Many scholars and mathematicians in history vividly expounded the beauty of mathematics from different aspects. Plocque Ras has long asserted: "Where there is mathematics, there is beauty." Aristotle once said: "Although mathematics does not explicitly mention goodness and beauty, goodness and beauty cannot be completely divorced from mathematics. Because the main forms of beauty are "order, symmetry and certainty", these are the principles of mathematical research. "
Hua, a famous mathematician in China, said: "As far as mathematics itself is concerned, it is magnificent, colorful and fascinating ... People who think that mathematics is boring only see the rigor of mathematics, but they don't realize the inner beauty of mathematics." Mathematician Xu Lizhi said: "Mathematics, as a scientific language, has the beauty of ordinary language and art, that is, mathematics also has its own beauty in content, structure and method, which is the so-called beauty of mathematics. The meaning of mathematical beauty is rich, such as the simplicity and unity of mathematical concepts, the coordination and symmetry of structural relations, the generality, typicality and universality of mathematical propositions and mathematical models, and the singularity in mathematics. "
As can be seen from the above discussion, mathematics is full of beautiful factors. Mathematical beauty is the perceptual and rational presentation of the essential strength of mathematical science. It is not illusory and unpredictable, but has its definite objective content. Mathematical beauty is different from other beauty. It has no bright colors, pleasant sounds and dynamic pictures, but it is a unique beauty.
Klein, a German mathematician, once described the beauty of mathematics as follows: "Music can stimulate or soothe feelings, painting can make people pleasing to the eye, poetry can impress people, philosophy can make people gain wisdom, science and technology can improve material life, but mathematics can provide all of the above."
The difference between mathematical beauty and other beauty lies in the beauty contained in it. For example, everyone must feel this way. Most students are interested in the aesthetic feeling of sound and body, but not in mathematics. I think there are two main reasons for this: first, the beauty expressed by the beauty of sound and body is explicit, which makes it easier for students to feel, know and understand; Although some beauty in mathematics is manifested in the appearance of mathematical objects, such as exquisite graphics, beautiful formulas and ingenious solutions. Generally speaking, the beauty in mathematics is deeply hidden in its basic structure, and this inner rational beauty is often difficult for students to feel, know and understand, which is one of the main characteristics that distinguish mathematics from other disciplines. Second, for a long time, our mathematics textbooks have overemphasized the logical system and logical deduction, ignoring the role of mathematical aesthetics and mathematical intuition. In the long run, students equate mathematics with logic. Paying attention to the logic of mathematics and ignoring the beauty of mathematics itself will make you feel dull and uninterested in the process of learning.
Most mathematicians will get aesthetic pleasure from their work and general mathematics. They express this joy by describing that mathematics is beautiful. Sometimes, mathematicians will describe mathematics as an art form, or at least as a creative activity. Usually compared with music and poetry.
The beauty of mathematics also lies in its accurate expression of life and perfect deduction of logic. It can be said that it is this precision that has made a good life in modern society. Bertrand Russell used the following words to describe his feelings about the beauty of mathematics:
Correctly looking at mathematics, it not only has the truth, but also has the supreme beauty-a kind of cold and simple beauty, just like sculpture, which has no attraction to our weaker nature, has no gorgeous decoration of painting or music, but is extremely pure and can achieve strict perfection that only the greatest art can express. The true spirit of happiness, excitement, transcending human feelings, is the touchstone of the highest Excellence, which can certainly be found in mathematics and poetry. Mathematical research, mysticism and logic, and other papers, ch. 4. London: Longmans, Green, 19 18. )
Mathematics, if viewed correctly, has ... supreme beauty-just like the beauty of sculpture, it is a kind of cold and serious beauty. This beauty is not the weak aspect that attracts our nature. This beauty has no ornate decoration of painting or music. It can be pure to sublime, and it can reach a perfect state that only the greatest art can strictly express. A real spiritual pleasure, a spiritual excitement, a sense of superiority-these are the standards of perfection and beauty, which can be obtained in poetry and mathematics. Study mathematics, mysticism and logic, and other papers, induction. 4. London: Romantic Bookstore, green, 19 18. )
Erd?s· Parr described his unspeakable views on mathematics. He said, "Why are numbers beautiful? It's like asking why Beethoven's Ninth Symphony is so beautiful. If you don't know why, no one can tell you why. I know these numbers are beautiful. If they are not beautiful, nothing in the world is beautiful. " The best thing about it is that it makes your mind agile invisibly. When considering things, it is not so extreme and single.
As a citizen, not knowing is another matter, at least it should not be denied, especially for students.
Let's look at the following formula first:
1 x 8 + 1= 9
12 x 8 + 2= 98
123 x 8 + 3= 987
1234 x 8 + 4= 9876
12345 x 8 + 5= 98765
123456 x 8 + 6= 987654
1234567 x 8 + 7= 9876543
12345678 x 8 + 8= 98765432
123456789 x 8+9 = 98765432 1
1 x 9 + 2= 1 1
12 x 9+3 = 1 1 1
123 x 9+4 = 1 1 1 1
1234 x 9+5 = 1 1 1 1 1 1
12345 x 9+6 = 1 1 1 1 1 1 1
123456 x 9+7 = 1 1 1 1 1 1 1 1
1234567 x 9+8 = 1 1 1 1 1 1 1 1 1 1
12345678 x 9+9 = 1 1 1 1 1 1 1 1 1 1 1
123456789 x 9+ 10 = 1 1 1 1 1 1 1 1 1 1 1 1
9 x 9 + 7= 88
98 x 9 + 6= 888
987 x 9 + 5= 8888
9876 x 9 + 4= 88888
98765 x 9 + 3= 888888
987654 x 9 + 2= 888888
9876543 x 9 + 1= 8888888
98765432 x 9 + 0= 88888888
1 x 1= 1
1 1 x 1 1 = 12 1
1 1 1 x 1 1 1 = 1232 1
1 1 1 1 x 1 1 1 1 = 123432 1
1 1 1 1 1 x 1 1 1 1 1 = 12345432 1
1 1 1 1 1 x 1 1 1 1 1 1 1 = 12345654332 1
1 1 1 1 1 1 x 1 1 1 1 1 1 1 1 = 1234567 1
1 1 1 1 1 1 1 x 1 1 1 1 1 1 1 1 1 1 1 1 = 123456538
1 1 1 1 1 1 1 1x 1 1 1 1 1 1 1 1 1 1 1 165438
3 x 4= 12
33 x 34= 1 122
333 x 334 = 1 1 1222
3333 x 3334 = 1 1 1 12222
33333 x 33334 = 1 1 1 1 1 122222
333333 x 333334 = 1 1 1 1 1 1 122222
142857 x 1 = 142857
142857x 2=2857 14
142857x 3=42857 1
142857x 4=57 1428
142857x 5=7 14285
142857x 6=857 142
142857x 7=999999
1 1x 10 1 = 1 1 1 1
12x 10 1 = 12 12
13x 10 1 = 13 13
14x 10 1 = 14 14
15x 10 1 = 15 15
16x 10 1 = 16 16
17x 10 1 = 17 17
18x 10 1 = 18 18
19x 10 1 = 19 19
20x 10 1=2020