Thoughts on the evolution from irrational numbers to super real numbers

Intuitively, we have a deep impression on the PI of the irrational number circle rate, as shown in;

In general, the appearance of the irrational number is like this. That is to say, the irrational number cannot be a finite decimal or a repeating
Decimal ). The logic is to drag a long tail and cannot see the head at a glance. We imagine placing all rational numbers and irrational numbers on a geometric straight line, so that they each have their own positions (coordinate points) and constitute the so-called "continuum, it is also called "real line" or a real number axis ). In 1870s, the founder of the set theory, Cantor, proved the fact that the number of irrational numbers is far greater than the number of rational numbers, and the dense and rational numbers are surrounded by irrational numbers. This is an intuitive picture of "continuity.

In 1948, the US edwinhewitt used the so-called "superapower" technology to construct a "continuous system" that is rich in infinitely small numbers, revealing the infinite number of non-repeating decimals (that is, irrational numbers) the long tail contains many "super real numbers" (hyperreals need to use our imagination ). These super real numbers are grouped around a real number. They are separated by an infinitely small number, that is, they are infinitely close to the real number. This scenario is very similar to the preceding scenario where irrational numbers are surrounded by rational numbers.

This kind of super real number "group structure" (also called "list" structure, the English name of a list is "monad "). It is easy to prove that there is only one traditional real number in each list, because the distance between two different real numbers cannot be infinitely small. The traditional real number in the same "super real number list" is the standard part of each super real number in the list.
Part), expressed:

ST (x) = R (x ε monad (r ))

Someone guessed that the number of super real numbers (also called set 3 "base", cardinality) must be much larger than the number of traditional real numbers. Otherwise, there are as many super real numbers as the "Number" of real numbers, that is, there are as many elements as in the real number continuous system. In fact, the carrying capacity of geometric straight lines is very strong. Embedding real or super real numbers doesn't matter for geometric straight lines. I don't know how mathematicians think in their heads. It's strange.

For mathematics, the expansion of the number series is a revolution. Physical measurements are sufficient. However, to build a suitable mathematical theory, we need to expand the number system. In particular, for calculus, we often need to deal with the "infinitely close" quantitative phenomenon, it is more convenient (suitable) to adopt a "continuous system" rich in infinitely small values ). In essence, both irrational numbers and super real numbers are constructed through an "infinite process" and are "ideal numbers". The truth is the same, it is the same level, and we do not need to "be considerate ". Our starting point is as follows: There are many Chinese people and a large number of students. It is of practical significance to try to reduce the difficulty of Calculus Teaching and reduce the teaching hours of calculus. To be honest, if you don't think about these children, you don't need to do so.