"Real System" 02-Real number Construction

People are skilled in working and living, and as long as they are calculated according to the arithmetic, they do not have to doubt the result is correct. In the face of those seemingly natural algorithms, the average person will not think more, and see no tricks. Even when it comes to specious notions, most people choose to turn a blind eye, as they do not seem to affect the final outcome. However, as the language of nature, mathematicians are not willing to just use it as a general tool of dialogue, but rather to communicate with the world through it and transform it into a weapon to explore the world.

It turns out that in-depth thinking about simple problems, sometimes subverting people's traditional understandings, many of the major discoveries in the history of mathematics come from "basic problems", and new theories always make one step closer to the truth. We do not rush to see the truth, but from the beginning of the story, the pursuit of the pace of ancestors, one day, you will be out of your own pace. In the "set theory" we have already felt the axiomatic definition of the natural number, this one we will continue to make a number of structures, I believe that we are mentally prepared and full of curiosity.

1. Natural numbers

Defining natural numbers with collections although perfect, but not in line with intuition, the average person is not easy to think. The earlier definition in history comes from the axiom system of Peano (Peano), which we have mentioned in passing, and now take a look at its specific content. The axiom of the skin is that the natural number set \ (\bbb n\) satisfies the following axioms:

　　(I)\ (1\in \bbb{n}\). \ (1\) is the natural number, but \ (1\) does not define.

　　(II)\ (a\in \bbb{n}\rightarrow a^+\in \bbb{n}\). Any natural number has a " successor ", which is also a natural number.

　　(III)\ (A^+=b^+\rightarrow a=b\). The subsequent equal two natural numbers are also equal.

　　(IV)\ (\forall A (a^+\ne 1) \). \ (1\) is not the successor of any natural number.

　　(V)\ (M\subset \bbb{n}\wedge 1\in m\wedge (a\in m\rightarrow a^+\in M) \rightarrow m=\bbb{n}\). inductive axiom .

The axiom of the skin is more intuitive and understandable, and there is no difficulty in understanding it. \ (1\) and "successor" do not define, the first two axioms determine the natural number of the chain structure, the latter two to avoid the generation of the ring, the final summary of the axiom ruled out the superfluous elements, it is the basis of mathematical induction method . Many of the concepts here are very similar to those defined in set theory, and they can also be used to derive the nature of natural numbers, and to strictly define addition, multiplication, size, and the proving of various arithmetic laws.

It is necessary for you to try these proofs and definitions yourself, because through research and exploration you will marvel at the many qualities that can be introduced from these simple axioms, and many seemingly obvious conclusions are not easy to prove. By practicing, you are more able to feel the charm of abstraction and research, to feel the nature of the evidence, and to recognize the need for a strict definition of numbers. Here are some simple questions for the reader to practice:

• Any natural number does not take its own as a successor;
• Any natural number other than \ (1\) is the successor of others;
• Define addition, multiplication, proof of exchange law, binding law, distribution rate, elimination law;
• Defines the size, proves its transitivity, the ambiguity, the Operation Monotonic;
• There is no other number between \ (n\) and its successor;
• Minimum number principle .

The intentional reader may notice that in set theory we are starting with \ (0\) as the natural number, and here is the beginning of \ (1\). In fact, purely from the structure of natural numbers, the starting point is not important, the difference occurs mainly in the addition and multiplication theorem. \ (0\) complicates the definition of operations, and here we choose to defer the introduction so that we can focus on the principles and methods of construction.

2. Rational number

The natural numbers have addition and multiplication, but their inverse are not always meaningful, and the natural numbers need to be extended according to this requirement. Both the difference and the quotient can be represented by a natural number pair, so that both negative and fractional numbers can be defined. To reduce the impact of \ (0\), you can define a positive fraction first. According to the nature of division, the "equality" of natural number pairs can be defined, and the equivalent natural number pairs are defined as positive fractions, and the natural numbers are embedded in them. Then the size, arithmetic, arithmetic, and reciprocal are defined in turn, and the work is not difficult and can be used as an exercise.

Positive fractions are a lot more than natural numbers, and some new properties need to be clearly identified. One is the apparent density , which means that there is another positive fraction between any two positive fractions, which is relatively easy to construct. The other is Archimedes : for any positive fraction \ (x,y\), there are natural numbers \ (n\), so \ (x<ny\). You may think Archimedes is also very obvious, but consider the "set theory" in the overrun number, where it is not tenable. Archimedes is frequently used, you cannot even perceive it as a property, but it is not so obvious from the axiom that you can try to prove it. The density and Archimedes are also established in rational and real numbers, where they are not repeated.

