Reading notes: "From one to Infinity"

Reading notes: "From one to Infinity"

This book was found in the beauty of mathematics, the original is an old book in the 80 's, but the scope of the book is quite wide, from digital to infinity, then to four-dimensional space, and then to the theory of relativity, then to the microcosm, and then to the macro-world, some of the content with some simple way to let people understand, with high school knowledge of people can understand, In terms such as complex functions or analysis of the function, the majority of people are frightened away.

Read the comments of the watercress, the original book is not a gamma-a person wrote, inside also used a lot of other people's results, also do not go to textual research.

http://book.douban.com/people/57526326/annotation/1102715/

Chapter One large number

In ancient times, it was impossible to represent a large number, so the scientific notation was a great invention.

The story of the wheat on the chess plate has been heard many times, the total grain is: 2^64–1 = 18,446,744,073,709,551,615.

The number of Hanoi of 64 slices is also this 18,446,744,073,709,551,615 times.

A never-ending automatic printing machine The probability of writing a 65-character Shakespeare poem is 1/(50 ^ 65), now there is a computer is good, calculate 50^65=2.7e+110, every atom in the world is a printing machine (10^74), From the time of the birth of the Earth began printing (to now work for 3 billion years), or the atomic vibration frequency (1 seconds Printing 10^15 line) to work, in order to print 3.0E+106 lines.

Comparing the size of two infinity, the original mathematician Cantor (Georg Cantor) had already thought about the problem.

The comparison of the two infinite numbers is explained by the one by one corresponding method, which is easy to understand. All the integers and all the scores turned out to be just as much.

2--1/1

3--1/2 2/1

4--1/3 2/2 3/1

......

In the infinite World, part may be equal to all.

It is ingenious to prove that the number of points on the line is as many as the points on the plane.

Three-level Infinity number: N0 all the integers, N1 all the geometric points, N2 all the curves.

Chapter II Natural numbers and artificial numbers

To now feel that number theory is still a place of application, such as in large numbers of mass factor decomposition successfully applied in cryptography.

The method of proving that there is no maximum prime number is quite ingenious, and junior high school students can understand it. 1*2*3*5*7*11*13*...*n+1, to disprove the law.

Fermat number, or called Fermat Prime, Fermat prime, such as this form, but only found the first 5 (3, 5, 17, 257, 65537) is prime, the back is composite, see Baidu Encyclopedia http://baike.baidu.com/view/443594.htm

Goldbach conjecture, remember in the university heard a Pan disciple held lectures, understand what is called Chen Jingrun proof of "1+2", originally from "" "only one step away from the conjecture so far can not solve.

The distribution theorem of prime numbers: the percentage of all prime numbers from 1 to any natural number n, approximately represented by the reciprocal of the natural logarithm of N.

X^n + Y^n = = Z^n There is no integer solution when n>2, and the famous Fermat equation has not been proved yet.

-1 square root, the introduction of imaginary I, with geometric rotation to understand the complex plane!

Chapter III The unusual nature of space

An important theorem in topology (Euler's theorem): v + F = E + 2, where V is the number of vertices, E is the number of edges, and F is the number of polygons, where the polyhedron is empty.

The proof of this theorem is also very interesting, the first step of thinking is quite worthy of reference, cut a face, change into a plane problem. To prove the network v-e+f=1 on the plane.

The well-known four-color theorem, which was previously heard to prove the theorem with a computer, seems to have something to do with the Euler theorem.

Turn the space over! Imagine a space for an apple inside a black worm and a white worm tunnel.

about how a worm-eaten Apple transforms into a topological transformation of a doughnut, and after some removal and bonding, it really needs some space imagination.

Fourth four-dimensional world

We understand the four-dimensional space in three-dimensional space, you can try to view the three-dimensional world from the perspective of two-dimension flat people.

This chapter is a good understanding of the tired ah.

The fifth chapter the relativity of Time and space

Speaking of Einstein's theory of relativity, the object of motion is actually shortened in length, and the inner angle of the triangle is not necessarily 180 degrees.

This chapter is more difficult to understand. Einstein was indeed a God, not to be in the four-dimensional space to expand the imagination.

Sixth chapter

This chapter comes to the microscopic world of chemistry and tells a simple test that can measure the size of the oil molecule.

After a few chapters from the micro-world to the macro-world, need to have time to read slowly, although as far as possible to write in a more easy-to-understand way, but the scope of coverage is too wide, including physical, chemical, biological content, needs to be based on personal interest and slowly pondering.

It seems that this book and "Out of control" are the digest books that need to be read in a section.

Reading notes: "From one to Infinity"