980 --> 00:26:06,940
when I divide by a piece

360
00:26:06,940 --> 00:26:10,740
which is of zero size,
I'll have infinitely many pieces.

361
00:26:10,740 --> 00:26:14,900
So for Bhaskara,
one divided by zero is infinity.

362
00:26:23,220 --> 00:26:27,020
But the Indians would go further
in their calculations with zero.

363
00:26:28,180 --> 00:26:32,500
For example, if you take three
from three and get zero,

364
00:26:32,500 --> 00:26:35,940
what happens when you take
four from three?

365
00:26:35,940 --> 00:26:38,180
It looks like you have nothing,

366
00:26:38,180 --> 00:26:40,980
but the Indians recognised
that this

367
00:26:40,980 --> 00:26:44,060
was a new sort of nothing -
negative numbers.

368
00:26:44,060 --> 00:26:47,780
The Indians called them "debts",
because they solved equations like,

369
00:26:47,780 --> 00:26:51,380
"If I have three batches
of material and take four away,

370
00:26:51,380 --> 00:26:53,540
"how many have I left?"

371
00:26:57,220 --> 00:26:59,180
This may seem odd and impractical,

372
00:26:59,180 --> 00:27:01,740
but that was the beauty
of Indian mathematics.

373
00:27:01,740 --> 00:27:05,020
Their ability to come up
with negative numbers and zero

374
00:27:05,020 --> 00:27:08,420
was because they thought of
numbers as abstract entities.

375
00:27:08,420 --> 00:27:11,700
They weren't just for counting
and measuring pieces of cloth.

376
00:27:11,700 --> 00:27:15,340
They had a life of their own,
floating free of the real world.

377
00:27:15,340 --> 00:27:19,340
This led to an explosion
of mathematical ideas.

378
00:27:31,220 --> 00:27:34,900
The Indians' abstract approach
to mathematics soon revealed

379
00:27:34,900 --> 00:27:38,780
a new side to the problem of
how to solve quadratic equations.

380
00:27:38,780 --> 00:27:42,340
That is equations including
numbers to the power of two.

381
00:27:43,860 --> 00:27:47,860
Brahmagupta's understanding of
negative numbers allowed him to see

382
00:27:47,860 --> 00:27:51,060
that quadratic equations
always have two solutions,

383
00:27:51,060 --> 00:27:52,940
one of which could be negative.

384
00:27:55,460 --> 00:27:57,380
Brahmagupta went even further,

385
00:27:57,380 --> 00:28:00,340
solving quadratic equations
with two unknowns,

386
00:28:00,340 --> 00:28:04,380
a question which wouldn't be
considered in the West until 1657,

387
00:28:04,380 --> 00:28:06,260
when French mathematician Fermat

388
00:28:06,260 --> 00:28:08,940
challenged his colleagues
with the same problem.

389
00:28:08,940 --> 00:28:12,100
Little did he know that they'd
been beaten to a solution

390
00:28:12,100 --> 00:28:15,020
by Brahmagupta
1,000 years earlier.

391
00:28:20,340 --> 00:28:24,980
Brahmagupta was beginning to find
abstract ways of solving equations,

392
00:28:24,980 --> 00:28:28,140
but astonishingly,
he was also developing

393
00:28:28,140 --> 00:28:31,460
a new mathematical language
to express that abstraction.

394
00:28:32,780 --> 00:28:36,980
Brahmagupta was experimenting with
ways of writing his equations down,

395
00:28:36,980 --> 00:28:40,460
using the initials
of the names of different colours

396
00:28:40,460 --> 00:28:43,020
to represent unknowns
in his equations.

397
00:28:44,980 --> 00:28:47,740
A new mathematical language
was coming to life,

398
00:28:47,740 --> 00:28:50,180
which would ultimately lead
to the x's and y's

399
00:28:50,180 --> 00:28:53,220
which fill today's
mathematical journals.

400
00:29:06,780 --> 00:29:11,180
But it wasn't just new
notation that was being developed.

401
00:29:13,660 --> 00:29:16,180
Indian mathematicians
were responsible for making

402
00:29:16,180 --> 00:29:19,900
fundamental new discoveries
in the theory of trigonometry.

403
00:29:22,740 --> 00:29:26,980
The power of trigonometry
is that it acts like a dictionary,

404
00:29:26,980 --> 00:29:30,220
translating geometry into numbers
and back.

405
00:29:30,220 --> 00:29:33,420
Although first developed by the
ancient Greeks,

406
00:29:33,420 --> 00:29:36,060
it was in the hands
of the Indian mathematicians

407
00:29:36,060 --> 00:29:38,100
that the subject truly flourished.

408
00:29:38,100 --> 00:29:42,620
At its heart lies the study
of right-angled triangles.

409
00:29:44,300 --> 00:29:48,340
In trigonometry,
you can use this angle here

410
00:29:48,340 --> 00:29:52,580
to find the ratios of the opposite
side to the longest side.

411
00:29:52,580 --> 00:29:55,340
There's a function
called the sine function

412
00:29:55,340 --> 00:29:58,380
which, when you input the angle,
outputs the ratio.

413
00:29:58,380 --> 00:30:02,060
So for example in this triangle,
the angle is about 30 degrees,

414
00:30:02,060 --> 00:30:06,060
so the output of the sine function
is a ratio of one to two,

415
00:30:06,060 --> 00:30:10,660
telling me that this side is half
the length of the longest side.

