Astrology is always thought-provoking and fascinating.
Since ancient times, understanding astronomical phenomena has always been the driving force and lofty goal of geometry, arithmetic, philosophy and religion research.
Geometry and algebra are rational knowledge of the space where the universe is located, and they are the basis and forerunner of science.
According to Zhou Li, as early as the Spring and Autumn Period and the Warring States Period, many branches of geometric calculation such as "square field (land survey)", "small area (long area)" and "commercial work (earthwork)" were formed in China. The ancients in China mastered the area formulas of rectangles and triangles and the volume formulas of cubes and cones long ago. During the Three Kingdoms period, Zhao Shuang made a pythagorean diagram and proved the pythagorean theorem.
If China's ancient mathematical achievements are the summary of production practice, then in ancient Greece thousands of miles away at the same time, its mathematical system can be said to be a rational deduction based on the axiomatic system.
The Elements of Geometry written in 300 BC established a complete and self-consistent plane geometry system with five postulates.
These five hypotheses are:
Starting from these five concise and highly generalized axioms, Euclid established a huge and complete mathematical system, including similar shapes, circles, polygons, drawings, solid geometry and even a large number theory and algebraic theorems. In the next few thousand years, The Elements of Geometry became a masterpiece after the Bible. Although this is not his own credit, Euclid can still be described as a master of geometric algebra in ancient Greece. 1
The tenth volume of this book inherits and develops the predecessors' ideas about incommensurable ratio (irrational number) and limit, which is of milestone significance.
The story begins here:
When studying the basic theory of plane geometry quantitatively, the ancient Greek mathematicians tried their best to be meticulous in the face of the basic concept of "length". "Length" is a relative concept, and there is no absolute definition of "length" in the era of lack of understanding of cosmological constants. Mathematicians put forward the pure theory of "commensurability":
The principle of "commensurability" relatively defines the "length" of natural quantity and expands the number system from integer to rational number. On this basis, mathematicians strictly demonstrated other important formulas of geometry, such as area formula, similar triangles formula, Pythagorean theorem and so on. The Greeks "firmly believe" that all numbers in the world can be expressed by integers and commensurability ratios, and regard commensurability principle as the "first axiom" of geometry.
As we all know, Pythagoras (Greek: π υ θ α γ? ρα? 570 ~ 495 BC) also discovered and proved Pythagorean theorem (Pythagorean theorem). The Greeks "firmly believe" that all numbers in the world can be expressed by integers and commensurability ratios. However, it is this basic and beautiful theorem that has triggered an unprecedented crisis.
Pythagoras student hippasus (Greek:? ππασο? , about the 5th century BC) is the direct initiator of this crisis.
"1" is a simple and wonderful number. If the lengths of two right angles of a right triangle are "1", what is the chord length?
3/2 ? No, it's too big;
7/5? No, it's too small, but it's closer;
With the deepening of thinking, an idea made him shudder. It seems that no commensurability ratio can accurately express this chord length, but this chord length does exist!
Another fact is:
The ratio of diagonal to side length of a regular Pentagon cannot be expressed by rational numbers.
As shown in the figure, it can be seen from the properties of regular pentagons that triangles and triangles are two congruent isosceles triangles.
Remember that the diagonal length is b, the side length is a, and the ratio of diagonal to side length is k, then b = a+r.
On a whim, hippasus extended each part by an R, and he saw a new regular pentagon at a glance.
So we get an equation.
Hippasus proved by geometric method that this equation has no rational number solution.
These two seemingly ordinary but groundbreaking discoveries shocked the ancient Greek academic circles. The problem of incommensurability, which urgently needs to be explained, was a severe and urgent challenge faced by Greek academic circles at that time.
In fact, mathematicians have found that the process of finding the solution of the problem through trial and error is the process of constantly approaching the final answer, and it is an effective method to constantly approach this unknown with rational numbers.
