From 0 to Infinity in 26 Centuries (9 page)

I
NDIAN
M
ATHEMATICS

In 1920 archaeological excavations in north-western India unearthed the Indus Valley civilization, which existed from
c.
3500
BC
to 2000
BC
. These Bronze Age settlements, contemporary to the first urban areas in Egypt and Mesopotamia, indicated the Ancient Indians had a good understanding of basic
mathematical concepts, and possessed a standardized system of weights and measures.

Ancient Indian religious texts also contain evidence of mathematical knowledge; in Hinduism, mathematics, astronomy and astrology were considered to be in the same field, and they each had
important religious implications. It was a religious requirement that all altars should occupy the same amount of floor space, even if they weren’t the same shape or used different
configurations of bricks – all of which required a good knowledge of geometry. Texts from 700
BC
show the Ancient Indians possessed knowledge of Pythagoras’
theorem, irrational numbers and methods for calculating them.

Astronomical discoveries

Brahmagupta (
AD
598–668), an astronomer, was the first person to treat zero as a number. The Hindu numeral system, predecessor to the Hindu-Arabic
numeral system that we use today (see
here
), developed over time and was fully established by the end of the first millennium
AD
. Up until Brahmagupta’s treatment
of zero as a number, it had been used merely as a place-holder within various number systems in order
to show a gap. Brahmagupta, however, thought of 0 simply as a whole
number or
integer
that lies between 1 and -1. He wrote down rules for its use in arithmetic, alongside rules for using negative numbers.

Useful Functions

Aryabhata (
AD
475–550) was an astronomer who is credited with being the first person to introduce
trigonometry
, which
we use to work out lengths and angles in triangles, and the concept of the
sine
,
cosine
and
tangent
functions.

Brahmagupta recognized that an equation could have a negative solution and, as a result, that any positive integer would have a positive and negative square root. For example, the square roots
of 36 are 6 and -6, because, as Brahmagupta himself stated, a negative multiplied by a negative gives a positive.

Brahmagupta is also famous for developing
Brahmagupta’s formula
, which tells us the area of a
cyclic quadrilateral
– a four-sided shape, the corners of which lie on a
circle:

Modern Indian Mathematics

Srinvasa Ramanujan (1887–1920) was an Indian mathematical genius. After dropping out of university, he became an accounting clerk at a government
office, from where he sent papers to various British mathematicians for consideration. The English mathematician Godfrey Hardy (1877–1947) recognized Ramanujan’s genius and arranged
for him to have a research post at the University of Madras.

In 1914 Ramanujan joined Hardy at Cambridge University and remained in England for five years, in which time he became one of the youngest ever members of the Royal
Society, had work published and finally gained a degree. However, Ramanujan was often ill.

During one bout of illness, Hardy visited him and mentioned that the number of his taxi, 1729, was ‘rather dull’. Ramanujan replied instantly that 1,729 was
the lowest number that could be written as the sum of two cubes in two different ways, and as such, was actually quite interesting:

1
3
+ 12
3
= 1 + 1728 = 1729

9
3
+ 10
3
= 729 + 1000 = 1729

There are lower numbers that can be written as the sum of two cubes, but 1,729 is the lowest number that can be written like this in two ways, and Ramanujan’s
instant recognition of this was nothing short of miraculous.

In his short life Ramanujan came up with nearly 4,000 theorems, equations and identities that still inspire mathematical research to this day.

If you find half the perimeter of the quadrilateral (let’s call it ‘s’) then the area of the shape can be found using Brahmagupta’s
formula:

√(s-a)(s-b)(s-c)(s-d)

Although the Indians were clearly excellent mathematicians, when the British began to take control of the country in the 1700s they assumed the backward pagan Hindus had nothing
of worth to contribute beyond vast natural resources and cheap labour. It has only been in the last hundred years that we have come to appreciate the mathematical heritage of the sub-continent.

I
SLAMIC
M
ATHEMATICS

Mohammed, the founder of Islam, was born in
AD
570. In the two centuries following Mohammed’s birth the Islamic Empire came to dominate all of the
Middle East, Central Asia, North Africa and what would become Spain and Portugal. This Islamic Golden Age saw much important mathematical progress emerge from the countries in the empire, while
Europe remained still in its Dark Ages.

The Islamic religion itself is particularly open to the idea of science, which contrasted strongly with the ideas prevalent in medieval Europe, where it was often considered heretical to
question or investigate the workings of a world made by God.

The Islamic Empire too was committed to gathering the knowledge of the ancient world. Texts in Classical Greek and Latin, Ancient Egyptian, Mesopotamian, Indian, Chinese
and Persian were all translated by Islamic scholars, broadening their availability to the empire’s scientists and mathematicians.