416
00:30:13,140 --> 00:30:17,140
The sine function
enables you to calculate distances

417
00:30:17,140 --> 00:30:21,420
when you're not able to make
an accurate measurement.

418
00:30:21,420 --> 00:30:25,500
To this day, it's used
in architecture and engineering.

419
00:30:25,500 --> 00:30:28,340
The Indians used it
to survey the land around them,

420
00:30:28,340 --> 00:30:33,180
navigate the seas and, ultimately,
chart the depths of space itself.

421
00:30:35,140 --> 00:30:38,100
It was central to the work
of observatories,

422
00:30:38,100 --> 00:30:39,940
like this one in Delhi,

423
00:30:39,940 --> 00:30:42,820
where astronomers
would study the stars.

424
00:30:42,820 --> 00:30:45,340
The Indian astronomers
could use trigonometry

425
00:30:45,340 --> 00:30:48,460
to work out the relative distance
between Earth and the moon

426
00:30:48,460 --> 00:30:49,900
and Earth and the sun.

427
00:30:49,900 --> 00:30:53,700
You can only make the calculation
when the moon is half full,

428
00:30:53,700 --> 00:30:56,900
because that's when it's
directly opposite the sun,

429
00:30:56,900 --> 00:31:01,420
so that the sun, moon and Earth
create a right-angled triangle.

430
00:31:02,980 --> 00:31:04,820
Now, the Indians could measure

431
00:31:04,820 --> 00:31:08,140
that the angle between the sun
and the observatory

432
00:31:08,140 --> 00:31:09,980
was one-seventh of a degree.

433
00:31:11,220 --> 00:31:14,500
The sine function of
one-seventh of a degree

434
00:31:14,500 --> 00:31:17,820
gives me the ratio of 400:1.

435
00:31:17,820 --> 00:31:23,580
This means the sun is 400 times
further from Earth than the moon is.

436
00:31:23,580 --> 00:31:25,460
So using trigonometry,

437
00:31:25,460 --> 00:31:28,740
the Indian mathematicians
could explore the solar system

438
00:31:28,740 --> 00:31:31,780
without ever having
to leave the surface of the Earth.

439
00:31:39,340 --> 00:31:42,940
The ancient Greeks had been the
first to explore the sine function,

440
00:31:42,940 --> 00:31:47,300
listing precise values
for some angles,

441
00:31:47,300 --> 00:31:50,940
but they couldn't calculate
the sines of every angle.

442
00:31:50,940 --> 00:31:55,460
The Indians were to go much further,
setting themselves a mammoth task.

443
00:31:55,460 --> 00:31:57,540
The search was on to find a way

444
00:31:57,540 --> 00:32:01,540
to calculate the sine function
of any angle you might be given.

445
00:32:18,260 --> 00:32:21,780
The breakthrough in the search for
the sine function of every angle

446
00:32:21,780 --> 00:32:24,820
would be made here in
Kerala in south India.

447
00:32:24,820 --> 00:32:27,900
In the 15th century,
this part of the country

448
00:32:27,900 --> 00:32:31,700
became home to one of the most
brilliant schools of mathematicians

449
00:32:31,700 --> 00:32:33,500
to have ever worked.

450
00:32:34,940 --> 00:32:38,900
Their leader was called Madhava,
and he was to make

451
00:32:38,900 --> 00:32:42,660
some extraordinary
mathematical discoveries.

452
00:32:45,460 --> 00:32:49,420
The key to Madhava's success
was the concept of the infinite.

453
00:32:49,420 --> 00:32:53,020
Madhava discovered that you could
add up infinitely many things

454
00:32:53,020 --> 00:32:54,860
with dramatic effects.

455
00:32:54,860 --> 00:32:58,180
Previous cultures had been nervous
of these infinite sums,

456
00:32:58,180 --> 00:33:00,660
but Madhava
was happy to play with them.

457
00:33:00,660 --> 00:33:03,220
For example,
here's how one can be made up

458
00:33:03,220 --> 00:33:05,660
by adding
infinitely many fractions.

459
00:33:07,180 --> 00:33:11,540
I'm heading from zero
to one on my boat,

460
00:33:11,540 --> 00:33:15,780
but I can split my journey up
into infinitely many fractions.

461
00:33:15,780 --> 00:33:18,540
So I can get to a half,

462
00:33:18,540 --> 00:33:22,260
then I can sail on a quarter,

463
00:33:22,260 --> 00:33:25,260
then an eighth, then a sixteenth,
and so on.

464
00:33:25,260 --> 00:33:29,660
The smaller the fractions I move,
the nearer to one I get,

465
00:33:29,660 --> 00:33:34,060
but I'll only get there once I've
added up infinitely many fractions.

466
00:33:36,380 --> 00:33:38,500
Physically and philosophically,

467
00:33:38,500 --> 00:33:41,980
it seems rather a challenge
to add up infinitely many things,

468
00:33:41,980 --> 00:33:46,020
but the power of mathematics is
to make sense of the impossible.

469
00:33:46,020 --> 00:33:47,580
By producing a language

470
00:33:47,580 --> 00:33:49,940
to articulate and manipulate
the infinite,

471
00:33:49,940 --> 00:33:52,820
you can prove
that after infinitely many steps

472
00:33:52,820 --> 00:33:54,780
you'll reach your destination.