After the natural number expands on the division, it needs to be extended on the subtraction. Similar to the definition of positive fractions, the "equality" of positive fraction pairs is defined in the sense of subtraction, then the equivalent positive fraction pairs are defined as rational numbers and the positive fractions are embedded in them. It then similarly defines size, arithmetic, arithmetic, inverse number, absolute value. The set of rational numbers is recorded as \ (\bbb{n}\), and the extension of the natural number under the subtraction operation is called integer \ (\bbb{n}\).

The rational number is completely closed on addition and subtraction, and it seems to have been filled with a number of axes, and we don't seem to need any more. Pythagoras was also so confident, but in the classic \ (\sqrt{2}\) is the proof of irrational numbers, and ultimately helpless. There is still an unknown number between a dense set of points, and people are suddenly wary of seemingly simple axis numbers, and how are these ghosts defined?

Now is the intermission, think about how to prove that \ (\sqrt{d}\) (\ (d\) is a non-square natural number) is irrational number? \ (\sqrt{2}\) is the proof of irrational numbers, although classic, but not universal, if confined to the elementary method, we need a way. Here the proof thought is called the Infinite Degradation method , it invented in the ancient Greece period, now still has the widespread application in the number theory. If \ (\sqrt{d}\) can be expressed as both about the score \ (\dfrac{p}{q}\), set \ (0<p-mq<q\), use \ (p^2=dq^2\) can construct the \ (\dfrac{nq-mp}{p-mq}=\dfrac{p}{ q}\), its denominator is smaller than \ (q\), which is not possible.

3. Real number 3.1 de gold Division

The most successful real definition in history comes from Dedekind, which divides the rational number set \ (\bbb{n}\) into two non-empty sets \ (\xi,\bar{\xi}\), where the elements in the right set \ (\bar{\xi}\) are greater than the elements in the left set \ (\xi\), and the left set has no maximum value. Dedekind \ (\xi\) is defined as a real number, and the set of real numbers is written as \ (\bbb{n}\). When the right concentration has the minimum value \ (r\), the knife is tangent to the rational number \ (r\), the division is the corresponding real of \ (r\). When there is no minimum value in the right concentration, it is of course the irrational number we need. Before moving on, review the concept with a few questions and warm up.

• The "right-set element is greater than the left-set element" is equivalent to: \ (R\in\xi\wedge R ' <r\rightarrow R ' \in\xi\);
• \ (\sqrt{2}\in\bbb r\);
• For any \ (\varepsilon>0\), there is \ (\exists r\in\xi,\bar r\in\bar\xi (\bar{r}-r<\varepsilon) \).

The following first defines the size of the real number: when \ (\xi\subset\eta\), define \ (\xi<\eta\), it is easy to prove the ambiguity and transitivity of the definition. The next step is to define addition and multiplication, as well as their operational laws. By personally defining and proving that these concepts will enhance your sense of well-defined identity, here is the definition of addition, and hopefully you will begin to explore the legitimacy that proves it.

\[\xi+\eta=\{a+b|a\in\xi,b\in\eta\}\]

Our main question is: Can this definition of real numbers be filled with a number of axes? In other words, on the axis of a knife cut down, it must be cut to the real number? Similar to the Dedekind partition, the real set \ (\bbb{n}\) is divided into left and right sets \ (x,\bar{x}\), we have to prove that: \ (\bar{x}\) must have the minimum value, because it is a split point. Examine all left-set \xi=\cup (x\), prove that \ (\xi\) is a real number, and then prove that it is not less than any of the numbers in \ (x\) and not greater than any number in \ (\bar{x}\), so it must be the minimum value of \ (\bar{x}\). This proves our conclusion that the real number has no space on the axis, and this property is called completeness or continuity of the real number. The completeness of the real number is the biggest characteristic which distinguishes it from other numbers, it is the foundation of the analytic Science, we will discuss it in depth in the next article, and give some basic theorems of real numbers which are equivalent to completeness.