This number to be defined (called P for the time being here) cannot be expressed by a ratio, but its relationship with any rational number is clear, which is the so-called "comparison principle".
Eudoxus (Greek: ε? δοξο? , 408 BC to 355 BC) put forward:
For any positive integer n, there is a positive integer m, so
In this way, the incommensurability ratio p is limited between two rational numbers with a difference of1/n. With the increase of denominator n, the difference between the upper and lower bounds of inequality becomes smaller and smaller, and eventually tends to zero, and the left and right sides of inequality will be unified into one equation. This is the idea of "limit".
Moreover, if there are two unequal incommensurable ratios p and q, there must be a fixed value between them that is not zero, which is denoted as d, and 1/N is a quantity that can be arbitrarily small. As long as n is large enough and 1/N can be less than |d|, then p and q cannot be limited between m/N and (m+ 1)/N at the same time. This is called uniqueness. Therefore, the incommensurability ratio can be defined.
For any incommensurable ratio that can be expressed by an equation, we can always get accurate results by calculating and comparing it many times with the ideas and methods put forward by Odoksos.
Suppose there is an infinite straight line in the space, let a point on the straight line be the reference point O, and then set a point E which is not coincident with the point O, we can remember that the length of the line segment OE is 1. The distance from any point P to point O on a straight line can be expressed by commensurability ratio or incommensurability ratio. Conversely, any commensurability ratio or incommensurability ratio can be intuitively expressed as the relative length on a straight line. Commensurability ratio (rational number) and incommensurability ratio (irrational number) are collectively called real numbers. Then it can be said that there is a one-to-one correspondence between real numbers and points on a straight line. If the real number is complete, the straight line is continuous and the space is continuous. 2
With such a simple and precise idea, eudoxus added irrational numbers to the number system and reconstructed the foundation of geometry and algebra. He himself may not have imagined what a great scientific revolution will be born after 1000 years.
Two thousand years ago, the ancient Greeks had a profound understanding of the nature of space, which was amazing.
In ancient China in the East, the achievements made by mathematicians were equally amazing.
During the Three Kingdoms period (the third century A.D.), Liu Hui put forward secant technique, which established a strict theoretical basis and perfect algorithm for deducing the formula of circular area and calculating pi, and also contained profound "limit" thought.
On the basis of a circle inscribed with a regular n polygon, a regular 2N polygon can be inscribed, and the area of the latter must be larger than the former. Remember that the side length of a polygon inscribed with positive N is L(N), the perimeter is C(N) and the area is S(N).
According to the principle of "complementary in and out" summarized by Liu Hui (that is, digging and patching method), the inscribed regular 2N polygons can be reassembled and spliced into a rectangle as shown in the figure, and the area is calculated as follows.
Similarly, for a circle circumscribed by a regular polygon, the area of a regular hexagon is smaller than that of a regular triangle, the area of a regular dodecagon is smaller than that of a regular hexagon, and so on.
The area T(N) of the circumscribed circle of a regular N polygon can also be obtained by digging and filling method. Remember that the side length is M(N) and the perimeter is D(N).
The inscribed area of a polygon is no larger than the circular area, and the circumscribed area of a polygon is no smaller than the circular area. Remember that the circumference of a circle is 0 and the area of a circle is 0, so we get the inequality.
On the other hand, with the increase of the number of sides, the outlines of circumscribed polygons and inscribed polygons are closer to the circumference, that is, the circumference of polygons is closer to the circumference of circle 3. Can be expressed in modern mathematical language as:
The ratio of the circumference to the diameter of a circle is defined as π, that is.
therefore
This is the well-known formula for calculating the area of a circle.
Liu Hui didn't stop there. Through secant technique, he challenged the calculation of pi.
For a circle with a radius of 1, the side length of the inscribed regular hexagon is also exactly 1, so we start here. Make an inscribed regular dodecagon and a regular quadrilateral. ...