A
L
-K
HWARIZMI
(
c.
790–
c.
850)

Mathematician Al-Khwarizmi hailed from an area situated in present-day Uzbekistan, and he is credited with providing several significant contributions to mathematics. Although
some of his original works have survived, he is familiar to us through editions of his work translated into Latin for use later in Europe.

The new number system

One of Al-Khwarizmi’s significant legacies was what is now known as the Hindu-Arabic numeral system, which we still use to this day. Derived from his
Book of Addition
and Subtraction According to the Hindu Calculation
, Al-Khwarizmi’s system of numbers, developed over time in India from
c.
300
BC
and passed through into
Persia, revolutionized arithmetic.

Up to this point, no culture had a system of numerals with which it was really possible to use in arithmetic. Numbers would always be converted into letters or symbols (either mentally or using
counters, abacuses or other such tools), the calculation
performed and the result reconverted back into numerals. Lots of symbols were often needed to show a number, many of
which were difficult to decipher at a glance.

The Hindu-Arabic system contains just ten symbols – 0 1 2 3 4 5 6 7 8 9 – that could be used to write any number. It is important to note that these symbols were exactly that –
they were not associated with the value they represented through stripes or dots. The zero (from the Arabic
zifer
, meaning ‘empty’) meant that the symbols could have a different
value depending on where they were positioned in the number – which freed people of the difficulty the Mesopotamians had faced. Today, the concept of place-value is taken for granted. But the
idea that the 8 in 80 is worth eight tens, and yet could be used, with the help of those friendly zeros, to also mean 800 or 8 million was revolutionary at the time. In fact, some European scholars
were deeply suspicious of this heathen method of calculating, despite its advantages.

In the
Book of Addition and Subtraction According to the Hindu Calculation
Al-Khwarizmi describes how to do arithmetic using these new numbers. His translators referred to him by the
Latinized name Algorism. Over time Al-Khwarizmi’s methods of calculation became known as
algorithms
,
a word still in use today and which refers to a set of instructions to
perform a calculation – which is exactly what Al-Khwarizmi provided.

Transforming Mathematics

Al-Khwarizmi also wrote
The Compendious Book on Calculation by Transformations and Dividing
, which set out to show how to solve different types of
quadratic equations
(equations in which the unknown numbers are squared). ‘Transformations’ in Arabic is
Al-Jabr
, from which we derive (via Latin) the English term
algebra
. While Al-Khwarizmi himself did not replace unknown numbers with letters, he did pave the way for this to happen.

O
MAR
K
HAYYÁM
(1048–1131)

Persian scholar Omar Khayyám is best known for his
The Rubaiyat of Omar Khayy
á
m,
a selection of poems that were later translated into English in the
nineteenth century by the poet Edward Fitzgerald. Multi-talented, Khayyám spent a great proportion of his life as a court astronomer to a sultan, while also working as a scientist and
mathematician.

Khayyám’s mathematical works were far-reaching. He expanded on Al-Khwarizmi’s earlier work in algebra, and he was one of the first mathematicians to use the replacement of
unknown numbers with letters to make solving equations easier. He also devised techniques for solving
cubic equations
, where the unknown term has been cubed. Khayyám’s insight
enabled him to be one of the first people to connect geometry and algebra, which had until that point been separate disciplines.

Endless possibilities

Khayyám also investigated something now called the
binomial theorem
. This has many applications in mathematics, many of which involve rather tricky algebra. One
side product of binomial theorem is something called
Pascal’s triangle
, named after the seventeenth-century French mathematician Blaise Pascal, who borrowed the triangle from
Khayyám, who in turn borrowed it from the Chinese (see
here
). Unlike the binomial theorem, Pascal’s triangle is simple to understand: the number in each cell of a triangle is made
by adding together the two numbers above it.

Pascal’s triangle is useful because each horizontal row shows us the
binomial coefficients
that the binomial theorem spits out. These can tell us how many
combinations of two different things it is possible to have.

For example, imagine you have planted a row of four flower
bulbs. It says on the packet that the flowers can be blue or pink, with an equal chance of having either.

There is one way for you to grow four blues:

BBBB

There are four ways for you to grow three blues and one pink:

BBBP

BBPB

BPBB

PBBB

There are six ways for you to end up with two of each:

BBPP

BPPB

PBBP

PPBB

BPBP

PBPB

There are four ways for you to have three pinks and one blue:

PPPB

PPBP

PBPP

BPPP

And one way for you to have four pinks:

PPPP

If you look across the fourth row of the triangle, it says 1, 4, 6, 4, 1, which corresponds to the number of ways worked out in the example above. Because there is an equal
chance of a flower being either pink or blue, you can also see that you’re most likely to end up with two of each colour because there are 6 out of 16 total ways this could happen.

A new geometry

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