On the Other Side of the Planet

The Genius of the East is the second installment of BBC’s Story of Maths. It tells the untold story of Mathematics of the East that somehow transformed the west and gave birth to the modern world.  Mark du Sautoy, the host, talked first about the emergence of Mathematics in China. The Great Wall of China is an amazing piece of engineering that in some ways made the ancient Chinese to realize they needed to make calculations. The Chinese has a simple number system that served as a foundation on how the west and other parts of the world count today. They use bamboo rods to represent the number from one to nine and were then placed in columns representing units, tens, hundreds and thousands. These rods also make the calculations very quickly. The Chinese only use the decimal system when calculating with rods. When writing the numbers down, they use special symbols to represent tens, hundreds, thousands and so on. Like the Egyptians, Babylonians and the Greeks, the Chinese have no concept of zero as well. Using the counting rods, the Chinese would use a blank space to represent a zero. The Chinese also believed that the numbers held cosmic and is still evident up today. Odd numbers would represent male, even numbers for female, 4 is a number one must certainly avoid and the number 8 brings good fortune. The Chinese also developed their own early version of Sudoko called the Magic Square. All the numbers in each line- horizontal, vertical and diagonal- add up to the same number 15. The Magic Square has great religious significance and showed the fascination of the Ancient Chinese to numbers.

Mathematics also played a vital role in running of the Emperor’s court. The calendar and movements of the planets are important influences/factors to the emperor’s decisions. The Geometric Progression also emerged at this time and was made as basis for the Emperor’s harem.  The Geometric progression is a series of numbers which you get from one number to the next by multiplying the same number each time. Legend has it that in the space of 15 nights the Emperor slept with 121 women. The empress on the first day, three senior consorts on the second day, the nine wives on the third day, the 27 concubines on the 4^th – 6^th day, 9 each night then finally the 81 slaves which were dealt with in groups of nine on the 7^th up to the 15^th night. The objective of this is to procure the best possible imperial succession and for the emperor to sleep with the lady of highest rank closest to the moon. The emperor’s court was not alone in its dependence on mathematics.  It was central to the running of the state.  A mathematical textbook called The Nine Chapter was used to educate civil servants making them competent in mathematics. The book is a compilation of 246 problems in practical areas like trade, payment of wages and taxes and at the heart of these problems lies the central themes of mathematics -  how to solve equations.  The Chinese went on to solve even more complicated equations which involves far larger numbers and what’s became known as the Chinese remainder theorem. The Chinese remainder theorem was used to measure planetary movement and in internet cryptography at the present time. The most important mathematician was Qin Jiushao who solved equations like quadratic equations where numbers are squared and cubic equations in which the numbers are cubed. He found a way of solving cubic equations that can also be applied to even more complicated equations.  China had made great mathematical leaps, but the next great mathematical breakthroughs were to happen in a country lying to the southwest of China which is India.

India is a country that had a rich mathematical tradition. Their first mathematical gift lay in the world of numbers. Like the Chinese, the Indians use the decimal place – value system and perfected it creating the ancestors for the nine numerals used across the world now. The Indian system of counting was ranked as one of the greatest intellectual innovations of all time developing into closest thing we could call a universal language. The Indians also introduced the concepts of zero. There are two possible reasons to the invention of zero. First, it may came from the calculations they did with the stone in the sand and second, their cultural belief. The ancient Indians believe the concepts of nothingness and eternity. Brahmagupta, an Indian mathematician proved some of the essential properties of zero which are now taught in schools all over the world. Brahmagupta’s rule runs like this: 1+0 = 1, 1-0 = 1, and 1*0 = 0. But Brahmagupta came a cropper when he tried to do 1/0. The Indians also introduced the concept of negative number which they call as “debts”. Brahmagupta’s understanding of negative numbers allowed him to see that quadratic equations have two solutions one of which could be negative. He also developed a new mathematical language that lead to the x’s and y’s that filled mathematical journals today. Indian mathematicians were also responsible in the theory of trigonometry. Trigonometry was used by Indian mathematicians to explore the solar system. The sine function in trigonometry for instance enables one to calculate distances. At present, it is used in architecture and engineering. In the 15^th century at Kerala in the southern part of India, a leader named Madhava gave birth to the concept of infinite and to the exact formula of pi.

