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Ancient Math |
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| Monday, February 24, 2003 | ||
| Found while tracking down Xeno's paradox. | ||
The Foundation of Everything in the Understanding of Nothing "The Greeks were the most impressive of all civilizations, the most influential in Western culture, and the most decisive in founding mathematics as we know it" (Allen Greek). Such is one of many common misconceptions that result from the modern-day scholarly bias towards ancient Western civilizations as opposed to Indian civilization (Vasudeo 2). Modern mathematics has its origins in three main ancient cultures: Egyptian, Babylonian, and Indian (based upon the Hindu religion) (Allen General). Many of the mathematical developments of Indian origin are often unjustly ignored, overlooked, or attributed to ancient Egyptian or Babylonian developments. However, the influence Hindu mathematical developments have exhibited on modern systems is far greater than that of ancient Sumerian cultures. Many of the developments in India, including an advanced notion of zero (McQullin) and hundreds of practical mathematical algorithms (Jhunjhunwala 13), are unique to India in the ancient mathematical arena. The oldest recorded mathematics originated in Egypt. Evidence of such antiquity comes from the Rhind papyrus, which is dated roughly 2000 BCE to 1800 BCE, and some sources claim finding mathematically-related Egyptian hieroglyphics dating as far back as 2700 BCE (Allen Egyptian). Perhaps because of this antiquity, the importance of Egypt's mathematical progress is overestimated. Egyptian math was very practical, and in that sense primitive (Allen Egyptian). In addition, the Egyptians used a relatively primitive numbering system. It was a decimal system (base ten) like the modern numbering system, yet it lacked the fundamentals of place-value and the decimal point which are important parts of the modern numbering system (ThinkQuest). As a result, the system had seven main numerals, or figures, one for each of the first seven powers of ten. To represent a number, one would simply line up as many of each figure as needed. Thus the number 203 would be represented by two 100-numerals and three 1-numerals lined up. Numbers greater than a few million were tedious to represent, because there were no numerals for numbers greater than one million. However, because the Egyptians were very practical, they seldom needed numbers as large as one million. Another result of this practicality was that the Egyptians were not very abstract in their mathematics (Allen Egyptian). Their fractions were limited to unit fractions (1/2, 1/3, 1/4, etc…). They only had a basic understanding of area and volume, with the exception of a complex formula for the volume of a pyramid. In addition, they used binary multiplication (a tedious process involving lots of sequential additions), and their division was limited to multiplying by reciprocals looked up in tables (Allen Egyptian). In other words, despite some claims that the Egyptians were the greatest mathematicians in Western history (Ramsey), they were somewhat lacking to say the least. Babylonian mathematics were more advanced. Babylonians drew symbols using the top of a stylus on wet clay, then let the clay bake in the hot sun (Melville). This writing medium made it very difficult to draw curved lines (Ramsey), so the Babylonians utilized a system with cuneiform (literally, "wedge-shaped") symbols. Cuneiform is very old, generally dated as far back as 2500 BCE, although claims have been made that it was developed even before Egyptian hieroglyphics (around 3000 BCE). Because the medium was clay, hundreds of thousands of these tablets have survived today (Allen Babylonian), making Babylonians sources very clear and complete. Tablets with mathematics on them date back to a period known as the Old Babylonian period of mathematics, estimated between 1800 BCE and 1600 BCE (Melville). The main reason that Babylonian mathematics was so much more advanced than Egyptian mathematics was its sexigesimal (base sixty) number system (Allen Babylonian). The number 60 has many common, small, integer divisors (2, 3, 4, 5, 6, 10, 12, 15, 20, and 30), making it a very flexible number for arithmetic, and easy to make fractions with (Fowler). This base sixty system is further represented by the fact that their culture utilized 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle (Ramsey). In addition, the Babylonians had a limited concept of the place-value system (Humez 21), in which a numeral to the left of another represents a larger number, an incredible improvement over the Egyptians' summation method of representing numbers. The Babylonians did make some significant advances in mathematics. One famous tablet demonstrates at least some knowledge of the Pythagorean Theorem, dated 1200 years before Pythagoreas lived, 1900 BC, found in the ancient city of Harmal:
