No Man Is An Island

Mathematics had flourished through time. From the humble beginnings of Egypt, Babylonia, and Mesopotamia, great ideas emerged which are of great help to us at present. In the proceeding paragraphs, I will tackle about what the East had done; what discoveries and knowledge they contributed to the world of mathematics.

            The Great Wall of China which is thousands of miles long was strategized with great concepts of engineering. This defensive wall requires the calculation for distances, angles of elevation and amounts of material which required them to have much deeper understanding of mathematics. The decimal- place value system or the use of units, hundreds and thousands is Chinese’ own simple number system aided by rods. Rods represent the numbers from one to nine and for each rod’s position, indicates a unique numerical value. But in writing numbers, this number system is not used, instead, they use symbols which represent tens, hundreds, thousands and more. With the Chinese, zero does not exist as a number and is represented by a blank space. In playing with numbers, Chinese have its own version of Sudoku which they call the magic square and have originated from a sacred turtle with numbers on its back. Numbers in a magic square always equal to a sum of fifteen in each different line: horizontal, vertical and diagonal.

In Chinese’ legends, one of the Yellow Emperor’s deities,  created mathematics in 2800 BC with cosmic significance: even numbers for females, odd numbers for males, avoidance of the number four and good fortune for number eight. On the other hand, astronomers were important members of the imperial court and these astronomers were mathematicians. Whatever the emperor does, he consults the calendar and do his concerns with mathematical precision. Even with his problem with how will he sleep with a lot of women in his harem, he asked his mathematical advisers to help him. To solve this problem, his mathematical advisers based it on geometric progression. Having 121 women in his harem (the empress, three senior consorts, nine wives, twenty-seven concubines and eighty-one slaves), the emperor should be able to sleep with them in a span of fifteen nights.

Mathematics is a very big use for civil servants. In fact, they were educated by a mathematical textbook which was perhaps written in around 200 BC, “The Nine Chapters”. This textbook contains 246 practical problems which served as guides for the Chinese in solving equations. Also, the Chinese have practiced the use of small numbers in apprehending large numbers. Back then, ancient Chinese astronomy used Chinese remainder theorem to measure the planetary movement, but at present, it is already used in internet cryptography. During the golden age of Chinese maths, thirty schools of math were all around the country. Qin Jiushao was the most significant mathematician that time. He was not just a mathematician, but also an imperial administrator. He was corrupt and very cruel in his time that he poisoned anyone who hinders his plans. Despite this, mathematics was his true passion which led him to be interested with cubic equations. Qin’s method of solving cubic equations which was the approximation method was just yet to be discovered by Isaac Newton in the seventeenth century. This method can be used even in highly complex mathematics like numbers with the power of ten. Qin realized that he was only doing inexact solutions; he cannot derive formulas to obtain an exact answer to complicated equations.

Like the Chinese, Indians have also discovered the advantages of decimal place-value system. The Indians invented zero, they transformed it from a mere place holder to a number that does make sense. Invention of zero was influenced by their religion with the belief of nothingness and eternity. This mathematical term, zero, is represented by the word “shunya”. In the seventh century, Brahmagupta, an Indian mathematician, verified some vital properties of zero which are now used all over the world.  But, dividing with zero is another concept. In twelfth century, Bhaskara II, also an Indian mathematician, worked with it and found out that one divided by zero is infinity. Indians later on discovered the negative numbers which they treat as another type of nothing. Seeing numbers as abstract entities, Indians made a new way of solving quadratic equations. Brahmagupta have seen quadratic equations to always have two solutions, and one of it can be negative. As he went further, he was solving quadratic equations with two unknowns, a problem which was not accepted in the west until Fermat, a French mathematician challenged his colleagues in 1657.

Brahmagupta began to solve equations through abstraction and this led him to develop a new mathematical language. He represented the unknowns in his equations using the initials of the names of colors. The x’s and y’s which we use today emerged from Brahmagupta’s mathematical language. Aside from this notation, Indian mathematician also pioneered the discoveries in the theory of trigonometry which involves the study of right-angled triangles. In trigonometry, there is a function called the sine function which enables you to calculate the distances even if there is no accurate measurement. Indian astronomers use trigonometry in determining the relative distance between Earth and the moon and the moon and the Earth and the sun. This is only calculated whenever the moon is half full; by this time, the sun, moon and the Earth create a right-angled triangle. The Indians now know that the angle between the sun and the observatory was one-seventh of a degree. Sine function was first used by the Greeks but the problem was they could not calculate the sines of every angle. That is why the Indians find  a solution to this problem. Kerala, located in South India, was the home of one of the most brilliant schools of mathematicians. They call their leader Madhava. Madhava made extraordinary mathematical discoveries during his time. He discovered the concept of the infinite and later on the infinite series which he wanted to connect with trigonometry. Madhava’s principle made him realize that he could capture pi. For centuries, no one gave the precise value for pi until the 6^th century came. In India, a mathematician named Aryabhata discovered  a very accurate approximation for pi, 3.1416. Not so soon, Madhava discovered its exact value by using infinity. He also used this concept to get the sine formula in trigonometry.

