Pythagoras was an ancient Greek mathematician and philosopher. Born into an aristocratic family on the island of Samos in the Aegean Sea (present-day a small island in eastern Greece), Pythagoras was intelligent and studious from an early age, studying geometry, natural sciences and philosophy under a famous teacher.
The religious sect founded by Pythagoras mystified numbers, and the Pythagorean school believed that numbers were the origin of all things, and that numbers were the principles that played a decisive role behind all things, just like the mysterious fate that played a decisive role behind tragic heroes, and even God could not escape fate.
Numbers were given a specific meaning in the eyes of Pythagoras:
1, symbolizes reason, is the mother of all things, because it is the most basic number;
2, symbolize the opinion, because it is wobbly and has two hearts;
4 and 9, symbolize justice, because they are squares of the first even number 2 and the first odd number 3, square and square drops;
5, symbolize marriage, because it is the sum of the first even number and the first odd number;
8, symbolize love and friendship, because the octave scale is the most harmonious;
10, the symbol of perfection, is the sum of 1, 2, 3, 4, and is considered to have the highest order of the number.
Interestingly, Lao Tzu (chapter 42) has clouds: "The Tao gives birth to one, life to two, two to three, and three to all things." The Tao is considered in Taoist philosophy to be the fundamental observance of the formation of all things in heaven and earth, which I think is similar to Heraclitus's "Logos". The Tao, which preceded heaven and earth, is the origin and return of all things. Lao Tzu once used the metaphor of "Mother of Heaven and Earth" to "Tao", that is, the starting point of all things under the heavens, so "The Tao gives birth to one, one life to two, two to three, three to all things..."
"Tao Sheng Yi" is the transformation of abstract rules into concrete things. The "one" here, as I understand it, refers to the initial state of a concrete thing. So Lao Tzu's "one" is different from Pythagorean's "1." Corresponding to Pythagoras's "1" (the mother of all things) is Lao Tzu's "Tao" (Mother of Heaven and Earth). By the way, I checked the birthdays of these two people and wanted to confirm the priority, but I did not succeed. The two philosophers coexisted on both the eastern and western ends of the earth at about the same time, with Pythagoras (c. 580–c. C. 500 BC) slightly older than Lao tzu (571–471 BC).
The Pythagoreans attached great importance to the theoretical study of numbers, often depicting them as grains of sand or small stones on the beach, and studying natural numbers in the shapes they arranged.
Especially the study of form numbers, also known as mimetic numbers, stacked numbers, is a kind of figure-related numbers, the ancient Greek Pythagorean school in the study of number theory is very concerned about the relationship between form and number, shape number is a concept of the combination of number and form, they use dots to arrange triangles, squares, pentagons. What are the characteristics of these numbers?
How to understand the square number, we use a diagonal line (the red line in the figure) to divide the square number into two parts, so it is not easy to see that a square number is obtained by adding two consecutive triangle numbers: the upper left is the third (n-1) triangle number, the lower right is the nth triangle number, and the result is the nth square number.
After re-dividing the number of forms into two parts, each part is reduced to a simpler number of shapes, which is a very good method. The Pythagoreans knew how to divide square numbers into two consecutive triangle numbers to add up, but they did not move forward.
Later Nicomachius (who lived around 100 AD) had his greatest achievement was to study arithmetic and algebra independently, without relying on geometry. His book Introduction to Arithmetic is an arithmetic book that is completely divorced from the geometry of the teachings. The importance is similar to the importance of Euclid's Original for geometry. After developing the Research of the Beech School, he should have similarly divided the pentagon numbers into straight lines similar to the above, and as a result, the law appeared, as shown in the following figure:
You can see that no, the left side of the image above is still the number of the (n-1) triangle, and the right side is exactly the number of the nth square. Well, we've just put aside for the general formula for unsolved pentagon numbers, and now we can get it by the above division:
Put the graph of the above pentagon number here, you can use the above formula to verify that the first 4 pentagon numbers that have been drawn are 1, 5, 12, 22 are all correct. The 5th pentagon number not drawn in the figure and the subsequent pentagon number can be easily derived from the above formula.
