Mathematics

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Mathematics in Antiquity

The Babylonians of ancient Mesopotamia and the ancient Egyptians left the earliest records of organized mathematics. Arithmetic dominated their mathematics. In geometry, measurement and calculation were emphasized, with no trace of concepts such as axioms or proofs.

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Babylonia

Our knowledge of Babylonia comes from well-preserved clay tablets on which people wrote with wedge-shaped marks known as cuneiform. The earliest tablets date from about 3000 bc. Much of the mathematics on the tablets involved commerce. The Babylonians used arithmetic and simple algebra to exchange money and merchandise, compute simple and compound interest, calculate taxes, and allocate shares of a harvest to the state, temple, and farmer. The building of canals, granaries, and other public works also required using arithmetic and geometry. Calendar reckoning, used to determine the times for planting and for religious events, was another important application of mathematics.

The division of the circle into 360 parts and the division of the degree and the minute each into 60 parts originated in Babylonian astronomy. The Babylonians also divided the day into 24 hours, the hour into 60 minutes, and the minute into 60 seconds. Their number system was based on the number 60. The wedge-shaped symbol for 1 was repeated for numbers up to 9. Any number from 11 to 59 could be written as a combination of the symbol for 10 and the symbol for 1. For numbers 60 and higher the Babylonians used a symbol for position. The principle of position, or place value, was a significant advance in calculation. It enabled the same symbol to represent different numerical values depending on its position. The Babylonian system enabled them to represent fractions as well, but they lacked a symbol for zero, which led to ambiguities.

The Babylonians devised tables of reciprocals (numbers that yield 1 when multiplied, such as 3 and €), tables of squares and square roots, tables of cubes and cube roots, and tables of compound interest. They had a good approximation of Ã. Cuneiform tablets dealing with algebraic and geometric problems show that the Babylonians had, in effect, the quadratic formula for solving equations and could solve problems that involved ten unknowns in ten equations.

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Beginning about 700 bc the Babylonians used mathematics to study the motion of the Moon and planets. This enabled them to predict the daily positions of the planets, knowledge as important for astrology as for astronomy.

In geometry, the Babylonians knew a few relationships such as the proportionality of corresponding sides in similar triangles. They could solve problems using the Pythagorean theorem and were aware that an angle inscribed in a semicircle is a right angle. They had rules for the areas of simple plane figures including regular polygons and for the volumes of simple solids. They used 3 as the value of the ratio known as pi (p), a mathematical constant that is equal to about 3.14.

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Egypt

Although the Egyptians carved hieroglyph numerals onto some of their monuments, the earliest texts we have for Egyptian mathematics are two papyruses composed about 1800 bc. The papyruses contain problems in arithmetic and geometry, including practical problems dealing with the amount of grain needed to make a given quantity of beer and the amount of grain of one quality needed to give the same result as grain of another quality.

We also know that the Egyptians used mathematics to determine wages, find the areas of fields and the volumes of granaries, assess taxes, and calculate the number of bricks needed for particular structures. In addition, the Egyptians used mathematics in astronomy for calendar reckoning. Their calendar helped them set the times of religious holidays and predict the annual flooding of the Nile.

The Egyptians based their number system on the number 10, using separate hieroglyph symbols for the successive powers of 10 (1, 10, 100, and so forth). They wrote the symbol for 1 five times to represent the number 5, the symbol for 10 six times to represent the number 60, and the symbol for 100 three times to represent the number 300. Together, these 14 symbols represented the number 365. Addition was done by totaling the units—10s, 100s, and so forth—separately in the numbers to be summed. Multiplication was based on successive doublings, and division was the inverse of the process. The Egyptians represented fractions by using the symbol for mouth, meaning “part,” above the number symbols.

After the Egyptians began writing on papyrus in a script called hieratic, they developed individual symbols for every number from one to 10, every tenth number up to 100 (20, 30, and so forth), every 100th number up to 1,000 (200, 300, and so forth), and every 1,000th number. Although this system meant more symbols to memorize, it enabled the Egyptians to write numbers more compactly.

In geometry the Egyptians had rules for areas of rectangles, triangles, trapezoids, and the circle, as well as formulas for certain volumes, including rectangular prisms (such as a brick), cylinders, and pyramids. To find the area of a circle (pr²), the Egyptians used a value of about 3.16 for p, which was closer than the Babylonian value of 3.

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Maya Mathematics

Most records of Maya achievements in mathematics were destroyed after the arrival of Spanish conquerors in Middle America during the 1500s, but four codices (manuscript volumes) remain. Although the dates of Maya achievements in mathematics are difficult to determine, these accomplishments merit attention. The Maya used a base-20 number system, which probably descended from early times when people counted on both fingers and toes. The Maya may have been the first people to employ a special symbol for zero. They used two types of systems for numerals. One employed hieroglyphs, while the more commonly used system employed a dot for 1, a bar for 5, and a shell-like symbol for zero.

The calendar was extremely important in Maya civilization, and the Maya developed two. One was based on the Sun and had 365 days. The other was a sacred almanac with 260 days divided into 13 months of 20 days each. The almanac was used for predicting lucky and unlucky days. Scholars have speculated on the reasons for the 260-day calendar and believe it may be related to other astronomical data compiled by the Maya. The Maya calculated the length of the lunar month and the solar year with remarkable precision. See also Maya Civilization.

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Greek Mathematics

The Greeks adopted elements of mathematics from the Babylonians and the Egyptians. The new element in Greek mathematics was the invention of an abstract mathematics founded on a logical structure of definitions, axioms (propositions accepted as self-evident), and proofs. According to later Greek accounts, this development began in the 6th century bc with Thales of Miletus and Pythagoras of Sámos. The mathematics that had existed before their time was a collection of conclusions based on observation. In the Greek system of deductive proof, on the other hand, a new statement was logically derived from accepted premises.

The Greeks’ insistence on deductive proof was an extraordinary step. No other civilization had conceived the idea of establishing conclusions exclusively by deductive reasoning based on explicitly stated axioms.