473
00:33:57,980 --> 00:34:02,220
Such infinite sums are called
infinite series, and Madhava

474
00:34:02,220 --> 00:34:04,860
was doing a lot of research
into the connections

475
00:34:04,860 --> 00:34:07,900
between these series
and trigonometry.

476
00:34:08,900 --> 00:34:12,540
First, he realised that
he could use the same principle

477
00:34:12,540 --> 00:34:15,180
of adding up infinitely many
fractions to capture

478
00:34:15,180 --> 00:34:19,700
one of the most important
numbers in mathematics - pi.

479
00:34:21,220 --> 00:34:26,020
Pi is the ratio of the circle's
circumference to its diameter.

480
00:34:26,020 --> 00:34:30,220
It's a number that appears
in all sorts of mathematics,

481
00:34:30,220 --> 00:34:32,700
but is especially useful
for engineers,

482
00:34:32,700 --> 00:34:36,940
because any measurements
involving curves soon require pi.

483
00:34:38,540 --> 00:34:43,140
So for centuries, mathematicians
searched for a precise value for pi.

484
00:34:48,660 --> 00:34:52,660
It was in 6th-century India
that the mathematician Aryabhata

485
00:34:52,660 --> 00:34:57,500
gave a very accurate approximation
for pi - namely 3.1416.

486
00:34:57,500 --> 00:34:59,180
He went on to use this

487
00:34:59,180 --> 00:35:02,340
to make a measurement
of the circumference of the Earth,

488
00:35:02,340 --> 00:35:05,820
and he got it as 24,835 miles,

489
00:35:05,820 --> 00:35:09,820
which amazingly is only 70 miles
away from its true value.

490
00:35:09,820 --> 00:35:12,700
But it was in Kerala
in the 15th century

491
00:35:12,700 --> 00:35:15,580
that Madhava realised
he could use infinity

492
00:35:15,580 --> 00:35:18,020
to get an exact formula for pi.

493
00:35:21,300 --> 00:35:25,140
By successively adding
and subtracting different fractions,

494
00:35:25,140 --> 00:35:28,660
Madhava could hone in
on an exact formula for pi.

495
00:35:30,020 --> 00:35:34,500
First, he moved four steps
up the number line.

496
00:35:34,500 --> 00:35:36,860
That took him way past pi.

497
00:35:38,380 --> 00:35:41,420
So next he took
four-thirds of a step,

498
00:35:41,420 --> 00:35:44,740
or one-and-one-third
steps, back.

499
00:35:44,740 --> 00:35:46,900
Now he'd come too far
the other way.

500
00:35:48,140 --> 00:35:51,860
So he headed forward
four-fifths of a step.

501
00:35:51,860 --> 00:35:56,660
Each time, he alternated between
four divided by the next odd number.

502
00:36:03,380 --> 00:36:06,500
He zigzagged up and down
the number line,

503
00:36:06,500 --> 00:36:08,980
getting closer and closer to pi.

504
00:36:08,980 --> 00:36:12,340
He discovered that if you went
through all the odd numbers,

505
00:36:12,340 --> 00:36:15,860
infinitely many of them,
you would hit pi exactly.

506
00:36:20,260 --> 00:36:22,980
I was taught at university
that this formula for pi

507
00:36:22,980 --> 00:36:26,820
was discovered by the 17th-century
German mathematician Leibniz,

508
00:36:26,820 --> 00:36:30,220
but amazingly, it was actually
discovered here in Kerala

509
00:36:30,220 --> 00:36:32,100
two centuries earlier by Madhava.

510
00:36:32,100 --> 00:36:34,700
He went on to use
the same sort of mathematics

511
00:36:34,700 --> 00:36:36,620
to get infinite-series expressions

512
00:36:36,620 --> 00:36:38,980
for the sine formula
in trigonometry.

513
00:36:38,980 --> 00:36:42,420
And the wonderful thing is that
you can use these formulas now

514
00:36:42,420 --> 00:36:46,380
to calculate the sine of any angle
to any degree of accuracy.

515
00:36:57,100 --> 00:37:00,860
It seems incredible that
the Indians made these discoveries

516
00:37:00,860 --> 00:37:03,740
centuries before
Western mathematicians.

517
00:37:06,500 --> 00:37:11,100
And it says a lot about our attitude
in the West to non-Western cultures

518
00:37:11,100 --> 00:37:15,060
that we nearly always
claim their discoveries as our own.

519
00:37:15,060 --> 00:37:19,100
What is clear is the West has
been very slow to give due credit

520
00:37:19,100 --> 00:37:22,660
to the major breakthroughs
made in non-Western mathematics.

521
00:37:22,660 --> 00:37:25,860
Madhava wasn't the only
mathematician to suffer this way.

522
00:37:25,860 --> 00:37:28,940
As the West came into contact
more and more with the East

523
00:37:28,940 --> 00:37:30,820
during the 18th and 19th centuries,

524
00:37:30,820 --> 00:37:33,460
there was a widespread dismissal
and denigration

525
00:37:33,460 --> 00:37:35,540
of the cultures
they were colonising.

526
00:37:35,540 --> 00:37:38,340
The natives, it was assumed,
couldn't have anything

527
00:37:38,340 --> 00:37:40,580
of intellectual worth
to offer the West.

