Excerpts from the original text

① Comparative example

In fact, in an infinite world, part may be equal to the whole!

② comparison method

This is the method proposed by Cantor to compare two "infinite numbers": we can pair two groups of infinite numbers, and one element in each group corresponds to one element in the other group. If they happen to be one-to-one in the end, and there are no redundant elements in any set, then the two numbers are equal in size;

Georg Cantor, the founder of infinite mathematics, suggested that we can use Hebrew letters? (aleph) to describe an infinite number, and the corner mark in the lower right corner of the letter represents the position of the number in the infinite series.

Today, almost all branches of pure mathematics have become tools for scientists to explain the physical world, including those theories that people once thought were too pure to have any practical value, such as group theory, noncommutative algebra, non-Euclidean geometry and so on. However, even today, there is still a huge system in the field of mathematics that has always adhered to the lofty position of "uselessness". Its only function is to help people exercise their intelligence. Such detachment is definitely worthy of the title of "King of Purity". This system is the so-called "number theory" (here "number" refers to integers), and it is one of the oldest and most complicated pure mathematical ideas. Strangely, although number theory is indeed the purest mathematics, in a way, it is a science based on experience and even experiments.

In fact, most of the propositions of number theory come from practice-people try to do various things with numbers, and then get some results, thus forming a theory. This process is no different from physics, except that physicists try to deal with real objects instead of theoretical numbers. There is another similarity between number theory and physics: some of their propositions have been proved mathematically, but others are still in the empirical stage, waiting for the most outstanding mathematicians to prove them.

1) Goldbach conjecture

So we haven't listed a general formula that can only calculate prime numbers. There is another interesting problem in number theory, which has neither been proved nor falsified, and it is called "Goldbach conjecture". This conjecture was put forward in 1742, claiming that any even number can be expressed as the sum of two prime numbers. [

② the theorem of average distribution of prime numbers

The theorem of average distribution of prime numbers is one of the most important discoveries in the whole mathematical field, which can be simply expressed as follows: in the range from 1 to any natural number n greater than 1, the percentage of prime numbers is approximately equal to the reciprocal of the natural logarithm of n, and the larger n is, the more accurate the result of this formula is.

③ Fermat's last theorem

Fermat wrote a short note in the margin. He proposed that the equation x2+y2 = z2 has infinite integer solutions, but for the equation xn+yn = zn [22], if n is greater than 2, then the equation has no solution.

It is proved that equations x3+y3 = z3 and x4+y4 = z4 cannot have integer solutions. Dirichlet proved that x5+y5 = z5 has no integer solution, and with the efforts of several other mathematicians, we also proved that this equation has no integer solution as long as n is less than 269.

④ imaginary number

People chose a modifier used by cardano to name such a number, so it is now called "imaginary number". Since the birth of imaginary number, mathematicians began to use this concept more and more frequently.

For such numbers, perhaps we can only say that they are not zero, but they are neither greater than zero nor less than zero, so they are completely fictional numbers, or impossible numbers.

By analogy, every real number has a corresponding imaginary number. You can also combine real numbers and imaginary arrays into a formula and write it in (abbreviated) form. This mixed expression invented by cardano is usually called a complex number.

It was not until two amateur mathematicians gave it a simple geometric meaning that the imaginary number was proved to be justified.

The three-dimensional space we are used to can be combined with time to form a unified coordinate system that conforms to four-dimensional geometry.

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① Introduction

Objects without symmetry plane can be divided into two categories-left hand and right hand.

The thread on one snail shell is clockwise and the other is counterclockwise. Even the basic particles (so-called "molecules") that make up all matter often have left-handed and right-handed forms. For example, sugar has left-handed and right-handed forms. Believe it or not, there are two kinds of bacteria that feed on sugar, and each kind of bacteria can only eat the corresponding chiral sugar.

② How to convert the two?

However, if you let a donkey leave the plane, flip it in space 180 degrees, and then let it return to the plane, then it will become exactly like the other donkey. By analogy, we can say that if the right-handed condom leaves the three-dimensional space, flips it in a proper way in the fourth dimension, and then returns it to our space, then it can also become a left-handed condom.

