What is a magical natural number?

Speaking of the fun of numbers, there is a legend here: once upon a time there was a wise king. One day, an old scholar respected by the king came to see him and they talked for a long time. The king said, "Old man, I have an interesting arithmetic self-test question here." Now think about the month when you were born, but don't tell me. "The old scholar immediately thought that he was 60 years old and was born in 65438+February.

"Multiply your birth month by 2, and then add 5."

The king said, "Well, multiply it by 50." "It's over, too."

"Plus your age." "it's over."

"negative 365." "It's also reduced."

"Well, tell me how to count." Said the king.

" 1 145。" The old scholar replied.

"Thank you," said the king. "You are 60 years old and were born in 65438+February."

"How do you know?" The old scholar asked in surprise. "You told me the number was 1 145, and I added 1 15, and the number was 1260, which means your birth month is 12, and the last two digits of 60 are your age."

Another example: please think of any positive integer with four digits, so that its four digits are not all the same, such as 46 17. Now rearrange the four bits in this four-bit integer to get the largest four bits 764 1 and the smallest four bits 1467 respectively. 764 1 subtract 1467 to get 6 174. This is a strange number.

Let's change another number, such as 1987, and follow the above process. The number is not the strange number we expected 6 174. But please don't worry, repeat the above process twice, and the strange phenomenon will appear again: the number is 6 174 again. We write the whole operation process as follows:

1987987 1- 1789=80828820-0288=85328532-2358=6 174

Now let's arbitrarily change a number: 4959, and carry out the above operation process: 49599954-4599 = 53555553-3555 =19989981899 = 80828820-0288 =

It can be proved that any four-digit positive integer, as long as its four digits are not all the same, will get 6 174 if it is used at most seven times according to the above operation steps. If you don't believe me, please try it yourself.

With regard to the magic of numbers, there is a report in the February issue of World Sci-tech Translation News 1987+0 1 that Jiang, a sixth-grade primary school student in Taipei County, found a wonderful change in the arrangement of numbers while playing video games, which attracted the attention of people in the mathematics field. Xiao Xiong Zhi once played a number game on the computer. He started with 1 and pressed the keys to 8 o'clock in sequence. He accidentally pressed 9, so the numbers were arranged in the order of "12345679". He randomly multiplied by 9, and a wonderful number "1 1 16" appeared on the screen. He found it very interesting and wanted to know more about whether the multiplication of these eight numbers with other numbers would produce other wonderful phenomena.

After his continuous research, it is found that the multiplicand "12345679" will show the result of three-digit cycle as long as it is a multiple of "3". When the multiplier is a multiple of 9, the product is "1111111" It is found that the above phenomenon disappears after the multiplier is "8 1".

For example, "12345679" multiplied by 3 equals "37037037", multiplied by 6 equals "74074074" and multiplied by 9 equals "111/. Multiplied by 12 equals "148148", multiplied by 15 equals "185 185", multiplied by185 ".

When the calculation of the eight-digit number came to an end, Jiang then tried to solve the mystery of the seven-digit number "1234568", and found that when this number was multiplied by a multiple of 3, the mantissa of its product would appear cyclic phenomena such as "4,8, 12, 16". By analogy, the calculation of two digits has been completed successively, and it is found that there will be repeated digital phenomena. Jiang's class teacher guided him to complete the record and provided this discovery for mathematicians' reference, hoping that mathematicians could solve this mystery.

There is also a mystery of natural numbers, which is not only a scientific number, but also a beautiful number. Almost where there is it, there is beauty-the most harmonious and pleasing rectangle, such as books, wardrobes, doors and windows. The ratio of short side to long side is 0.618; The most beautiful figure, such as Venus, Athena and Amanda, the ratio of upper body to lower body is 0.618; The best position for the announcer to stand is 0.6 18 times the width of the stage; Toronto TV Tower, the tallest building in the world, has a height ratio of 0.6 18. The famous Eiffel Tower, the ratio of the second floor below to the second floor above is 0.618; To get the best timbre, the "weight" of erhu must be 0.6 18 times of the chord length;

What's even more surprising is that Bartok's two piano sonatas, the total number of beats in three movements is 6432 octaves, and its "golden section"-6432× 0.618 = 3975 octaves, conforms to the structure of the works (slow-fast+slow-fast) and falls in the first place so accurately.