This tablet also demonstrates a common aspect of all instructional tablets found. Babylonian mathematics was taught by example, and in word problems; no general algorithms for solving problems were ever given, and the somewhat more advanced concept of symbolic mathematics had not dawned upon them (Melville). Another famous tablet, known as YBC 7289, or more commonly "the root-two tablet", demonstrates the Babylonian knowledge of square roots (as well as another demonstration of the Pythagorean Theorem, though less direct in that respect). A right triangle was given. The lengths of the two legs were both labeled one, and the hypotenuse was labeled with a number representing the square root of two, correct to a remarkably precise seven decimal digits (Ramsey). Despite these developments, the Babylonian number system had a major problem, namely the lack of the full use of zero. As mentioned above, the Babylonians had established a limited place-value system, and a symbol, which can be called zero for convenience, was used to represent a placeholder. That is, the number 208 (written using Babylonian symbols, of course, as opposed to the Arabic numerals being used here) would represent (in decimal notation) 2x(60x60) + 0x(60) + 8. In this case, the zero simply holds a place to make the 2 signify a larger number. However, the Babylonian concept of this place-value system was not fully developed. They did not realize that zeros could be tacked on to the end of a number to increase total order of magnitude (Humez 21). So, numbers such as 2080 or 20800 would never be written. They would both be written as 208 and the order of magnitude would be assumed from context. Even though the zero numeral was understood as an intermittent placeholder, Babylonian mathematicians never understood the concept of zero as a number (Allen Babylonian). They merely did not know what a number subtracted from its equivalent was. Although the advances of the Babylonian culture in the field of mathematics were noteworthy, the Hindus were much more influential to modern-day mathematics. In fact, the Hindus were more influential to the field than any other ancient culture. This, in part, arises from their full understanding of zero and the place-value system, an understanding that no other culture had attained. An organization known as Neo-Tech makes the following claim:
Other evidence backs up this claim. As stated previously, the Babylonian culture did partially understand the use of zero as a placeholder, but not at all as a lone number. Ironically, the Egyptians, though never thinking of a place-value system, completely understood the number zero as a null value, a representation of a number subtracted from an equivalent value, and a reference point for counting (Arsham). The Hindus, however, understood both uses of zero, and understood them well. It is also important to note that the Hindu use of zero is what eventually influenced the modern numbering system, which uses zero in the exact same way. According to the scholar Kristin McQuillin, the "discovery of the symbol for nothingness had an enormous significance upon subsequent humanity" (McQuillin). It was a Buddhist astronomer that introduced the Hindu usage of zero to China. The Arabs incorporated it into their own system as well, and ended up spreading it throughout the rest of Europe (ThinkQuest). The modern-day usage of an open circle to represent zero was derived from the Indian numeral zero, which was created to look like a hole in the ground (McQuillin). The roots of the English word "cipher", which means zero, originated in India as well. The Hindu word for void is "sunya", which became the Arabic "sifr", the Roman "cifra", and finally the English "cipher" (Arsham). It is clearly evident that it was the Hindu culture and its use of zero, not a Western culture, that influenced the modern world's mathematics. However, India was not so influential on modern day mathematics merely because of its complete grasp on the concept of zero. The mathematician Gurjar Laxman Vasudeo goes as far as to claim that "it can legitimately be said that India led the whole world in the field of mathematics as far as the beginning of the late 17th century" (2). Although that assertion is a bit extreme, ancient Indian mathematics has had an important influence upon much of modern mathematics. Mathematics in ancient India developed during two distinct time periods: the Sulba, or Súlvasutra, period, which ranges from 800 BCE to 500 BCE (Bidyarana 6), and the Vedic, or Brâhmana, period, which most scholars trace to between 1500 BCE and 750 BCE, though claims have been made that it extends as far back as 2000 BCE or 3000 BCE (Vasudeo 1). The mathematics of each period differ, though sources from the Vedic period show more clearly that many fundamentals of mathematics were developed in India (Vasudeo 1). The Vedic period is so called because that was the period when