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Mathematics from India • • • • • • • We know very little about early Indian mathematics. We do know that they had a workable number system used for astronomical and other calculations. They also had a practical interest in geometry. Early sources show an interest in very large numbers. After the fall of the Greeks, there was a time referred to as the “Dark Ages” throughout Europe and North Africa. During this time, mathematics in India flourished. Indian mathematicians contributed to astronomy, trigonometry, algebra, and combinatorics. Their texts do not describe how any of their methods or results were found. Their number system was invented sometime before 600 A.D. and has been refined over centuries. They had used nine symbols for numbers from an earlier system, 1 through 9. They introduced place value, and a small dot or circle to represent a place holder. We now refer to this system as the Hindu-Arabic system. Arabic Mathematics • • • • • • • • The number system in India, invented by the Hindus, was picked up by the Arabs during the Islamic expansion into India in the 7th and 8th centuries. (The Europeans later learned it from the Arabs, which is why we call it the Hindu-Arabic system.) By 750 A.D. the Islamic Empire stretched all the way from the western part of India to parts of Spain. Bagdad (now, central Iraq) became the cultural center of this Empire. Being located on the Tigris River made it a natural crossroads, the place where the East and West could meet. Like the Greek, the Arabic mathematical tradition is marked by the use of a common language. Muhammad Ibn Musa Al-Khwarizmi, during the 9th century, was one of the earliest Arabic mathematicians. He wrote a book describing India’s number system and computation with the system. He also wrote a book titled “al-jabr w’al-muqabala”, meaning “restoration and comprensation.” This book described how to solve equations, amongst other things. When translated into Latin, the word “al-jabr” became “algebra”. Another famous Arabic mathematician, scientist, philosopher, and poet, is “Umar AlKha-yammi, known in the West as Omar Khayyam (1048-1131). In his algebra book, he tried to find a way to solve 3rd degree polynomials. Though he couldn’t do it algebraically, he found a way to do it geometrically. Moving on to Europe • • • • • • • • At the time, much of Europe was using the Roman numeral system. This “new” number system finally found its way into Europe not only through books, but also through trade between European and Arabic merchants in the ports of Italy. When the Arabic merchants calculated so quickly, the European merchants were amazed/intrigued. One such merchant, Leonardo of Pisa (also called Fibonacci), was a mathematician in his spare time. He wrote a book called Liber Abaci around 1200 A.D. that popularized this new Arabic arithmetic. Because there was no such thing as a printing press at the time, handwritten copies were written and spread across Europe. The system met initial resistance, however, and it took several centuries for the HinduArabic system to replace Roman numerals in Europe. This was mostly because people were avoiding change. It was also partly for practical purposes. People were concerned about how easy it was for numbers to be changed to look like other numbers. Because of this, there were laws passed saying that legal documents had to have the numbers spelled out in words. One Italian community even outlawed their use. Fibonacci’s Liber Abaci • • • • • • The title has two common translations, The Book of the Abacus or The Book of Calculation. The first section introduces the Hindu-Arabic numeral system, including lattice multiplication and methods for converting between different repesentation systems. The second section presents examples from commerce, such as conversions of currency and measurements, and calculations of profit and interest. The third section discusses a number of mathematical problems. One example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots. The book also includes Euclidean geometric proofs, and a study of linear equations. Lattice Multiplication To multiply the two numbers 345 by 12, first create a grid that is a three by two grid, like the one below: Now write the number 345 above it and 12 to the right. Then carry out the multiplication of each digit with another, putting the tens digit in the top triangles and the units digit on the bottom. Then sum along the diagonal tracts and carry as needed to get the answer. Fractions • • • • • • • • Hindu manuscripts, as early as the 7th century A.D. showed them writing fractions with the number of times we have the part over the size of the part. This is basically the same notation we use, but they did not use the fraction bar. This way of writing fractions became common centuries later in Europe. The Latin word for counter, “numerator” was used to describe the number on top. The Latin word for namer, “denominator” was used for the bottom number. The horizontal bar to separate the two numbers was inserted by the Arabs during the 12th century. When the printing press was invented, during the late 15th century, the fraction bar was omitted, due to typesetting problems. It gradually came back in use during the 16th and 17th centuries. Although the “slash” notation, such as 5/7, would have been easier to type, that notation was not thought of until about 1850. The use of decimal fractions was not commonly in use until after Simon Stevin’s 1585 book, The Tenth. He showed how writing numbers as decimals made computation much easier. He ignored the use of infinite decimal expansions, though. Many symbols had been used for the decimal point, such as a dot, comma, apostrophe, raised dot, wedge, and left parenthesis. Today, there still is no standard notation for a decimal point. For example, much of Europe uses a comma.