| A Brief History of MathematicsPeople seem compelled to organize. They also have a practical need to count certain things: cattle, cornstalks, and so on. There is the need to deal with simple geometrical situations in providing shelter and dealing with land. Once some form of writing is added into the mix, mathematics cannot be far behind. It might even be said that the symbolic approach precedes and leads to the invention of writing.Archaeologists, anthropologists, linguists and others studying early societies have found that number ideas evolve slowly. There will typically be a different word or symbol for two people, two birds, or two stones. Only slowly does the idea of 'two'become independent from the things that there are two of. Similarly, of course, for other numbers. In fact, specific numbers beyond three are unknown in some lesser developed languages. A bit of this usage hangs on in our modern English when we speak, for example, of a flock of geese, but a school of fish.The Maya, the Chinese, the Civilization of the Indus Valley, the Egyptians, and the region of Mesopotamia between the Tigris and Euphrates rivers -- all had developed impressive bodies of mathematical knowledge by the dawn of their written histories. In each case, what we know of their mathematics comes from a combination of archaeology, the references of later writers, and their own written record.Mathematical documents from Ancient Egypt date back to 1900 B.C. The practical need to redraw field boundaries after the annual flooding of the Nile, and the fact that there was a small leisure class with time to think, helped to create a problem oriented, practical mathematics. A base-ten numeration system was able to handle positive whole numbers and some fractions. Algebra was developed only far enough to solve linear equations and, of course, calculate the volume of a pyramid. It is thought that only special cases of The Pythagorean Theorem were known; ropes knotted in the ratio 3:4:5 may have been used to construct right angles.What we know of the mathematics of Mesopotamia comes from cuneiform writing on clay tablets which date back as far as 2100 B.C. Sixty was the number system base -- a system that we have inherited and preserve to this day in our measurement of time and angles. Among the clay tablets are found multiplication tables, tables of reciprocals, squares and square roots. A general method for solving quadratic equations was available, and a few equations of higher degree could be handled. From what we can see today, both the Egyptians and the Mesopotamians (or Babylonians) stuck to specific practical problems; the idea of stating and proving general theorems did not seem to arise in either civilization.Chinese mathematics -- a vast and powerful body of knowledge --, although mainly practical and problem oriented, did contain general statements and proofs. A method similar to Gaussian Reduction with back-substitution for solving systems of linear equations was known two thousand years earlier in China than in the West. The value of was known to seven decimal places by 500 A.D., far in advance of the West.In India mathematics was also mainly practical. Methods of solving equations were largely centered around problems in astronomy. Negative and irrational numbers were used. Of course, India is noted for developing the concept of zero, that was passed into Western mathematics via the Arabic tradition, and is so important as a place holder in our modern decimal number system.The Classic Maya civilization (250 BC to 900 AD) also developed the zero and used it as a place holder in a base-twenty numeration system. Again, astronomy played a central role in their religion and motivated them to develop mathematics. It is noteworthy that the Maya calendar was more accurate than the European at the time the Spanish landed in The Yukatan Peninsula.Ancient GreeceThe axiomatic method came into full force in Ancient Greek times; it has...

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