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Introduction to China

China is the fourth largest country in the world with an area of 9,596,960 km². It also has the largest population in the world with over 1.3 million billion people. It is bordered by 14 nations. There are various landscapes in China which include huge alluvial plains, grasslands, hills, mountain ranges, deltas and rivers.

Not much is known about Chinese Mathematics before 100 BC, but there are elements which are consistent. Chinese mathematics in early times was strongly related to astronomy and the calendar. Hence many of the earliest texts also involved astronomy. Many works simply listed equations but no exact proof. In other instances, evidence was given, but declared to be an established method after some fashion. Therefore it is very hard to be certain of the actual dates of when the mathematical methods were discovered. There have been many arguments about the Chinese discovering the Pythagorean theorem.

The Nine Chapters on the Mathematical Art is a Chinese mathematics book, believed to be written in the 1st century AD or even earlier. Most scholars believe that Chinese and ancient Mediterranean mathematics had developed independently before the Nine Chapters was completed.

Gougu Theorem

While this theorem has been named Pythagoras' theorem, it is quite likely that Pythagoras was not actually the first to discover it. In Pythagorean Theorem, it states that in a right angled triangle with legs a, b and hypotenuse c, a² + b² = c².

The chinese text, "Chou Pei Suan Ching", written in the Han dynasty (500 BC-200 AD), proved Pythagoras' Theorem numerically using a right angle triangle with side lengths of 3, 4 and 5. However, it is believed by some to be written up to 300 years before the Han dynasty. In China the theorem is known as the 'Gougu Theorem' and is based on the numerical evidence in the Chou Pei Suan Ching. Below is a visual proof of Gougu theorem.

Nine Chapters on the Mathematics Arts

Nine Chapters on the Mathematics Arts in Chinese is called Jiu Zhang Suan Shu. It is a very influential book in the history of Chinese mathematics. Being the earliest specialized mathematical work in China, it is fortunate to survive till the present day. However, It is uncertain when this book was produced. However it is estimated that the book was first written during the Han dynasty (around 200 B.C). An important role has been played by it in the development of mathematics in China. Basically, it deals with practical problems in daily life like finding the areas of cultivated land and different types of calculations for construction and taxation. Nine Chapters on the Mathematical Art has dominated the history of Chinese mathematics. It served as a textbook not only in China but also in neighbouring countries until western science was introduced to the Far East in around 1600 AD.

Nine Chapters on the Mathematics have had numerous achievements in Arithmetic, Geometry and Algebra. In Arithmetic, it devised a systematic way of solving arithmetic operations with fractions. In Geometry, it introduced the method of calculating the areas and volumes of different shape, which were very useful in construction. In Algebra, the method of extracting square roots and cubic roots in this book was discovered several hundred years earlier than the west. The concepts of positive and negative numbers were introduced by this book. It also generated a method of solving a particular type of quadratic equation.

Here is a brief description of each chapter.

• Chapter 1: Land Surveying.
  • Method for calculation of areas of land
  • Deals with computation with fractions
• Chapter 2: Millet and Rice
  • Concerns with proportions (for exchange of cereals, millet, or rice)
• Chapter 3: Distribution by Proportion
  • Problems on proportional distribution
• Chapter 4: Short Width
  • Finds the length of a side when given the area or volume
  • Finds the square root or cubic root of a number
• Chapter 5: Civil Engineering
  • Concerns with calculation for constructions of solid figures
  • Finds the volumes of various shapes of solid figures
• Chapter 6: Fair Distribution of Goods
  • Deals with calculation on how to distribute grain and labour
• Chapter 7: Excess and Deficit
  • Uses of method of false position to solve difficult problems
• Chapter 8: Calculation by Square Tables
  • Problems on simultaneous linear equations
  • Introduces concept of positive and negative numbers
  • Addition and subtraction of positive and negative numbers
• Chapter 9: Right angled triangles
  • Discusses the Gougu theorem(Pythagorean theorem)and properties of the right-angled triangle
  • Problems on similar right-angled triangles
  • Introduces general methods of solving quadratic equations

Introduction to Greece

Greece is a country in southern Europe, situated on the southern end of the Balkan peninsula. Regarded as the cradle of western civilization and being the birthplace of democracy, philosophy, the Olympic Games and of the arts and drama, Greece has a very long and remarkably rich history during which its culture has proven to be especially influential in Europe, North Africa and the Middle East. Today, Greece is a developed nation, member of the European Union and a member of the Eurozone .

Ancient Greek mathematicians were more focused on geometry, and used geometric methods to solve problems though others used algebra. Greek mathematicians were also very interested in proving that certain mathematical ideas were right so they spent a lot of time using geometry to prove that things were always true, even though people like the Egyptians and Babylonians already knew that they were true most of the time anyway.

Greece's contribution to Mathematics

A significant contribution was made by Greek-speaking people around the Mediterranean. They calculated Pi, discovered Pythagorean Theorem and published various mathematical books.