528
00:37:40,580 --> 00:37:43,500
It's only now, at the beginning
of the 21st century,

529
00:37:43,500 --> 00:37:45,380
that history is being rewritten.

530
00:37:45,380 --> 00:37:49,740
But Eastern mathematics was to have
a major impact in Europe,

531
00:37:49,740 --> 00:37:53,380
thanks to the development
of one of the major powers

532
00:37:53,380 --> 00:37:55,060
of the medieval world.

533
00:38:17,780 --> 00:38:21,300
In the 7th century,
a new empire began to spread

534
00:38:21,300 --> 00:38:22,660
across the Middle East.

535
00:38:22,660 --> 00:38:25,460
The teachings
of the Prophet Mohammed

536
00:38:25,460 --> 00:38:28,900
inspired a vast
and powerful Islamic empire

537
00:38:28,900 --> 00:38:31,260
which soon stretched
from India in the east

538
00:38:31,260 --> 00:38:35,500
to here in Morocco
in the west.

539
00:38:42,300 --> 00:38:46,820
And at the heart of this empire
lay a vibrant intellectual culture.

540
00:38:51,740 --> 00:38:56,500
A great library and centre of
learning was established in Baghdad.

541
00:38:56,500 --> 00:38:59,980
Called the House of Wisdom,
its teaching spread

542
00:38:59,980 --> 00:39:02,180
throughout the Islamic empire,

543
00:39:02,180 --> 00:39:05,420
reaching schools
like this one here in Fez.

544
00:39:05,420 --> 00:39:08,700
Subjects studied included astronomy,
medicine,

545
00:39:08,700 --> 00:39:10,580
chemistry, zoology

546
00:39:10,580 --> 00:39:12,260
and mathematics.

547
00:39:13,820 --> 00:39:18,500
The Muslim scholars collected
and translated many ancient texts,

548
00:39:18,500 --> 00:39:20,940
effectively saving
them for posterity.

549
00:39:20,940 --> 00:39:24,220
In fact, without their intervention,
we may never have known

550
00:39:24,220 --> 00:39:27,820
about the ancient cultures of
Egypt, Babylon, Greece and India.

551
00:39:27,820 --> 00:39:30,780
But the scholars at the
House of Wisdom weren't content

552
00:39:30,780 --> 00:39:33,700
simply with translating
other people's mathematics.

553
00:39:33,700 --> 00:39:36,420
They wanted to create
a mathematics of their own,

554
00:39:36,420 --> 00:39:38,260
to push the subject forward.

555
00:39:42,420 --> 00:39:46,420
Such intellectual curiosity
was actively encouraged

556
00:39:46,420 --> 00:39:49,660
in the early centuries
of the Islamic empire.

557
00:39:51,780 --> 00:39:54,860
The Koran asserted
the importance of knowledge.

558
00:39:54,860 --> 00:39:58,980
Learning was nothing less
than a requirement of God.

559
00:40:01,700 --> 00:40:05,620
In fact, the needs of Islam
demanded mathematical skill.

560
00:40:05,620 --> 00:40:08,260
The devout needed to calculate
the time of prayer

561
00:40:08,260 --> 00:40:10,980
and the direction of Mecca
to pray towards,

562
00:40:10,980 --> 00:40:13,980
and the prohibition
of depicting the human form

563
00:40:13,980 --> 00:40:15,860
meant that they had to use

564
00:40:15,860 --> 00:40:18,860
much more geometric patterns
to cover their buildings.

565
00:40:18,860 --> 00:40:22,420
The Muslim artists discovered all
the different sorts of symmetry

566
00:40:22,420 --> 00:40:26,660
that you can depict
on a two-dimensional wall.

567
00:40:34,380 --> 00:40:37,380
The director of the House of Wisdom
in Baghdad

568
00:40:37,380 --> 00:40:40,740
was a Persian scholar
called Muhammad Al-Khwarizmi.

569
00:40:43,860 --> 00:40:48,780
Al-Khwarizmi was an exceptional
mathematician who was responsible

570
00:40:48,780 --> 00:40:53,020
for introducing two key
mathematical concepts to the West.

571
00:40:53,020 --> 00:40:56,020
Al-Khwarizmi recognised
the incredible potential

572
00:40:56,020 --> 00:40:57,860
that the Hindu numerals had

573
00:40:57,860 --> 00:41:00,820
to revolutionise
mathematics and science.

574
00:41:00,820 --> 00:41:03,380
His work explaining
the power of these numbers

575
00:41:03,380 --> 00:41:06,340
to speed up calculations
and do things effectively

576
00:41:06,340 --> 00:41:09,740
was so influential that it wasn't
long before they were adopted

577
00:41:09,740 --> 00:41:13,580
as the numbers of choice amongst the
mathematicians of the Islamic world.

578
00:41:13,580 --> 00:41:16,340
In fact, these numbers
have now become known

579
00:41:16,340 --> 00:41:18,660
as the Hindu-Arabic numerals.

580
00:41:18,660 --> 00:41:21,700
These numbers -
one to nine and zero -

581
00:41:21,700 --> 00:41:25,500
are the ones we use today
all over the world.

582
00:41:30,020 --> 00:41:34,980
But Al-Khwarizmi was to create
a whole new mathematical language.

583
00:41:36,620 --> 00:41:38,580
It was called algebra

584
00:41:38,580 --> 00:41:43,100
and was named after the title of
his book Al-jabr W'al-muqabala,

585
00:41:43,100 --> 00:41:46,460
or Calculation By Restoration
Or Reduction.