But the so-called "Mobius surface". The name of this surface comes from the German mathematician who first studied it more than 100 years ago. Making Mobius noodles is very simple: take a long piece of paper and wrap it in a circle; Then twist one end of the note by 180 degrees, and finally stick the two ends together. Look at Figure 23 and you will know what to do. Mobius plane has many strange features, one of which is easy to find: take a pair of scissors and cut it completely along the direction parallel to the edge of Mobius plane (as shown by the arrow in Figure 23). Of course, according to your expectation, we should eventually get two independent rings. But after doing it, you will find that you are wrong: instead of cutting two rings, we cut a big ring, which is twice as long as the original one, but the width is only 1/2!

What happens when a shadow donkey walks on Mobius. The donkey found himself in a dilemma and somehow fell apart! Of course, it can turn over and stand up again, but in that case, it will become a donkey on the right. In short, our "left" donkey turned into a "right" donkey after bypassing the Mobius belt.

On a twisted surface, a right-handed object can only be transformed into a left-handed object by twisting, and vice versa. Mobius ring actually represents a part of another more common surface, namely Klein bottle.

But if you think about it again, you will find that the fourth dimension is not mysterious. In fact, there is a word that most of us use every day. It can be regarded as, or actually the fourth dimension in the physical world. The word is "time".

In the four-dimensional space-time geometry, the line representing the life history of each independent matter particle is called the "world line". Similarly, a bundle of world lines that make up a composite object is called a "world belt".

Therefore, if we can find a recognized standard speed, we can describe the time span in units of length.

Through the term "light year", we have turned time into a practical dimension, and the unit of time has therefore become a unit that can be used to measure space. Conversely, we can also create another term "light mile" to describe the time required for light to travel 1 mile. Using the light speed value introduced above, we can calculate that 1 light mile is equal to 0. 0000054 seconds.

We only need to generalize Pythagoras theorem to calculate four-dimensional distance; Studying the physical relationship between events, four-dimensional distance is a more basic value than independent space interval and time interval.

The difference between space and time is completely erased, which means that we admit that space can be transformed into time, and vice versa.

We can define the fourth coordinate as a pure imaginary number.

Since we think that the space distance is always a real number and the time distance is always a pure imaginary number, it may be said that the four-dimensional distance of the real number is more closely related to the ordinary space distance, and the four-dimensional distance of the imaginary number is more closely related to the time interval. In Minkowski's terminology, the first four-dimensional distance is called "space" and the second is called "time".

Space-like distance can be transformed into ordinary space distance, and time-like distance can be transformed into ordinary time interval. However, these two distances are real and imaginary, and there is an insurmountable barrier between them, so they cannot be transformed into each other. It is for this reason that we can't turn a ruler into a clock and vice versa.

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Same:

This is the Rutherford model.

Different:

According to the existing knowledge of physics, if the internal structure of an atom is really the same as that of a planetary system, then it can only last for a billionth of a second. In other words, such atoms cannot exist for a long time. However, although we put forward such a pessimistic prospect in theory, the reality tells us that the atomic structure is very stable, and the electrons inside the atom revolve around the central nucleus happily and tirelessly, never losing any energy and showing no signs of falling!

Rutherford's model is incorrect because electrons do not revolve around the nucleus.

(1) Nucleons and Electrons

Although there are many known substances, they are actually different combinations of two basic particles: 1. Nucleon, the basic particle of matter, may be electrically neutral (neutron) or positively charged (proton); 2. Electrons, free negative charges.

In fact, there are positrons in nature, which are very similar to ordinary electrons with negative charges, but the electrical properties are opposite. Charged protons may also exist, but physicists have not detected such particles yet. In our physical world, positrons and negative protons (if they exist) are not as common as negative electrons and positive protons, because these two groups of particles are "opposite to each other". As we all know, if two charges have opposite electrical properties, they will cancel each other out once they touch. Therefore, since positrons and negative electrons represent positive and negative free charges respectively, they cannot exist in the same space region. Such annihilation will produce strong electromagnetic radiation (gamma rays) where they meet. The process of "annihilation" of two electrons with opposite electrical properties and the process of "creation" of a pair of electrons out of thin air by strong gamma rays are mirror images of each other.

As far as we know, there may be a planetary system composed of antimatter in the universe. If you throw an ordinary stone from the solar system into an anti-galaxy, or vice versa, the stone will become an atomic bomb as soon as it falls to the ground.

② neutrinos

The existence of neutrinos is deduced from the reduction to absurdity in mathematics. This exciting achievement didn't start with what people found, but we found something missing in some physical processes. These "missing things" are energy.