3.2 Cantor Definition

As early as the first half of the 19th century, Cauchy, Weierstrass and others began to work on the rigorous analysis, including the strict definition of real numbers. Cauchy defined the real number as the limit of the sequence of rational numbers, but he did not realize that his definition had previously acknowledged the existence of the "number" of the limit, so it was a cyclic definition. Before we move on, we need a few concepts. The sequence \ (a_0,a_1,a_2,\cdots\) is called a sequence number (sequence), which is recorded as \ (\{a_n\}\), \ (a_n\) is called a sequence of entries. A sequence can be seen as a mapping of a set of numbers to a set of natural numbers, which, if the sequence satisfies the following conditions, is called a basic sequence or Cauchy series . Cauchy is the definition of the real number as the limit of the basic sequence, this article does not intend to introduce the limit, and all replace it as the basic sequence, in the next article we will see that they are actually equivalent.

\[\forall\varepsilon>0\exists N (m,n>n\rightarrow\left|{ A_m-a_n}\right|<\varepsilon) \]

If the two basic sequences \ (\{a_n\},\{a ' _n\}\) meet the following conditions, they are called equivalent and are recorded as \ (\{a_n\}\sim\{a ' _n\}\). It is easy to prove that \ (\sim\) is an equivalence relation, and Cantor defines the equivalence class \ ([\{a_n\}]\) of the basic sequence of rational numbers as a real number. A series of only rational numbers \ (r\) is obviously the basic sequence in which the equivalence class is defined as the rational number \ (r\), and the different rational numbers are not equivalent.

\[\forall \varepsilon>0\exists N (n>n\rightarrow\left|{ A_n-b_n}\right|<\varepsilon) \]

For two real numbers \ (\alpha,\beta\), take its representative sequence \ (\{a_n\},\{b_n\}\), it is easy to prove that \ (n\) is large enough to have \ (a_n>b_n\) or \ (a_n<b_n\) established, which can be defined as the real size. The following is the definition of real addition, where multiplication and arithmetic are left to the reader.

\[[\{a_n\}]+[\{b_n\}]=[\{a_n+b_n\}]\]

As mentioned above, the basic sequence that is not equivalent to a certain rational number should be an irrational number, can they fill the space of the axis? If any real number can be approximated, the language in this section is whether the real number basic sequence is necessarily equivalent to a real number? The concept of a real distance, which can be defined as the distance of the sequence, will appear here. For the real number basic sequence \ (\alpha_0,\alpha_1,\alpha_2,\cdots\), take it to represent the rational sequence \ (\{a_{0n}\},\{a_{1n}\},\{a_{2n}\},\cdots\). To examine the Rational series (a_{00},a_{11},a_{22},\cdots\), it is proved that it is the basic series, and then proves that the real number represented by it is to be searched, which proves the completeness of the real numbers.

3.3 Outlook

The one by one mappings between the Dedekind partition and the real number defined by Cantor can be established, they are isomorphic , and historically have a high status. However, due to the complexity of the expression and the definition of differences, not easy to use directly in the argument, the next chapter will introduce the real number of the basic theorem and they are equivalent, and easier to understand and use, they will be used in the future as the equivalent of the completeness of the real.

The real number represents a point on a straight line, so can each point on the plane be represented in numbers? Gauss answers this question by matching the points of the complex and two-dimensional planes. You may be optimistic that every point in space can be expressed as a number, which is not a mistake, because we don't even have the definition of a number! With the maturity of abstract algebra, people think of numbers as an algebraic system satisfying certain algorithms. Unfortunately, the plural is already the largest system to satisfy the existing algorithms, and if you want to be big, you have to sacrifice some algorithms, such as Hamilton's four-dollar number , which does not satisfy the Exchange law.

When it comes to algebraic systems, rational numbers can be considered as a subsystem of real numbers, and subsystems are completely closed to arithmetic. There are other subsystems in the real number, the more important one is the study of the polynomial. The rational number can be regarded as the real number satisfying the whole coefficient equation \ (a_0x+a_1=0 (a_0\ne 0) \), extending the concept, which is generally called satisfying equation \ (\sum\limits_{i=0}^n{a_ix^i}= 0 (a_n\ne 0) \) The number of algebraic numbers , the minimum satisfying \ (n\) order equation, is called \ (n\) Order algebra number. It is easy to prove that the algebraic number is a number, the natural real algebraic number is also a number, so the real number is not real algebraic number, it is called the transcendental number , the typical representative is \ (\pi\) and \ (e\). You might think that algebraic numbers are numbers with various root-types, and conversely, the opposite is true, but most algebraic numbers cannot be expressed in finite expressions! This wonderful proposition is put in the abstract algebra.

"Real System" 02-Real number Construction