586
00:41:51,300 --> 00:41:56,420
Algebra is the grammar that
underlies the way that numbers work.

587
00:41:56,420 --> 00:41:58,820
It's a language
that explains the patterns

588
00:41:58,820 --> 00:42:01,980
that lie behind
the behaviour of numbers.

589
00:42:01,980 --> 00:42:05,900
It's a bit like a code
for running a computer program.

590
00:42:05,900 --> 00:42:09,580
The code will work whatever the
numbers you feed in to the program.

591
00:42:11,380 --> 00:42:15,020
For example, mathematicians
might have discovered

592
00:42:15,020 --> 00:42:17,300
that if you take a number
and square it,

593
00:42:17,300 --> 00:42:19,580
that's always one more
than if you'd taken

594
00:42:19,580 --> 00:42:22,580
the numbers either side
and multiplied those together.

595
00:42:22,580 --> 00:42:25,780
For example, five times five is 25,

596
00:42:25,780 --> 00:42:29,700
which is one more
than four times six - 24.

597
00:42:29,700 --> 00:42:33,500
Six times six is always one more
than five times seven and so on.

598
00:42:33,500 --> 00:42:35,220
But how can you be sure

599
00:42:35,220 --> 00:42:38,420
that this is going to work
whatever numbers you take?

600
00:42:38,420 --> 00:42:41,380
To explain the pattern underlying
these calculations,

601
00:42:41,380 --> 00:42:43,660
let's use the dyeing holes
in this tannery.

602
00:42:51,620 --> 00:42:56,860
If we take a square of
25 holes, running five by five,

603
00:42:56,860 --> 00:43:01,100
and take one row of five away
and add it to the bottom,

604
00:43:01,100 --> 00:43:03,980
we get six by four
with one left over.

605
00:43:05,500 --> 00:43:08,900
But however many holes there
are on the side of the square,

606
00:43:08,900 --> 00:43:12,660
we can always move one row of holes
down in a similar way

607
00:43:12,660 --> 00:43:16,580
to be left with a rectangle
of holes with one left over.

608
00:43:19,220 --> 00:43:21,300
Algebra was a huge breakthrough.

609
00:43:21,300 --> 00:43:23,020
Here was a new language

610
00:43:23,020 --> 00:43:26,060
to be able to analyse
the way that numbers worked.

611
00:43:26,060 --> 00:43:28,220
Previously, the Indians
and the Chinese

612
00:43:28,220 --> 00:43:30,460
had considered
very specific problems,

613
00:43:30,460 --> 00:43:33,940
but Al-Khwarizmi went
from the specific to the general.

614
00:43:33,940 --> 00:43:37,540
He developed systematic ways
to be able to analyse problems

615
00:43:37,540 --> 00:43:41,140
so that the solutions would work
whatever the numbers that you took.

616
00:43:41,140 --> 00:43:44,900
This language is used
across the mathematical world today.

617
00:43:46,420 --> 00:43:51,140
Al-Khwarizmi's great breakthrough
came when he applied algebra

618
00:43:51,140 --> 00:43:52,820
to quadratic equations -

619
00:43:52,820 --> 00:43:55,900
that is equations including
numbers to the power of two.

620
00:43:55,900 --> 00:43:58,700
The ancient Mesopotamians
had devised

621
00:43:58,700 --> 00:44:02,460
a cunning method to solve
particular quadratic equations,

622
00:44:02,460 --> 00:44:06,580
but it was Al-Khwarizmi's
abstract language of algebra

623
00:44:06,580 --> 00:44:10,340
that could finally express
why this method always worked.

624
00:44:11,940 --> 00:44:14,540
This was a great conceptual leap

625
00:44:14,540 --> 00:44:18,260
and would ultimately lead to a
formula that could be used to solve

626
00:44:18,260 --> 00:44:22,500
any quadratic equation,
whatever the numbers involved.

627
00:44:30,820 --> 00:44:32,780
The next mathematical Holy Grail

628
00:44:32,780 --> 00:44:37,380
was to find a general method that
could solve all cubic equations -

629
00:44:37,380 --> 00:44:40,980
equations including numbers
to the power of three.

630
00:44:58,260 --> 00:45:00,980
It was an 11th-century
Persian mathematician

631
00:45:00,980 --> 00:45:04,340
who took up the challenge of
cracking the problem of the cubic.

632
00:45:08,780 --> 00:45:12,300
His name was Omar Khayyam,
and he travelled widely

633
00:45:12,300 --> 00:45:15,940
across the Middle East,
calculating as he went.

634
00:45:17,860 --> 00:45:21,780
But he was famous for another,
very different, reason.

635
00:45:21,780 --> 00:45:24,420
Khayyam was a celebrated poet,

636
00:45:24,420 --> 00:45:28,380
author of the great
epic poem the Rubaiyat.

637
00:45:31,260 --> 00:45:35,460
It may seem a bit odd that a poet
was also a master mathematician.

638
00:45:35,460 --> 00:45:38,900
After all, the combination
doesn't immediately spring to mind.

639
00:45:38,900 --> 00:45:42,540
But there's quite a lot of
similarity between the disciplines.

