| Egyptian Mathematics |
Article Index for Egyptian |
Website Links For Egyptian |
Information AboutEgyptian Mathematics |
| CATEGORIES ABOUT EGYPTIAN MATHEMATICS | |
| ancient egyptian society | |
| egyptian mathematicsancient egyptian society | |
| egyptian mathematics | |
| history of mathematics | |
|
Egyptian methods principally employed the method of doubling and halving a known number to approach the solution, together with the Method Of False Position . Allied with their decimal number systems, unit fractions, and tables of common results (such as the decomposition of non-unit fractions, such as 2/n) into unit fractions, these methods allowed complex manipulation of problems. The Egyptians seem to have confined themselves to applications of arithmetic in practical contexts: for example, many problems address how a number of loaves can be divided equally between a number of men. Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. The problems in the Moscow And Rhind Mathematical Papyri are expressed in a practical context, seem to be exercises to practice the techniques for the manipulation of numbers. OVERVIEW Circa , Egyptian construction techniques included precision Surveying . Also, the oldest mathematics text discovered so far is the Moscow Papyrus , which is an Egyptian Middle Kingdom papyrus dated circa 2050 BC - 1800 BC. Like many ancient mathematical texts, it consists of what are today called "word problems" or "story problems", which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a Frustrum : "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. Your are to take 28 twice, result 56. See, it is 56. You will find it right." The and Geometric Series {Link without Title} . Three geometric elements contained in the Rhind papyrus are basic to simplistic ; and (3) third, the earliest known use of a kind of Cotangent . Finally, the Berlin Papyrus (circa 1800 BC) shows that ancient Egyptians could solve a second-order Algebraic Equation {Link without Title} . NUMERALS ''Main article:'' Egyptian Numerals The Numeral System used in ancient Egypt is a Decimal System , written in Hieroglyph s and Hieratic . Both systems existed from at least the Early Dynastic Period . (It should be noted that the hieratic system does not differ from the hieroglyphic system beyond a use of simplifying Ligatures for rapid writing.) The Rhind Mathematical Papyrus is written in hieratic, and contains many examples of how the Egyptians did their mathematical calculations. MULTIPLICATION Egyptian Multiplication was done by repeated doubling of the number to be multiplied (the multiplicand), and choosing which of the doublings to add together (essentially a form of Binary arithmetic). The multiplicand would be written out next to the figure 1, then the multiplicand would be added to itself (i.e. doubled) and would be written out next to the number 2, and so on, until the doublings gave a number greater than half of the number to be multiplied by (the multiplier). Then, the doubled numbers (1, 2, etc.) would be repeatedly subtracted from the multiplier to select which of the results of the existing calculations should be added together to create the answer. As a short cut for larger numbers, the multiplicand can also be immediately multiplied by 10, 100, etc. So, for example, Problem 69 on the Rhind Papyrus provides the following illustration: The tick mark (/) denotes the intermediate results that are added together to produce the final answer. See also: Ancient Egyptian Multiplication . FRACTIONS ''Main article:'' Egyptian Fraction Rational Number s could also be expressed, but only as sums of Unit Fraction s, i.e. sums of Reciprocal s of positive Integer s, except for 2/3 and (rarely) 3/4. The hieroglyph indicating a fraction looked like a mouth, which meant "part", and fractions were written with this fractional solidus, i.e. the numerator 1, and the positive denominator below. Special symbols were used for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (used less frequently). Problem 25 on the Rhind Papyrus uses the Method Of False Position to solve the problem "a quantity and its half added together become 16; what is the quantity?" (i.e., in modern Algebra ic notation, what is ''x'' if ''x''+½''x''=16). Assume 2 1 2 / ½ 1 / Total 1½ 3 As many times as 3 must be to give 16, so many times must 2 be multiplied to give the answer. 1 3 / 2 6 4 12 / 2/3 2 1/3 1 / Total 5 1/3 16 So: 1 5 1/3 (1 + 4 + 1/3) 2 10 2/3 The answer is 10 2/3. Check - 1 10 2/3 ½ 5 1/3 Total 1½ 16 Problem 31 sets the problem "q quantity, its 1/3, its 1/2 and its 1/7, added together, become 33; what is the quantity?" (i.e. what is ''x'' if ''x'' + 1/3 ''x'' + 1/2 ''x'' + 1/7 ''x'' =33), giving the answer 14 1/4 1/56 1/97 1/194 1/388 1/679 1/776 (i.e. 14 28/97). GEOMETRY Problem 50 uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter (so 1/9 is subtracted from the diameter, and the resulting figure is multiplied by itself, using the doubling method). In essence, this assumes that π is 4×(8/9)&2 (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonian s (25/8 = 3.125, within 0.53 percent) and was not surpassed until Archimedes (211875/67441 = 3.14163, an error of just over 1 in 10,000). Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This was to give a second value for π of 3.111'. The two problems together indicate a range of values for Pi between 3.111' and 3.16.. There may be a play here on the missing 1/64th of the mathematical eye of Horus. SEE ALSO EXTERNAL LINKS
FURTHER READING
|
|
|