Although the Egyptians and Babylonians estimated the ratio between the circumferences to diameter before Greece, they did not prove it in a geometric way. By giving geometric arguments, Archimedes proved that Pi was constant and he got a more accurate value of Pi.

Pythagoras is not only famous for Pythagorean Theorem. Like Jesus, he had a lot of followers. They followed Pythagoras around and taught others in Greece and its colonies. He remained throughout his whole life. His enormous contribution to mathematics is undeniable.

Euclid's Elements is one of the most influential books to the development of math and other sciences. Its achievements are mainly about the logical development of geometry and other branches of mathematics. It has influenced all sectors of science. For 24 centuries, it has been studied by people all over the world and it has benefited numerous generations.

Euclid's Elements

Euclid's "Elements," was written in about 300 B.C. it was a widespread piece on geometry, proportions, and the theory of numbers. Euclid is known to almost every high school student as the author of The Elements, the long studied text on geometry and number theory. No other book other than the Bible has been so widely translated and circulated. From the time it was written, it was regarded as an extraordinary work already and was studied by all mathematicians, including one of the world's greatest mathematicians, Archimedes. Thus it has been studied by people all over the world in its following 24 centuries. It is unquestionably the best mathematics text ever written and will continue to benefit the world in the future. There are thirteen books in the Elements, mainly three parts.

There are thirteen books in the Elements, mainly three parts.

• Books 1 through 4 deal with plane geometry
• Books 5 through 10 introduce ratios and proportions
• Books 11 through 13 deal with spatial geometry

Introduction to Egypt

The Arab Republic of Egypt, commonly known as Egypt, is a Middle Eastern republic in North Africa. Egypt is the fifteenth most populous country in the world and is famous for its ancient civilization and some of the world's most ancient and important monuments.

Civilisation reached a high level in Ancient Egypt. The country was well suited for the people, with a fertile land thanks to the river Nile yet with a pleasant climate. Egypt enjoyed long periods of peace when society advanced rapidly. As the society became more complex, records and computations were required as the people started trading their goods. A need for counting arose, then writing and numerals were needed to record transactions. This gave rise to the development of mathematics in Ancient Egypt.

The ancient Egyptians were most likely the first civilisation to practice the scientific arts. The word chemistry is derived from the word Alchemy which is the ancient name for Egypt.

The Egyptians excelled in the application of mathematics. Although there is a large body of papyrus literature describing their achievements in medicine, there is hardly any records on their outstanding performance in Mathematics. However, they should have had a very advanced understanding of math because their achievements in engineering, astronomy and administration would not have been possible without it.

Ancient Egyptian Numerals

The Egyptians had a very interesting decimal system which used seven different symbols.
1 is shown by a single stroke.
10 is expressed by a drawing of a cattle hobble.
100 is represented by a coil of rope.
1,000 is shown by a drawing of a water lily.
10,000 is represented by a finger.
100,000 is expressed by a tadpole or frog.
1,000,000 is represented by the figure of a man with arms raised.

The way for reading and writing numbers is quite simple; the bigger number is always written in front of the smaller one and where there is more than one row of numbers the reader should add on to the top. Multiples of a certain value was expressed by repeating the symbol.

Ancient Egyptian Fractions

The Egyptians in 3000 BC had an interesting way to represent fractions.

Although they had a notation for 1/2 and 1/3 and 1/4 and so on , their notation did not allow them to write 2/5 or 3/4 or 4/7 as we would today.

Instead, they wrote any fraction as a sum of unit fractions where all the unit fractions were different.

For example,
5/8 = 1/2 + 1/8

11/12 = 1/2 + 1/3 + 1/12

A fraction written as a sum of distinct unit fractions is called an Egyptian Fraction.

The advantage of using Egyptian fractions is that the difference between two fractions is more obvious.

Which is larger: 3/4 or 4/5?
By using Egyptian fractions, we write each one as a sum of unit fractions:
3/4 = 1/2 + 1/4
4/5 = 1/2 + 3/10 and, expanding 3/10 as 1/4 + 1/20 we have
4/5 = 1/2 + 1/4 + 1/20
We can now see that 4/5 is the larger one by exactly 1/20.

Each fraction has an infinite number of Egyptian fraction forms.
To see why the second fact is true, consider this:
1 = 1/2 + 1/3 + 1/6 (*)
So if
3/4 = 1/2 + 1/4
then by diving the equation (*) by 4 we have:
1/4 = 1/8 + 1/12 + 1/24
which we can then feed back into our Egyptian fraction for 3/4:
3/4 = 1/2 + 1/4
3/4 = 1/2 + 1/8 + 1/12 + 1/24
But now we can do the same thing for the final fraction here, dividing equation (*) by 24 this time:
1/24 = 1/48 + 1/72 + 1/144
and so
3/4 = 1/2 + 1/8 + 1/12 + 1/48 + 1/72 + 1/144
Now we can repeat the process by again expanding the last term: 1/144 and so on for ever!
Each time we get a different set of unit fractions which add to 3/4!
This shows conclusively once we have found one way of writing T/B as a sum of unit fractions, then we can derive as many other representations as we want.