People once thought that this was the first experimental evidence that the law of conservation of energy failed, but Pauli suggested that this "Baghdad thief" who stole nuclear energy might be an imaginary particle called neutrino, which has no charge and its mass is less than that of ordinary electrons.

No existing physical equipment can detect this uncharged light particle, which can easily penetrate any substance. To block visible light, a thin metal film is enough; For more penetrating X-rays and γ-rays, a few inches of lead can significantly reduce their intensity; But neutrino beams can easily pass through the lead layer several light years thick! No wonder we can't observe neutrinos anyway.

③ Summary-Transformation between particles

Neutrinos can combine with electrons to form unstable mesons that we observe in cosmic rays. It also has an inappropriate name, "heavy electron":

4 more.

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① Temperature and thermal movement

Brownian motion is actually the result of invisible thermal motion of matter, and the temperature we usually talk about is actually only a standard to measure the intensity of molecular thermal motion.

When the temperature reaches. 273℃ (ie? 459℉), that is, absolute zero, the molecules of matter will completely stop thermal motion.

And if the temperature continues to rise, even the molecules themselves are in danger, because more and more violent collisions will tear the molecules into atoms. This thermal decomposition process depends on the strength of the molecule itself. Some organic molecules will decompose into independent atoms or atomic groups at a "low temperature" of several hundred degrees, but other more stable molecules (such as water) need a high temperature of more than 1000 degrees to collapse. But no molecule can survive at a temperature of several thousand degrees. In such a high temperature environment, the substance will become a gaseous mixture composed of pure chemical elements.

If the temperature rises to hundreds of thousands or even millions of degrees, this thermal ionization process will become more and more obvious. Such extreme high temperature exceeds the upper limit that we can reach in the laboratory, but it is common in stars, especially in the sun. Even atoms can't survive in such a hot environment, and their outer electrons will all be deprived. Matter will eventually become a mixture of bare cores and free electrons, and electrons will move at high speed in space and collide with each other with extremely powerful force.

We need at least several billion degrees of high temperature to use heat to completely decompose matter and split the nucleus into independent nuclei (protons and neutrons). Although we haven't found such a high temperature in the hottest star, it probably existed in the young universe billions of years ago.

② thermal movement and disorder law

The completely irregular characteristics of thermal motion can be described by a new law, which we call the law of disorder or the law of statistical behavior. To understand this embarrassing description, we might as well take a look at the famous "drunken man's sake is not wine" problem.

This formula means that the maximum possible distance between a drunk and a lamppost after making numerous random turns is equal to the average length of each straight line he walks multiplied by the square root of the number of line segments.

However, if a large number of drunks start from the same lamppost and do not interfere with each other, then you will find that after a long enough period of time, all drunks will be distributed in a certain area around the lamppost, and we can calculate the average distance between them and the lamppost by the method just introduced.

① Introduction

Any spontaneous process in the physical system will inevitably develop in the direction of increasing entropy until it finally reaches the equilibrium state with the maximum entropy. This is the famous law of entropy increase, also called the second law of thermodynamics (the first law is the law of conservation of energy), and the law of entropy increase is also called the law of disorderly increase.

② misunderstanding

1, the existence of life seems to completely violate the law of entropy increase.

Plants use negative entropy (order) from sunlight to build their own bodies with inorganic compounds as raw materials; Animals can only eat plants (or other animals) and get negative entropy in this way.

2、

But why can an ordinary steam engine convert heat into motion without violating the law of entropy increase? The mystery is that the steam engine only uses part of the energy generated by fuel combustion, and more energy is discharged in the form of exhaust gas or absorbed by specially installed cooling equipment. In this case, the entropy of the whole system has undergone two opposite changes: 1. Part of the heat is converted into mechanical energy of the piston, which is a process of entropy reduction; 2. Another part of the heat of the boiler flows into the cooling equipment, which is a process of entropy increase. The law of entropy increase only requires the total entropy of the system to increase, as long as the increased entropy of the latter part exceeds the decreased entropy of the former part.

3、

Another example can help us better understand the law of entropy increase. Suppose there is a 5-pound weight on a shelf 6 feet above the ground. According to the principle of conservation of energy, it is impossible for this weight to reach the ceiling by itself without external force. On the other hand, it may throw some of its weight on the floor to gain energy and make the rest fly. Similarly, we can allow the entropy of the local area of the system to decrease, as long as the entropy of the rest is increased enough to compensate for the difference. In other words, we can really make the molecular motion in some areas of the system more orderly, as long as we don't care that this operation will make the molecular motion in other areas more disorderly.