By 7^th century, Islamic empire began to spread across the Middle East and was inspired by the teachings of the Prophet Muhammed. Centering this empire is a flourishing intellectual culture. This can be seen in the great library of Baghdad called the House of Wisdom. Here, they studied astronomy, medicine, chemistry, zoology and mathematics. Islam demanded mathematical skill. According to them, learning was nothing less than a requirement of God. They also applied mathematics on a two-dimensional wall wherein their artists discovered all the different types of symmetry. Muhammad Al-Khwarizmi, the director of the House of Wisdom in Baghdad, was a Persian scholar who was exceptional in mathematics. He saw the potential of Hindu numerals in developing  mathematics and science. These numerals later became known as Hindu-Arabic numerals. Al-Khwarizmi was the one to create the new mathematical language called Algebra, originated from the title of his book “Al-jabr W’al-muquala ” or Calculation by Restoration or Reduction. Algebra was compared to a code for running a computer program that will work whatever numbers you substitute into the program. This breakthrough led to a formula that could be used to perform quadratic equation, whatever values you substitute. After unfolding the secrets of quadratic equation, the next would be to find out how to solve all equations involving numbers to the power of three, the cubic equations. A Persian mathematician, Omar Khayyam, was a famous poet and at the same time a great mathematician. He used a systematic analysis to unravel different sorts of cubic equations. But because he was influenced by the Greeks’ geometry, he couldn’t separate it from algebra and so he couldn’t arrive at a purely algebraic solution.

Italy had gone through Dark Ages freezing all intellectual life including the system of mathematics. It was in the 13^th century when their normal life came back. A son of a customs official became Europe’s first great medieval mathematician. While he travels with his father, he was also learning the improvements of Arabic mathematics and the advantages of the Hindu-Arabic numerals. After his travel, he wrote a book which made a great impact to Western mathematics. His name was Leonardo of Pisa or widely known as Fibonacci, the author of the book entitled “Book of Calculating”. He introduced a new number system which is much easier compared to the Roman numerals. And because it is too easy, people treated these numbers as fraud. They were afraid that the ordinary people would take the authority from the intelligentsia who knew how to handle the old type of numbers. These numbers were even banned in Florence in 1299 but later on, it still spread throughout Europe and the old Roman system was junked. Hindu –Arabic numerals were used and Fibonacci became known for his discovery of the Fibonacci sequence, which came up to him while trying to solve a riddle about the mating habits of rabbits. Fibonacci numbers can be found in every growth in nature. You can find it in petals of a flower, pineapples and the shell of a snail. Another mathematical breakthrough happened in the University of Bologna in 16^th century. This university was very fond of mathematical competitions attended by large audiences and mathematicians who fought each other in an intellectual fencing match. But for them, cubic equations were impossible to solve, until Tartaglia proved them wrong. He was twelve years old when a rampaging French army slashed him with a sabre across the face which caused him an awful facial scar and a disturbing speech impediment. Tartaglia means “the stammerer”. Being avoided by his schoolmates, he drowned himself in mathematics and found a formula to solve one sort of cubic equation. But he found out that Frior, a young Italian, also believed that he achieved the formula in solving the cubic. The two of them were invited in an intellectual fencing match, but Tartaglia got a problem, he only knew how to solve one type of cubic equation and Frior had given him different sorts of it. Tartaglia gave much effort a few days before the contest, learning how to solve Frior’s questions. On the day of the match, he solved all Frior’s questions  just within two hours. He then searched for the formula for solving all sorts of cubic equations until Cardano, a mathematician in Milan, heard the news. He encouraged Tartaglia to expose the secret with condition that it would be a secret and will never be published. But Cardano cannot resist discussing it with his student,  Ferarri. Ferarri then used the formula in solving the quartic equation, a more complicated one. Cardano broke his vow with Tartaglia by publishing his work together with Ferarri’s astonishing solution of the quartic as a reward to his brilliant student.  Tartaglia never recovered and died penniless. Tartaglia’s formula was credited to Cardano and is known as Cardano’s formula. Even if Tartaglia did not won the fame while he’s still alive, his work was the first great mathematical breakthrough in modern Europe.

After knowing all of these brilliant mathematicians of the east and their amazing contributions to the world of mathematics, I realized that I should be thanking them for making our lives today easier. Thanks to the Hindu-Arabic numerals for junking Roman numerals. I cannot help to laugh imagining Roman numerals in quadratic equations, cubic equations and with other equations. Perhaps it’ll blow my mind into pieces. Thanks to Tartaglia for discovering the formula for cubic equations. Also, thanks to Cardano for publishing Tartaglia’s work because if he did not break his vow with Tartaglia, we would not be able to play with cubic equations today. My gratitude to Qin for sharing with us his exact value for pi, because without him, we would not appreciate circles this great. Pertaining to all of these great mathematicians of the East, I just observed a trend in their lives. No matter how brilliant they were, there was always one thing that they can’t solve and whenever that happens, a man always came to crack it. That just implies that we, people, need each other to develop things and improve our simple ways of living.