the Vedas, Hindu religious documents, were composed (Vasudeo 1). The mathematics that developed during this period are known as "Vedic mathematics" (Jhunjhunwala 13). Hundreds of very practical and simple mathematical algorithms were created during this period, many of which are still in use today (Jhunjhunwala 13). Ashok Jhunjhunwala, commenting on the Vedic methods of mathematics, states, "Most of these techniques and algorithms follow from a deep understanding of the place value system in mathematics" (13), linking back to the full Hindu understanding of the value of zero. One example of these computational methods is the Navasesh method, in which a short series of simple arithmetic operations can be performed to check the results of much more complex arithmetic problems (Jhunjhunwala 7). Another method greatly simplifies the multiplication of values that have multiple units, such as feet and inches, in each number (Jhunjhunwala 2). India also had a highly developed abstract mathematical basis that did not rely on Vedic mathematics. An ancient Indian scholar named Brahmagupta described in detail negative numbers, a concept foreign to any other culture, including Egyptian and Babylonian cultures (MacTutor). The Babylonian mathematicians are often given credit for being the first to solve systems of equations, at about 400 BCE, yet there is no evidence of any written Babylonian equations (MacTutor). The Hindus, on the other hand, had developed methods for solving equations, and written evidence of equations exists. In fact, in these equations, the Hindus used letters to represent unknown quantities, known as variables (of which there were sometimes several)-a method still in use today (MacTutor). The Hindus even had a slight grasp of infinity as it relates to mathematics; Bhaskara, a Hindu mathematician, made the claim that three divided by zero equals an infinite quantity (Humez). Though a slightly incorrect assertion by modern rigorous mathematical standards, the concept is mostly correct, and a similar concept of the infinite is a major part of modern calculus. An ancient Indian manuscript, known as the Bahkshali Manuscript, named after the city where it was found, contains many references to high-level mathematics (Kaye 1). The manuscript, discovered in Northwest India in 1881, was written on about 70 pages of birch-bark. Many of these pages are missing parts or illegible due to aging, though at least 35 are fully comprehensible, and valuable information was salvaged from scraps of the others (Kaye 3). Just a few of the several topics the Manuscript covers include "Complex Series", "The Computation of the Fineness of Gold", "Approximate Evaluations of Square-Roots", and "Problems of the Type x(1-a1)(1-a2)…(1-an)=p" (Kaye 15). Keep in mind that Arabic peoples spread all these mathematical ideas, invented in India, throughout the rest of Europe (ThinkQuest). As a direct result of this, and the sheer level of mathematical achievement that India exhibited, much of modern mathematics is based upon Indian mathematics. In short, it was Hindu mathematical developments that spread throughout the world, and not Egyptian or Babylonian mathematics, as some claim. Western cultures had simply not advanced as far as India in the field of abstract mathematics. The modern universal number system itself, based upon zero and place-values, a system fundamental to mathematics, along with a host of other mathematical achievements, were developed by Indian mathematicians. The influence of ancient India is rather clear, so do not let yourself submit to the Western cultural bias against the mathematics of the Hindus. Works Cited Allen, Donald G. Babylonian Mathematics. Allen, Donald G. A General View of Mathematics before 1000 B.C. Allen, Donald G. Egyptian Mathematics. Allen, Donald G. The Origins of Greek Mathematics. Arsham, Dr. Hossein. The Zero Saga and Confusion with Numbers. Bidyaranya, Swami and Avadhesh Singh. History of Hindu Mathematics, a Source Book. Burney, Charles. The Ancient Near East. Fowler, Michael. Counting in Babylon. Hayden, Julia. The Ancient World Web. Hazeghi, Dara. Dara's Pi Pages. Humez, Alexander, Nicholas D. Humez, and Joseph Maguire. Zero to Lazy Eight: The Romance of Numbers. Jhunjhunwala, Ashok. Indian Mathematics. Kaye, G.R. The Bakhshali Manuscript. Lightspan Study Web. Studyweb Mathematics: History of Mathematics. MacTutor History of Mathematics Archive. History Topics Index. McQuillin, Kristen. A History of Zero. Melville, Duncan J. Mesopotopian Mathematics. Neo-Tech. The Discovery of the Zero. Office of ECSED, The. NIST Physics Laboratory. Ramsey, Dennis J. Babylonian Mathematics. RCN Corporation. Chronology of Mathematics. Sasson, Jack M., editor. Civilizations of the Ancient Near East. ThinkQuest. Mathematics History: Development. Vasudeo, Gurjar Laxman. Ancient Indian Mathematics and Vedha. downloaded: 24 Feb 2003 |
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