Natural numbers 1, 2, 3 ... are the starting points of mathematics, and other numbers are derived from natural numbers. The physical prototype of natural numbers may be ten fingers, otherwise we won't use decimal.

Natural numbers are all positive numbers, and the introduction of negative numbers solves the difficulty that decimals cannot be reduced by large numbers, such as 1-2 =- 1. Negative numbers also have prototypes. Isn't debt a negative asset? Therefore, the formation of the concept of negative number is probably related to the early commercial lending activities of human beings.

Zero is a great invention in the history of mathematics and is of great significance. First of all, zero stands for "nothing". Without "nothing", how can there be "nothing"? Therefore, zero is the basis of all numbers. Secondly, without zero, there is no carry system. Without the carry system, it is difficult to represent large numbers and mathematics can't go far. The characteristic of zero is also manifested in its operational function. Any number adds or subtracts zero, and its value remains unchanged. Multiply any number by zero to get zero; Any non-zero number divided by zero is infinite; Divide zero by zero to get any number. What is the prototype of zero? Is it "nothing" or "nothing"? ?

Zero-sum natural numbers and natural numbers with negative signs are collectively called integers. Centering on zero, arrange all integers equidistantly from left to right, and then connect them with a horizontal straight line. This is the "number axis". Each integer corresponds to a point on the number axis, and these points are separated from each other by an equal distance. Look! Negative numbers and positive numbers are arranged around like the wings of wild geese, and zero is in the center, which is quite a king's atmosphere.

The introduction of fraction solves the difficulty of divisibility, such as 1 ÷ 3 = 1/3. Of course, there are also prototypes of fractions. For example, three people share a watermelon equally, and each person gets a third.

You can insert an infinite fraction between two adjacent integers on the number axis to fill the blank space on the number axis. Mathematicians once thought that the whole number axis was finally filled. In other words, all the figures have been found. In fact, it's not that some numbers can't be expressed by integers or fractions at all. The most famous is pi. Fractions can only represent approximate values but not exact values. After people convert fractions into decimals, they find that there are two situations: one is that the number of decimal places is limited. Just like1/2 = 0.5; The other is infinite cyclic decimals, such as1/3 = 0.33333 ... Although they look different, they all contain limited information, because the cyclic part only repeats the original and does not contain new information. Pi is fundamentally different, 3.14153846 ... It has neither cycle nor end, so it contains infinite information. Think about it! The vast collection of books in Beijing Library contains a lot of information, but it is still limited, and pi contains unlimited information. How can it not be amazing? Mathematicians regard seconds that cannot be expressed as integers or fractions like pi as "irrational numbers", and irrational numbers are irrational numbers! I don't know why pi has such a bad reputation. I once wrote a poem called Pi, and I felt very wronged for it. I might as well quote its last paragraph to the reader:

……?

Like a long poem that can't be read?

Neither circulating nor exhausted?

Endless, always new

Mathematicians call it an irrational number?

The poet praised it as a lover?

Is Tao unreasonable but affectionate?

Heaven and earth may not last forever.

This rate is endless?

(Originally published in Poetry Journal No.8 1997)

Ever since Zu Chongzhi calculated the value of pi between "approximate ratio" 22/7 and "density" 355/ 1 13, someone has been calculating a more accurate value of pi, which was recently calculated by a computer to more than two million decimal places! But compared with "this rate is endless", it is not even a drop in the ocean. Even with the fastest supercomputer, it will be endless until the end of time! In addition, some people use a computer to digitize the calculated pi into a binary sequence, and then statistically analyze it, and find that it has the greatest uncertainty like a random number. Pi is a completely certain pi, but the infinite series it produces has the greatest uncertainty. We can't help being surprised and shocked by the mystery of nature.

With the addition of scores and irrational numbers, the kingdom of mathematics has expanded, and the lineup of wild geese wings lined up on both sides of the zero king is more magnificent.

With irrational numbers, the original integers and fractions are collectively called rational numbers. Is this the end of the logarithmic search? Mathematicians are not satisfied, so they continue to look for new numbers that have not yet been discovered. In fact, they have found them. The chance of discovery is to study the square roots of some numbers: the square root of 4 is 2 (2× 2 = 4), which is a positive integer that has long been known, not surprising; The square root of 2 is an irrational number, similar to pi, which is not new. What is the square root of-1? It's not easy to do! As we all know, the symbolic law of multiplication is: positive is positive, negative is positive, and the square of any number is positive, so the square root of-1 does not exist at all. But things that don't exist can be created! This is the innovative spirit of science. Mathematicians have created an "imaginary number" for this purpose, which is represented by the symbol I, and stipulated that the square of I is-1, and the square root of-1 is of course I, so that the square root problem of negative numbers is solved. For example, the square root of -4 is equal to 2i, which means 2 times i.

Although the introduction of imaginary number solves the square root problem of negative number, it also brings another difficulty-the imaginary number has nowhere to be placed on the number axis. This forced mathematicians to create an "imaginary axis", which was perpendicular to the original axis renamed as "real axis". A plane composed of imaginary axis and real axis is called a "complex plane". The points on the real axis are real numbers and the points on the imaginary axis are imaginary numbers. The remaining points on the complex plane are "complex numbers", including real numbers and imaginary numbers. The zero point is the intersection of the real axis and the imaginary axis, and is the center of the whole complex plane, which still occupies a very special position. No matter how the concept of number is expanded, the special position of zero remains unchanged, from "yan zhen" on the real number axis to "stars holding the moon" on the complex plane. No wonder zero was nominated when the most important invention of the Millennium was selected online recently. I have a poem by Shan Yongling:

Zero praise?

You have nothing of your own.

But give others ten times.

No wonder you are so beautiful.

Like a bright moon in the Mid-Autumn Festival.

(originally published in the 25 th and 26 th issues of Galaxy)

Who says math is boring? The world of mathematics is full of poetry and painting, which needs us to explore.

Do imaginary numbers and complex numbers have actual prototypes? At first glance, it seems that "emptiness" is not ethereal, and "complexity" is very complicated. In fact, both imaginary and complex numbers have prototypes; In electrotechnics, complex numbers represent alternating current and imaginary numbers represent virtual work, which greatly simplifies the calculation of electrotechnics. If the complex number is introduced into electrotechnics just for the convenience of calculation, it is ok not to use it, but it is a little troublesome. Look at quantum mechanics again: the wave function in quantum mechanics must be expressed by complex numbers, which is not a problem of simplifying calculation, but a substantive problem reflecting the essence of microscopic particles; In other words, the deep-rooted natural laws in the micro-world need plural numbers. Who says mathematics is too abstract? Even if it is abstracted as a complex number, its application is very practical.

From natural numbers to negative numbers and zeros, to fractions, irrational numbers and complex numbers, is there an updated chapter in the development history of numbers? We will wait and see.