Read From 0 to Infinity in 26 Centuries Online

**Authors: **Chris Waring

**INDIANMATHEMATICS**

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

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

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.

**ISLAMICMATHEMATICS**

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.

**AL-KHWARIZMI(**

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 Additionand Subtraction According to the Hindu Calculation*, Al-Khwarizmi’s system of numbers, developed over time in India from

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

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

**OMARKHAYYÁM(1048–1131)**

Persian scholar Omar Khayyám is best known for his*The Rubaiyat of Omar Khayy*á

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

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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