, A new empire began to spread across Middle East in the 7^th century.  At the heart of this empire lay a vibrant intellectual culture. The House of Wisdom is a great library and a center of learning was established in Baghdad. The Muslim scholars collected and translated many ancient texts and if it weren’t for them, the modern world may never have known about the ancient cultures of Egypt, Babylon, Greece and India. But the scholars were not contented with simply translating other people’s mathematics. They wanted to create a mathematics of their own. Such intellectual curiosity was actively encouraged in the early centuries of the Islamic empire. The Koran asserted the importance of knowledge. The needs of Islam demanded mathematical skill. Muslim artists discovered all the different sorts of symmetry. The director of the House of Wisdom in Baghdad was a Persian scholar called Muhammad Al – Khwarizmi. He was an exceptional mathematician who was responsible for introducing two key mathematical concepts and recognized the incredible potential that the Hindu numerals had to revolutionize mathematics and science.  His explanation that the Hindu numerals speed up calculations and do things effectively was so influential that it did not take long for it to be adopted by the mathematicians of the Islamic world. These numbers were now known as the Hindu-Arabic numerals with the numbers form 1-9 and 0. Also, Al-Khwarizmi was the one who introduce Algebra which he got its name after the title of his book Al-jabr W’ al – muqabala or Calculation by Restoration or Reduction. Algebra was a huge breakthrough that enables one to analyze the way numbers work.  But Al-Khwarizmi’s greatest breakthrough came when he applied algebra to quadratic equations – that is equations including the power of two.

The next mathematical Holy Grail was to find a general method that could solve all cubic equations. Omar Khayyam, a Persian mathematician in the 11^th century took up the challenge of cracking the problem of the cubic but it was to no avail. But the leap in finding a general solution to the cubic equation was made in the West – in Italy. When Europe had fallen under the shadow of the Dark Ages, all intellectual life, including the study of mathematics had stagnated. But by the 13^th century, things were beginning to change. Led by Italy, Europe was trading and with that contact came the spread of Eastern knowledge to the West and the birth of Europe’s first great medieval mathematician - Leanardo of Pisa or better known as Fibonacci. In his Book of Calculating, Fibonacci promoted the new number system, demonstrating how simple it was compared to the Roman numerals that were in use across Europe. But there was a widespread suspicion of these new numbers. Some believed that they would be more open to fraud or that they’d be so easy to use for calculations. But over time, common sense prevailed and the new number system spread throughout Europe and the old Roman system slowly became defunct. At last, the Hindu – Arabic numerals had triumphed. Fibonacci is also known for his discovery of some numbers now called the Fibonacci sequence that arose when he was trying to solve a riddle about the mating habit of rabbits.  The Fibonacci numbers are nature’s favorite numbers. Wherever you find growth in nature, you find the Fibonacci numbers.

The next breakthrough in European mathematics happened in the early 16^th century. It involved the discovery of the general method that would solve all cubic equations, and it happened in the Italian city of Bologna. The University of Bologna was the crucible of European mathematics. Pupils from all over Europe flocked here and developed a new form of spectator sport – the mathematical competition. Large audiences would gather to watch mathematicians challenge each other with number, a kind of intellectual fencing match. But some problems were just unsolvable and one scholar was to prove everyone wrong and his name was Tartaglia. Shunned by his schoolmates because of the terrible scar on his face and his stammering, Tartaglia lost himself in mathematics and it wasn’t long before he found the formula to solve one type of cubic equation. But Tartaglia soon discovered that he wasn’t the only one to believe he had cracked the cubic for there is one Italian called Fior who was also boasting that he too held the secret formula for solving cubic equations. The two mathematicians then had a competition with Tartaglia winning in the end by solving all the questions in less than two hours. Cardano, a mathematician in Milan was so desperate to find the solution that he persuaded Tartaglia to reveal the secret but on one condition - that Cardano keep the secret and never publish. But Cardano couldn’t resist and discussed Tartaglia’s solution with his student, Ferrari who used it to solve more complicated quartic equation. Cardano broke his vow of secrecy and published Tartaglia’s work together with Ferrari’s brilliant solution of the quartic. Tartaglia never recovered and died penniless and to this day, the formula that solves the cubic equation is known as Cardano’s formula. The Europeans now had in their hand the new language of algebra, the powerful techniques of the Hindu – Arabic numerals and the beginnings of the mastery of the infinite.