It is not difficult to prove by Pythagorean theorem that there is a recursive relationship between the side lengths of inscribed regular polygons.
For square root operation, it is necessary to use approximation method (that is, similar to eudoxus' method of finding irrational numbers) to calculate the numerical solution that meets the requirements of operation accuracy. Substituting the obtained side length into the formula for calculating the area of inscribed regular polygon (1.2), the approximate value of pi can be obtained. The more iterations, the smaller the difference between a regular polygon and a circle, and the higher the precision of pi. As Liu Hui said, "If you cut it carefully, the loss will be small. Cut and cut, and if it can't be cut, then it will fit the circle and there is nothing missing. "
In ancient times, when the calculation tools were simple, Liu Hui calculated 3072 polygons only by calculation, and obtained a satisfactory pi of 3. 145438+06. Later, the mathematician Zu Chongzhi of the Northern and Southern Dynasties continued to divide it into 12288 polygons, and got the area of 24,576 polygons, and got the approximate value of 3. 14 15926, which will become the most accurate pi in the world in the next millennium.
The thought of "limit" is an important step for human beings to enter the field of rational science.
They, mathematicians and thinkers from ancient Greece and China and born in other countries, have contributed hard work, sweat and sacrifice to the development of human civilization. They are praiseworthy and memorable.
In the discussion of eudoxus's limit thought, the viewpoint that "space is continuous" was put forward.
First of all, it should be clear that the real material space and abstract theoretical space are not equivalent concepts, but have long been regarded as equivalent, because the early scientific theories-geometry and algebra, whether in ancient Greece, China or other civilizations, all originated from people's observation of time and space, that is, they were abstracted from phenomena.
People "sum up" the space they live in.
These properties can't be explained by more basic theories, and they are considered as self-evident true propositions. Calling them axioms actually precedes all proved hypotheses. Eudoxus's limit theory and Euclid's geometric algebra system are self-consistent and reasonable under their respective axioms, and they are effective models for describing the material world at the cognitive level at that time. Classical physics is also built on a specific soil, and once it is separated from this soil, it is worthless.
Continuity is theoretically an important premise of limit theory and even calculus and mathematical analysis. In the real material world, is space continuous, or is the description of the objective world based on the theoretical system of continuity correct?
In modern times, with the improvement of observation level, people's understanding of time and space is getting deeper and deeper, and various theories have different opinions on the nature of real space, so far there is no conclusion.
In this column, the theoretical system used by the author is still dominated by classical physics. Therefore, the premise of making classical physics reasonable is also the default premise of the theory and logic involved in this column.
note:
The fifth postulate of 1 accords with the objective fact of plane geometry, but its expression is not as concise as the other four postulates. Later mathematicians proposed to replace the original postulate with the negative proposition of the fifth postulate in their research, but they all developed non-Euclidean geometry system.
Strictly speaking, the completeness of real numbers was formally put forward as an analytical theorem in the19th century. But ancient Greek mathematicians had a deeper understanding of "density" and spatial continuity.
The discussion here is not rigorous, you can see the following example.
Similarly, polygons are close to circles. Why do regular polygons approximate to get the correct pi, but the sawtooth in the figure is wrong? The reason is that the core of the approximation method is to make the difference between the approximation result and the target an infinitesimal amount. And what is the definition of infinitesimal? What can be called infinitesimal? This problem also triggered the crisis after the establishment of calculus. Through the unremitting efforts of Cauchy, Dirichlet, Dai Dejin and Cantor, a mature and rigorous analytical science was established and the answer to this question was obtained.
Every vertex of the regular polygon used by Liu Hui is on the circumference. This idea of replacing bending with straightness is reasonable and effective, so the correct result is obtained.
Part of the material in this paper comes from Eternal Mystery and Geometry, Astronomy and Physics in Two Thousand Years/Xiang Wuyi, Yao Heng, 20 10.2. Some pictures come from the internet.
Wechat official account original