Let's look at the hexagon number again:
The general formula for hexagonal numbers can be obtained by adding the number of triangles (n-1) to the number of nth pentagon:
By swiping a triangle number from the k-square number and continuing to do so, you can divide (k-2) triangle numbers, all of which are (n-1) triangle numbers, which can be represented by n(n-1)/2. Of course, a row of n pips is added at the end (the lowest row of pips in the graph). Thus, the formula for calculating the number of k-shaped numbers is:
It can be summarized as follows:
This method of representing numbers as graphs can provide a clever solution to some computational problems.
We are all familiar with the story of the great mathematician Gauss who asked for all numbers from 1 to 100 when he was a child. Gauss got the answer by summing in groups: he split all the numbers from 1 to 100 into two rows, with 1 to 50 in the first row in order from smallest to largest, and 51 to 100 in the following row from largest to smaller, as shown in the following figure.
Later, Gauss became the greatest mathematician of the 19th century, also because he was good at summarizing laws and finding better solutions to problems.
In fact, the problem of Gaussian summation can also be understood with the help of form numbers! We pay attention to the number of triangles, the nth triangle number is the sum of the sum from 1 to n, then to find the sum of 1 to n, we only need to count the number of points in the nth triangle number.
What happens when you place two triangle numbers side by side (one of them inverted)?
Obviously, this n row (n+1) of the two triangle numbers is a lattice of n(n+1) in total, so the nth triangle number has n(n+1)/2 points, so the sum formula for the first n numbers is:
Once you find a formula, other formulas usually appear in front of your eyes automatically. For example, if you multiply both sides of the above formula by 2 at the same time, you get the sum formula for the first n even numbers:
So, you can also think about it, what is the summation formula for the first n odd numbers?
It must be easy to think: with the summation formula of the first n numbers, and knowing the summation formula of the first n even numbers, then isn't the formula of the first n odd numbers the difference between the two formulas above? Good idea! Note, however, that the sum of the first 2n numbers minus the sum of the first n even numbers should be used here, and the sum of the first n odd numbers should be obtained, i.e., the sum of the first n odd numbers, i.e.:
Of course, this is a great way to make the most of what you already know! However, using the thinking of "shape numbers", there is a more ingenious way to understand the sum of all odd terms.
How many dots are in this square lattice?
As shown in the figure, this square lattice contains a total of 5 × 5 = 25 points. We can also count the number of points in this dot matrix in another way. Starting from the first point in the upper left corner, it is surrounded by 3, 5, 7, 9 points in turn, namely:
If the side length of this square is n, we can divide it into n - areas of size 1, 3, 5, 7, 9,..., ( 2n-1 ) respectively. Thus, we come up with the summation formula for the first n odd numbers:
Some people vividly summarize the summation formula of odd terms as: "The world is unparalleled, and the number of terms is squared!" "Did you remember it all at once?"
Knowing these, try a small bull knife:
As shown in the figure: Each square lattice is divided into two triangular lattices by a straight line, according to the information provided in the figure, the law in the fifth square lattice represented by an equation is _______
Analysis: Each quadrilateral lattice is divided into two adjacent triangle lattices, so 2² = 1 + 3, 3² = 3 + 6, 4² = 6 + 10, the fifth quadrilateral number 5² can be written as the sum of the fourth triangle number 10 and the fifth triangle number 15, that is, 5² = 10 + 15.
Pythagoras also gave interesting properties to the number of forms, such as: the sum of the numbers of two adjacent triangles is a square number, i.e. a square number
Pythagorean scholars even extended this idea of combining numbers and shapes to three-dimensional space, thus constructing three-dimensional numbers. For example, the first four triangular pyramids are numbered
"Mathematics is the gymnastics of the mind", and thinking is an intangible thing at the level of consciousness, invisible and untouchable. The abstraction, rigor and wide range of applications of mathematics determine that the mathematics classroom is a "dream factory" for thinking training.