640
00:45:42,540 --> 00:45:45,900
Poetry, with its rhyming structure
and rhythmic patterns,

641
00:45:45,900 --> 00:45:49,860
resonates strongly with constructing
a logical mathematical proof.

642
00:45:53,340 --> 00:45:55,660
Khayyam's major mathematical work

643
00:45:55,660 --> 00:46:00,420
was devoted to finding the general
method to solve all cubic equations.

644
00:46:00,420 --> 00:46:04,100
Rather than looking
at particular examples,

645
00:46:04,100 --> 00:46:08,980
Khayyam carried out a systematic
analysis of the problem,

646
00:46:08,980 --> 00:46:12,260
true to the algebraic spirit
of Al-Khwarizmi.

647
00:46:14,100 --> 00:46:16,620
Khayyam's analysis revealed
for the first time

648
00:46:16,620 --> 00:46:19,820
that there were several
different sorts of cubic equation.

649
00:46:19,820 --> 00:46:21,900
But he was still very influenced

650
00:46:21,900 --> 00:46:24,660
by the geometric heritage
of the Greeks.

651
00:46:24,660 --> 00:46:27,420
He couldn't separate the algebra
from the geometry.

652
00:46:27,420 --> 00:46:30,780
In fact, he wouldn't even consider
equations in higher degrees,

653
00:46:30,780 --> 00:46:34,180
because they described objects
in more than three dimensions,

654
00:46:34,180 --> 00:46:35,980
something he saw as impossible.

655
00:46:35,980 --> 00:46:37,860
Although the geometry allowed him

656
00:46:37,860 --> 00:46:40,460
to analyse these cubic equations
to some extent,

657
00:46:40,460 --> 00:46:43,620
he still couldn't come up
with a purely algebraic solution.

658
00:46:46,140 --> 00:46:51,740
It would be another 500 years before
mathematicians could make the leap

659
00:46:51,740 --> 00:46:55,060
and find a general solution
to the cubic equation.

660
00:46:56,580 --> 00:47:01,740
And that leap would finally be made
in the West - in Italy.

661
00:47:15,740 --> 00:47:19,220
During the centuries in which China,
India and the Islamic empire

662
00:47:19,220 --> 00:47:20,860
had been in the ascendant,

663
00:47:20,860 --> 00:47:25,100
Europe had fallen under
the shadow of the Dark Ages.

664
00:47:26,620 --> 00:47:30,900
All intellectual life, including the
study of mathematics, had stagnated.

665
00:47:36,100 --> 00:47:41,740
But by the 13th century,
things were beginning to change.

666
00:47:41,740 --> 00:47:47,020
Led by Italy, Europe was starting
to explore and trade with the East.

667
00:47:47,020 --> 00:47:51,460
With that contact came the spread
of Eastern knowledge to the West.

668
00:47:51,460 --> 00:47:53,460
It was the son of a customs official

669
00:47:53,460 --> 00:47:56,980
that would become Europe's first
great medieval mathematician.

670
00:47:56,980 --> 00:48:00,580
As a child, he travelled around
North Africa with his father,

671
00:48:00,580 --> 00:48:03,780
where he learnt about the
developments of Arabic mathematics

672
00:48:03,780 --> 00:48:07,060
and especially the benefits
of the Hindu-Arabic numerals.

673
00:48:07,060 --> 00:48:09,100
When he got home to Italy
he wrote a book

674
00:48:09,100 --> 00:48:10,980
that would be hugely influential

675
00:48:10,980 --> 00:48:13,580
in the development
of Western mathematics.

676
00:48:29,660 --> 00:48:32,140
That mathematician was
Leonardo of Pisa,

677
00:48:32,140 --> 00:48:34,780
better known as Fibonacci,

678
00:48:34,780 --> 00:48:37,420
and in his Book Of Calculating,

679
00:48:37,420 --> 00:48:41,060
Fibonacci promoted
the new number system,

680
00:48:41,060 --> 00:48:44,420
demonstrating how simple it was
compared to the Roman numerals

681
00:48:44,420 --> 00:48:46,420
that were in use across Europe.

682
00:48:46,420 --> 00:48:51,660
Calculations were far easier,
a fact that had huge consequences

683
00:48:51,660 --> 00:48:55,420
for anyone dealing with numbers -

684
00:48:55,420 --> 00:48:58,940
pretty much everyone,
from mathematicians to merchants.

685
00:48:58,940 --> 00:49:02,980
But there was widespread
suspicion of these new numbers.

686
00:49:02,980 --> 00:49:06,660
Old habits die hard, and the
authorities just didn't trust them.

687
00:49:06,660 --> 00:49:09,540
Some believed that they would
be more open to fraud -

688
00:49:09,540 --> 00:49:11,380
that you could tamper with them.

689
00:49:11,380 --> 00:49:14,860
Others believed that they'd be
so easy to use for calculations

690
00:49:14,860 --> 00:49:18,140
that it would empower the masses,
taking authority away

691
00:49:18,140 --> 00:49:22,140
from the intelligentsia who knew
how to use the old sort of numbers.

692
00:49:27,980 --> 00:49:31,540
The city of Florence
even banned them in 1299,

693
00:49:31,540 --> 00:49:34,740
but over time,
common sense prevailed,

694
00:49:34,740 --> 00:49:37,540
the new system spread
throughout Europe,

695
00:49:37,540 --> 00:49:41,300
and the old Roman system
slowly became defunct.

