The yang fai triangle, also known as the jaya (about 1050) triangle, is generally known in the west as the pascal 1623-1662 triangle, and our learning is now most closely linked to the two-dimensional extension coefficient. In the yang fai triangle, the third number in line 3 corresponds to the square formula of two digits.
Yang fai triangle
The essence of the yang fai triangle is that its two slopes consist of one number, while the rest is equal to the sum of two numbers on its shoulder.
Yang fai, a famous mathematician of the song generation, has made an important contribution in summing up the popular multiplication algorithm, the “mocking arts”, the vertical map and mathematics education. He was the first mathematician in the world to draw a wealth of vertical and vertical maps and discuss its composition patterns. It has also been argued that the arc target formula is known as "female." qin kwok, li shengjie and ju shijie also called "song yuan mathematics four." the main works are five 21 volumes of mathematical works, i. E., 12 volumes of the elaboration of nine chartered algorithms (1261), 2 volumes of the daily algorithms (1262), 3 volumes of the multiplication of the modified books (1274), 2 volumes of the field acre-speculation-screen-screen-screen-screech (1275) and 2 volumes of the screen-screen-screen-screen-screen-screen-screen-screen-screen (1275). The latter three are collectively referred to as the yang fai algorithm. Translations have been published in countries such as the democratic people's republic of korea and japan and are being circulated worldwide。
The leibniz, 1646-1716, a german mathematician, philosopher, natural scientist, who is an analogy to the yang fai triangle (known by them as the pascal triangle, bijahn, yanghui hundreds of years late) discovered the unit fraction triangle in the above question, which is characterized by the fact that the unit score is 1 molecule and the denominator is the fraction of the whole.
Nature of the yang fai triangle:
Each number equals the sum of two digits above it。
2. The round and round symmetry of numbers in each row begins to grow gradually from 1。
3. The figures in line n include n。
4. The number of preceding rows is 2。
5. The number of m in line n can be shown as c (n-1, m-1), i. E. The number of combinations of the m-1 elements taken from the different elements of n-1。
6. Line n equals the number of m and n-m+1 and is one of the characteristics of a combination。
7. Each number is equal to the sum of the two figures in the preceding line. This can be used to write the whole yang fai triangle. That is, the number of i in line n+1 equals the sum of i-1 and i in line n, which is also one of the characteristics of the combination. Name is c(n+1, i)=c(n), i)+c(n,i-1)。
8. The coefficients in (a+b)n's roll-out correspond to each of the rows (n+1) of the yang fai triangle。
Line 2n + 1, number 1 and line 2n + 2, number 3 and line 2n + 3, number 5... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The sum of these numbers is the 4n-2 fabonacci number。
10. Multiply the number in line n by 10^ (m-1), of which m is the column where the number is, and add each to 11^ (n-1). 11^0=1,11^1=1x10^0+1x10^1=11,11^2=1x10^0^10^1+10^1+1x10^2=121,11^3=1x10^0^0+3×10^1^2+1x3^3=1331,11^4=1x10^0+4x10^1+6^2^2+4^10^3^3^1^1,11^5=1x10^1415=1x10^0+5^1^10^2+10^10^10^3^10^3×3×5=1610051。
11. Line n numbers and 2^ (n-1). 1 = 2^ (1-1), 1 + 1 = 2^ (2-1), 1 + 2 + 1 = 2^ (3-1), 1 + 3 + 3 + 1 = 2 ^ (4-1), 1 + 4 + 6 + 4 + 1 = 2 ^ (5-1), 1 + 5 + 10 + 10 + 5 + 1 = 2 ^ (6-1)。
12. The sum of the numbers on the slash equals the number on the left side (from the top left to the lower right) or on the right side (from the top right to the lower left) and on the corner. 1 + 1 = 2, 1 + 1 + 1 = 3, 1 + 1 + 1 = 4, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10, 1 + 3 = 4, 1 + 3 + 6 = 10, 1 + 4 = 5。
Aligns the row numbers to the left, the top right to the bottom left diagonal and the number equal to the fibonacci series. 1,1,1+1=2,2+1=3,1+3+1=5,3+4+1=8,1+6+5+1=13,4+10+6+1=21,1+10+15+7+1=34,5+20+21+8+1=55。
It is therefore very important in the high examination to consider the yang fai triangle, and it is one of the topics especially popular with the subject。
Interested students can do it, and they won't be able to exchange messages.