Introduction to Arabia

The area highlighted on the map is the Arabian Peninsula otherwise known as Arabia. The Arabian Peninsula is a peninsula in Southwest Asia at the junction of Africa and Asia consisting mainly of desert. The Arabian peninsula is an important part of the greater Middle East, and plays a critically important geopolitical role due to its vast reserves of oil and natural gas.

The Arabian Peninsula is part of the Arab world which consists of twenty-two countries stretching from Mauritania in the west to Oman in the east. This is where the Arabs who are a large and heterogeneous ethnic group can be found. An arab can be thought to be someone whose first language is Arabic (including any of its varieties).

When the Arabs conquered Syria, Palestine and Egypt, they inherited much of the Greco-Roman mathematical heritage and did a good job of preserving it. While the Arab civilisation declined their enthusiasm for mathematics survived long enough to be passed to Christian Spain and from there to Italy and the rest of Europe. The Arabs adopted the Arabic number system which was imported from India and made significant advances in algebra.

Arabic Numerals

In the history of Arabic Mathematics, one of its greatest achievements is the introduction of "Arabic Numerals". The "Arabic" numerals were influenced by India's mathematics. It was probably developed in India, but because it was the Arabs who transmitted this system to the West, the numeral system is called Arabic Numbers. It is a system based on place values and a decimal system of tens. This system used a zero to hold a place. These numbers were much easier to use for calculation than the Roman system which used numbers, like I, V, X, L, C, M,. Addition, subtraction, multiplication and division became simpler. With Arabic numerals, simple fractions and decimal fractions were also possible. Fractions and decimal fractions were also developed by Muslim mathematicians during the middle Ages.

Top row : Western Arabic or Hindu-Arabic Numerals
Bottom row: Modern Arabic numerals which were developed from them

Developments in Algebra

In the ninth century, Arabic mathematician Al-Khwarizmi wrote one of the first Arabic algebras with both proofs and examples. He invented the word "algebra". Because of his contributions to Algebra, he is known as the "Father of Algebra." He was born in 790 AD in Baghdad and died in about 850 AD.

The word for "Algebra" comes from the Arabic word for "al-jabr" which means "restoration of balance". In the 7th and 8th centuries, Mohammed united the Arabs. The Arabs conquered many territories from India, across northern Africa, to Spain. From then on till the 14th centuries, they pursued the development of the arts and sciences. Therefore, they took over and improved the Hindu number symbols and the idea of positional notation. These numerals (the Hindu-Arabic system of numeration) and the algorithms for operating with them spread to Europe in around 1200 and are still in use throughout the world today. Al-Khwarizmi converted Babylonian and Hindu numerals into a workable system that is more user-friendly. He gave the name to his math as "al-jabr" which we now know as "algebra".

In the 12th century, a Latin translation of al-Khwarizmi's book on algebra surfaced in Europe. In the early 13th century the new algebra appeared in the writings of the famous Italian mathematician, Leonardo Fibonacci. In short, Algebra was brought from ancient Babylon, Egypt and India to Europe via Italy by the Arabs.

Introduction to Ancient Babylonia

Babylonia, named for its capital city, Babylon, was an ancient state in the south part of Mesopotamia (in modern Iraq), combining the territories of Sumer and Akkad. The earliest mention of Babylon can be found in a tablet of the reign of Sargon of Akkad, dating back to the 23rd century BC.

The Babylonian system of mathematics was sexagesimal, or a base 60 numeral system. The Babylonians were able to make great advances in mathematics. Babylonian mathematics was, in many ways, more advanced than Egyptian mathematics. They could extract square and cube root; work with Pythagorean triples 1200 years before Pythagoras, had knowledge of pi and possibly e (the exponential function), could solve some quadratics and even polynomials of degree 8, solved linear equations and could also deal with circular measurement.

Babylonian Numerals

The Babylonians, who were famous for their astrological observations and calculations, used a sexagesimal or base-60 numeral system.

The Babylonians used the concept of place value to write numbers larger than 60. So they had 59 symbols for the numbers 1-59, and then the symbols were repeated in different columns for larger numbers.

For example, a '2' in the second column from the right meant (2 x 60) =120, and a '2' in the column third from the right meant (2 x 602) =7200.

To use the sexagesimal notation in modern language, we separate the 'columns' by commas, so that the number
7267 = 2(602) + 1(60) + 7
would be written as 2, 1, 7.

The base 60 number system of the Babylonians is still very useful in the modern world. We use 60 seconds as a minute and 60 minutes as an hour. Moreover, the 24 hour clock is actually an inheritance from the ancient Babylonians.