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发表于 2011-6-17 08:31:05
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回复 34# 血猫
The history of numbers, especially the imaginary numbers The relation between the real and the imaginary number The combination of the real and the imaginary number Some conclusions Page ? 7 Contents of my presentation The history of numbers, especially the imaginary numbers The relation between the real and the imaginary number The combination of the real and the imaginary number Some conclusions Page ? 8 The history of numbers, especially the imaginary numbers The ancient Egyptians used the sign “|” to replace number “1”, and “||” –“2”,but before long they found the trouble---it was difficult to present a very large number; The ancient Romans used “Ⅰ,Ⅱ,Ⅲ,Ⅳ,Ⅴ,Ⅵ,Ⅶ,Ⅷ,Ⅸ,Ⅹ, ⅠⅡⅢⅣⅤⅥⅦⅧⅨⅩ L,C” to express the numbers “1,2,3,4,5,6,7,8,9, 10,50,100”. However, the same problem came again. Our ancestors—ancient Chinese people used our language to express the numbers, but is not convenient in the computational mathematics. Until the eighth century Indian invented the numbers we use today, which is called Arabic numerals. Page ? 9 The history of numbers, especially the imaginary numbers About 800 years after the invention of the numerals, it was just developed in the real, but how about the imaginary numerals? From our studying, we know that usually the problems come first, and then the solution and the new knowledge; It is the same with the imaginary numerals. In 1545 AD, the Italian mathematician Cardan gave the solutions of the cubic and the fourth equations in his famous book 《Arsmagna 》. x 3 + mx = n m, n ∈ N * n n 2 m3 3 n n 2 m3 x=3 + + + ? + 2 4 27 2 4 27 Page ? 10 The history of numbers, especially the imaginary numbers ?Then Cardan used his own solutions to solve the equation x3 ? 15 x = 4 ?According to the formula, he got the solution of the cubic equation. 3 3 x = 2 + ?121 + 2 ? ?121 In the time without imaginary numerals, like us in the middle school, people could easily say that the equation has no solutions, as the ?121. But he also found that x=4 was the solution. This problem left some questions to him: Why can this happen? Is there any relationships between the formula solution and 4? Page ? 11 The history of numbers, especially the imaginary numbers What’s more, he put forward another question. { m + n = 10 mn = 40 He searched for the solutions for a long time, and he found that if he could use the number ?15 just as usual, then he got (5 + ?15) + (5 ? ?15) = 10 (5 + ?15)(5 ? ?15) = 25 + 5 ?15 ? 5 ?15 + ( ?15) 2 = 25 ? (?15) = 40 { m = (5 + ?15) n = (5 + ?15) But he didn’t know why! Page ? 12 The history of numbers, especially the imaginary numbers It was not until several decades later that the problem was solved by Rafael Bombelli , another Italian mathematician . He found that x = 3 2 + ?121 + 3 2 ? ?121 = (2 + ?1) + (2 ? ?1) = 4 More important thing was that Rafael Bombelli defined the imaginary numerals and the operation rules of all the imaginary numerals. That was really a great achievement during the development of mathematics. Page ? 13 The history of numbers, especially the imaginary numbers In 1702, John Bernoulli first introduce the imaginary number into the mathematical analysis, such as (the imaginary number and the real began to combine) dx dx ∫ x 2 + a 2 = ∫ ( x + ia)( x ? ia) 1 dx dx =? (∫ ?∫ ) 2ia x + ia x ? ia 1 x + ia =? ln 2ia x ? ia In 1749, Euler delivered his important discovery---The Euler’s Formula e = cos x + i sin x ix Page ? 14 The history of numbers, especially the imaginary numbers In 1799, Gauss first proved algebra fundamental theorem . Every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. In 1843 a mathematical physicist, William Rowan Hamilton, extended the idea of an axis of imaginary numbers in the plane to a three-dimensional space of quaternion imaginaries. Page ? 15 The history of numbers, especially the imaginary numbers Maybe the birth of the numerals represents the entrance into the precise times of human beings, and from then on people can really look for the nature of science and the world. The birth of the imaginary numbers is a great leap people take in the development in science (especially used in physics), just like what Leibniz said: The imaginary number is fantastic shelter where the Gods have set foot, and it may be an amphibians existing both in existence and non-existence. |
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楼主: 血猫
狗仔卡
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求英文解釋……
哪來的……
