I hope it helps you.

The so-called real number is a number that can always compare sizes with each other; Coupled with the description of an overbearing point, all the numbers that can be compared with each other are real numbers.

This statement is too rough and simple. If we use a more accurate modern term, the so-called real number, it is the only orderly and complete field.

As the name implies, an ordered complete field is "ordered"+"complete"+"field", which defines the clear appearance of real numbers from three aspects, while "unique" means that any two ordered complete fields are isomorphic. The meanings of these terms are explained in turn below.

tidy/neat

Being able to compare sizes, more generally speaking, to determine the single relationship between two objects is the first condition for us to agree on the required counting system.

Let's express this condition in our intention more accurately in two sentences:

This relationship can be determined between any two different numbers;

This relationship has linear geometric properties, that is, it is assumed that this relationship between A and B is marked as A.

Perfection means nothing is left out. Any object that can establish the above relationship between the two must be represented by the counting system we need.

For example, if we take any fraction, we can compare it with the diagonal length of a square, so this counting system must be able to represent the diagonal length.

So how do we accurately express the meaning of "integrity", especially operability?

A sentence can be expressed in many ways. Similarly, integrity can be described in many ways, and those different descriptions are and should be equivalent to each other.

The main points of this description are:

You must first confirm the existence of such a counting system, which contains all the comparable values without omission;

Then how can I express this fact clearly? A is such a number that we can always know who it is bigger than and who it is smaller than. & lt/c, then a.