696
00:49:41,300 --> 00:49:46,260
At last, the Hindu-Arabic numerals,
zero to nine, had triumphed.

697
00:49:48,700 --> 00:49:52,060
Today Fibonacci is best known for
the discovery of some numbers,

698
00:49:52,060 --> 00:49:55,540
now called the Fibonacci sequence,
that arose when he was trying

699
00:49:55,540 --> 00:49:58,580
to solve a riddle
about the mating habits of rabbits.

700
00:49:58,580 --> 00:50:01,380
Suppose a farmer
has a pair of rabbits.

701
00:50:01,380 --> 00:50:03,860
Rabbits take two months
to reach maturity,

702
00:50:03,860 --> 00:50:07,580
and after that they give birth to
another pair of rabbits each month.

703
00:50:07,580 --> 00:50:09,420
So the problem was how to determine

704
00:50:09,420 --> 00:50:12,900
how many pairs of rabbits there
will be in any given month.

705
00:50:15,140 --> 00:50:20,340
Well, during the first month
you have one pair of rabbits,

706
00:50:20,340 --> 00:50:24,540
and since they haven't matured,
they can't reproduce.

707
00:50:24,540 --> 00:50:28,740
During the second month,
there is still only one pair.

708
00:50:28,740 --> 00:50:32,340
But at the beginning of the
third month, the first pair

709
00:50:32,340 --> 00:50:36,940
reproduces for the first time,
so there are two pairs of rabbits.

710
00:50:36,940 --> 00:50:39,060
At the beginning
of the fourth month,

711
00:50:39,060 --> 00:50:41,140
the first pair reproduces again,

712
00:50:41,140 --> 00:50:45,500
but the second pair is not mature
enough, so there are three pairs.

713
00:50:47,180 --> 00:50:50,340
In the fifth month,
the first pair reproduces

714
00:50:50,340 --> 00:50:53,820
and the second pair
reproduces for the first time,

715
00:50:53,820 --> 00:50:58,540
but the third pair is still too
young, so there are five pairs.

716
00:50:58,540 --> 00:51:00,460
The mating ritual continues,

717
00:51:00,460 --> 00:51:02,580
but what you soon realise is

718
00:51:02,580 --> 00:51:06,100
the number of pairs of rabbits
you have in any given month

719
00:51:06,100 --> 00:51:09,740
is the sum of the pairs of
rabbits that you have had

720
00:51:09,740 --> 00:51:13,460
in each of the two previous months,
so the sequence goes...

721
00:51:13,460 --> 00:51:17,620
1...1...2...3...

722
00:51:17,620 --> 00:51:21,460
5...8...13...

723
00:51:21,460 --> 00:51:26,260
21...34...55...and so on.

724
00:51:26,260 --> 00:51:30,020
The Fibonacci numbers are
nature's favourite numbers.

725
00:51:30,020 --> 00:51:31,940
It's not just rabbits that use them.

726
00:51:31,940 --> 00:51:36,220
The number of petals on a flower
is invariably a Fibonacci number.

727
00:51:36,220 --> 00:51:40,300
They run up and down pineapples
if you count the segments.

728
00:51:40,300 --> 00:51:43,300
Even snails use them
to grow their shells.

729
00:51:43,300 --> 00:51:47,260
Wherever you find growth in nature,
you find the Fibonacci numbers.

730
00:51:51,900 --> 00:51:55,220
But the next major breakthrough
in European mathematics

731
00:51:55,220 --> 00:51:58,340
wouldn't happen
until the early 16th century.

732
00:51:58,340 --> 00:51:59,940
It would involve

733
00:51:59,940 --> 00:52:04,580
finding the general method that
would solve all cubic equations,

734
00:52:04,580 --> 00:52:09,220
and it would happen here
in the Italian city of Bologna.

735
00:52:10,940 --> 00:52:14,380
The University of Bologna
was the crucible

736
00:52:14,380 --> 00:52:17,900
of European mathematical thought at
the beginning of the 16th century.

737
00:52:21,220 --> 00:52:25,060
Pupils from all over Europe
flocked here and developed

738
00:52:25,060 --> 00:52:29,780
a new form of spectator sport -
the mathematical competition.

739
00:52:31,460 --> 00:52:34,820
Large audiences would gather to
watch mathematicians

740
00:52:34,820 --> 00:52:40,020
challenge each other with numbers, a
kind of intellectual fencing match.

741
00:52:40,020 --> 00:52:43,260
But even in this
questioning atmosphere

742
00:52:43,260 --> 00:52:46,740
it was believed that some problems
were just unsolvable.

743
00:52:46,740 --> 00:52:51,460
It was generally assumed
that finding a general method

744
00:52:51,460 --> 00:52:55,100
to solve all cubic equations
was impossible.

745
00:52:55,100 --> 00:52:58,900
But one scholar
was to prove everyone wrong.

746
00:53:01,060 --> 00:53:03,380
His name was Tartaglia,

747
00:53:03,380 --> 00:53:05,380
but he certainly didn't look

748
00:53:05,380 --> 00:53:08,340
the heroic architect
of a new mathematics.

749
00:53:08,340 --> 00:53:11,500
At the age of 12, he'd been
slashed across the face

750
00:53:11,500 --> 00:53:14,060
with a sabre
by a rampaging French army.

751
00:53:14,060 --> 00:53:16,660
The result
was a terrible facial scar

752
00:53:16,660 --> 00:53:19,340
and a devastating speech
impediment.