Human beings live in the objective world, and everything in nature, from the universe and planets to neutrons and particles, has a mathematical relationship. These mathematical relationships have amazing secrets. In human life in nature, every move is related to numbers. Imagine, who can exist without numbers? Then we can't help but ask, who stipulated this mathematical relationship for us? In the objective world, what mathematical relationships have not been discovered by human beings? Numbers in nature are actually the most beautiful things and the most mysterious riddles, which are expected to be discovered and solved by human beings.

With regard to the mystery of numbers contained in nature, the mathematician Mr. Shen Zhiyuan wrote in Wen Wei Po on August 8, 1999 that natural numbers 1, 2, 3 … are the starting point of mathematics, and all other numbers are derived from natural numbers. The physical prototype of natural numbers may be ten fingers, otherwise we won't use decimal. Mr. Shen Zhiyuan said that 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 discovery of the concept of negative number is probably related to the early commercial lending activities of human beings, but whether it is discovered or not, negative number itself exists.

Zero is a great discovery in the history of mathematics, which is of great significance. First of all, zero stands for "nothing". Without "nothing", how can there be "nothing"? So 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 operation function; Any number that adds or subtracts zero has the same value; 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. On the number axis, negative numbers and positive numbers are lined up like the wings of a wild goose, and zero is in the center, which is quite king.

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.

Mr. Shen Zhiyuan said that an infinite number of fractions can be inserted between two adjacent integers on the number axis to fill the blank on the number axis. Mathematicians once thought that the whole number axis was finally filled, in other words, all the numbers were found. In fact, not all numbers can be expressed as integers or fractions. The most famous example is pi. On the other hand, fractions can only represent approximate values, but not exact values. When people convert fractions into decimals, they find two situations: one is finite decimals, such as1/2 = 0.5; The other is infinite cyclic decimal, such as 1/3 = 0.333 ... Although they are different in appearance, they all contain limited information, because the cyclic part only repeats the original and does not contain new information. Pi is fundamentally different, 3. 14 1592653589 ... There is neither cycle nor end point, so it contains infinite information.

Think about it! Beijing library has a vast collection of books, which contains a lot of information, but it is still limited, and pi contains unlimited information, how can it not be amazing! Mathematicians call numbers that cannot be expressed as integers or fractions like pi "irrational numbers". Unreasonable people are unreasonable! I don't know why pi has such a bad reputation. To this end, Mr. Shen once wrote a poem called Pi, which was published in the August issue of Poetry Journal 1997. Here, I quote the last paragraph for readers:

Like a long poem that can't be read.

Neither circulate nor exhausted.

Endless, always new

Mathematicians call it an irrational number.

The poet praised it as a lover.

Tao is unreasonable, but affectionate.

Heaven and earth may not last forever.

This speed is endless.

Since Zu Chongzhi calculated the value of pi between "approximate ratio" 22/7 and "approximate ratio" 355/ 1 13, some people have been calculating more accurate values of pi, and recently they have been calculated to more than two million decimal places by computers! 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 the ratio of pi to diameter, but the infinite series it produces has the greatest uncertainty. We have to be 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-there is no swing of imaginary number 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 point on the real axis is the buying number, and the point on the imaginary axis is the imaginary number. 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. From "flying geese" on the real number axis to "stars holding the moon" on the complex plane. No matter how the concept of number is expanded, the special status of zero remains unchanged. No wonder zero was nominated when the most important invention of 1000 was selected online. To this end, Mr. Shen Zhiyuan wrote a poem about zero, entitled "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)

Mr. Shen Zhiyuan said, do imaginary numbers and complex numbers have actual prototypes? At first glance, it seems that "virtual" is not ethereal, and "complex" 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 electrotechnics calculation. If the introduction of complex numbers in electrotechnics is just for the convenience of calculation, it is possible without it, but it is a little troublesome, then please look at quantum mechanics-the wave function in quantum mechanics must be expressed by complex numbers, which is not a problem of simplifying calculation, but reflects 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!

As Mr. Shen Zhiyuan said: from natural numbers to negative numbers and zeros, to fractions, irrational numbers and complex numbers, I wonder if there will be an updated chapter in the development history of numbers. Are there still unknown mysteries waiting for mankind to solve? This shows that the secret of "number" in objective things is not small!