753
00:53:19,340 --> 00:53:23,220
In fact, Tartaglia was the nickname
he'd been given as a child

754
00:53:23,220 --> 00:53:25,140
and means "the stammerer".

755
00:53:30,380 --> 00:53:32,220
Shunned by his schoolmates,

756
00:53:32,220 --> 00:53:37,660
Tartaglia lost himself
in mathematics, and it wasn't long

757
00:53:37,660 --> 00:53:43,420
before he'd found the formula
to solve one type of cubic equation.

758
00:53:43,420 --> 00:53:45,380
But Tartaglia soon discovered

759
00:53:45,380 --> 00:53:48,980
that he wasn't the only one
to believe he'd cracked the cubic.

760
00:53:48,980 --> 00:53:51,500
A young Italian called Fior
was boasting

761
00:53:51,500 --> 00:53:57,340
that he too held the secret
formula for solving cubic equations.

762
00:53:57,340 --> 00:54:00,180
When news broke
about the discoveries

763
00:54:00,180 --> 00:54:02,780
made by the two mathematicians,

764
00:54:02,780 --> 00:54:06,700
a competition was arranged to
pit them against each other.

765
00:54:06,700 --> 00:54:10,660
The intellectual fencing match
of the century was about to begin.

766
00:54:18,180 --> 00:54:20,220
The trouble was that Tartaglia

767
00:54:20,220 --> 00:54:23,140
only knew how to solve one sort
of cubic equation,

768
00:54:23,140 --> 00:54:25,140
and Fior was ready to challenge him

769
00:54:25,140 --> 00:54:27,460
with questions
about a different sort.

770
00:54:27,460 --> 00:54:29,820
But just a few days
before the contest,

771
00:54:29,820 --> 00:54:32,860
Tartaglia worked out how to
solve this different sort,

772
00:54:32,860 --> 00:54:36,220
and with this new weapon in his
arsenal he thrashed his opponent,

773
00:54:36,220 --> 00:54:38,580
solving all the questions
in under two hours.

774
00:54:42,180 --> 00:54:43,860
Tartaglia went on

775
00:54:43,860 --> 00:54:48,460
to find the formula to solve
all types of cubic equations.

776
00:54:48,460 --> 00:54:51,380
News soon spread,
and a mathematician in Milan

777
00:54:51,380 --> 00:54:55,140
called Cardano became so
desperate to find the solution

778
00:54:55,140 --> 00:54:59,700
that he persuaded a reluctant
Tartaglia to reveal the secret,

779
00:54:59,700 --> 00:55:01,540
but on one condition -

780
00:55:01,540 --> 00:55:05,340
that Cardano keep the secret
and never publish.

781
00:55:08,100 --> 00:55:09,940
But Cardano couldn't resist

782
00:55:09,940 --> 00:55:14,340
discussing Tartaglia's solution
with his brilliant student, Ferrari.

783
00:55:14,340 --> 00:55:17,220
As Ferrari got to grips
with Tartaglia's work,

784
00:55:17,220 --> 00:55:19,460
he realised that he could use it
to solve

785
00:55:19,460 --> 00:55:23,060
the more complicated quartic
equation, an amazing achievement.

786
00:55:23,060 --> 00:55:26,260
Cardano couldn't deny his
student his just rewards,

787
00:55:26,260 --> 00:55:29,620
and he broke his vow of secrecy,
publishing Tartaglia's work

788
00:55:29,620 --> 00:55:33,060
together with Ferrari's
brilliant solution of the quartic.

789
00:55:35,460 --> 00:55:39,420
Poor Tartaglia never recovered
and died penniless,

790
00:55:39,420 --> 00:55:42,940
and to this day, the formula
that solves the cubic equation

791
00:55:42,940 --> 00:55:45,580
is known as Cardano's formula.

792
00:55:54,420 --> 00:55:57,860
Tartaglia may not have won glory
in his lifetime,

793
00:55:57,860 --> 00:56:01,540
but his mathematics managed to
solve a problem that had bewildered

794
00:56:01,540 --> 00:56:06,060
the great mathematicians
of China, India and the Arab world.

795
00:56:08,260 --> 00:56:11,780
It was the first
great mathematical breakthrough

796
00:56:11,780 --> 00:56:13,780
to happen in modern Europe.

797
00:56:16,780 --> 00:56:21,220
The Europeans now had in their
hands the new language of algebra,

798
00:56:21,220 --> 00:56:24,460
the powerful techniques
of the Hindu-Arabic numerals

799
00:56:24,460 --> 00:56:27,460
and the beginnings
of the mastery of the infinite.

800
00:56:27,460 --> 00:56:29,260
It was time for the Western world

801
00:56:29,260 --> 00:56:31,740
to start writing
its own mathematical stories

802
00:56:31,740 --> 00:56:33,380
in the language of the East.

803
00:56:33,380 --> 00:56:36,260
The mathematical revolution
was about to begin.

804
00:56:39,940 --> 00:56:43,900
You can learn more about The Story
Of Maths with the Open University

805
00:56:43,900 --> 00:56:46,140
at open2.net.

806
00:57:06,420 --> 00:57:08,460
Subtitles by Red Bee Media Ltd

807
00:57:08,460 --> 00:57:12,020
E-mail subtitling@bbc.co.uk