① Introduction

In fact, the distribution of air molecules on the micro-scale is not uniform. If you zoom in enough times, you will see that the molecules in the gas keep gathering into small groups and then disperse quickly, but similar molecular groups will appear in other places. This effect is called density fluctuation. Ordinary liquid also has the fluctuation effect of density and pressure, but it doesn't look so obvious;

② Case1-Why is the sky blue?

Part of the reason why the sky is blue is that atmospheric scattering comes from suspended dust, and most of it is molecular scattering caused by density fluctuation.

It stands to reason that the pure sky is extremely uniform, and there is no "sky blue" with many molecules. Just like a very flat mirror, it only refracts or reflects and rarely scatters. In a uniform environment, the scattering of different molecules cancels each other out. But it is precisely because of the density fluctuation effect that "there are inevitable impurities in the air, that is, the fluctuation of the air itself." The scattering of solar light by density fluctuation forms the blue sky.

③ Case 2-Why is the water milky white after boiling?

So we can describe Brownian motion in another way: suspended particles in water are pushed around because the pressure they are subjected to in different directions is always changing rapidly. When the liquid is heated to near the boiling point, the density fluctuation becomes more obvious, making the liquid look slightly milky white.

Although life is complex, it is essentially no different from ordinary physical and chemical phenomena, so it is difficult for us to draw a clear line between life and non-life.

Extract raw materials from the surrounding media to generate structural units similar to themselves. These virus particles are both common chemical molecules and living things, so they are the "missing link" between living things and non-living things.

Genes are indeed the smallest biological unit (each independent gene is composed of about 654.38+00000 atoms). Genes seem to be the missing link between living things and nonliving things.

① Genetic characteristics

The genetic characteristics of color blindness need to be influenced by two chromosomes to show obvious traits, so we call it "recessive genetic characteristics"

Dominant inheritance is just the opposite of recessive inheritance, and this genetic feature can be affected by only one chromosome.

In addition to dominant inheritance and recessive inheritance, there is also a "neutral" genetic feature.

Of course, even under the most advanced microscope, all genes still look similar, and their different functions are deeply hidden inside the molecular structure.

② Others

However, before the start of division, pairs of chromosomes are often intertwined, so there may be some exchange. This kind of hybridization (as shown in Figure 99a and B) will lead to the confusion of parental gene sequences, resulting in mixed genetic traits.

Characters that are independent and do not affect each other must be far apart on the chromosome.

If only one eye is used, it is difficult to judge the distance between the needle nose and the thread end; But if both eyes are open, you can easily thread the thread through the needle nose, or at least learn it easily. When observing an object with two eyes, you will unconsciously let both eyes focus on an object at the same time.

You can try to close one eye first and then change the other. You will find that the position of the object (in this case, the needle) has changed relative to the distant background (such as the window opposite the room). This effect is parallax displacement.

The farther the object is, the smaller the parallax displacement is, and we can use this to judge the distance.

1, we don't really need to make a device that can pull your eyes that far, for example, the left eye is in Washington and the right eye is in new york, so we can shoot the moon on the starry background from these two cities at the same time. Put these two photos in the stereo.

2. Use the size of the earth itself to measure the size of the earth's orbit.

Measure the distance between the stars by the size of the orbit (of course, this means that we have to wait for half a year to complete two observations, so why not? )

What if it's further?

1, ranging method based on pulsating star

Harlow shapley, an astronomer at Harvard University, has discovered a new "ruler" that can measure the distance of distant stars. It is the so-called pulsating star, or Cepheid variable star.

If you find a Cepheid variable whose distance exceeds the upper limit of parallax displacement measurement, you only need to observe it through a telescope, record its pulsation period, and then calculate its actual brightness; Compare your observed brightness with its actual brightness, and you will know how far it is from you at once. Using this ingenious method, shapley successfully measured those very distant distances in the Milky Way; This method is also particularly useful in estimating the approximate size of the Milky Way.

2. Others

At present, we can only judge the distance of a galaxy according to its visible size; According to past experience, all galaxies of the same type are about the same size, which is very different from stars. If you know that everyone in the world is exactly the same height, neither tall nor short, then you can judge the distance between him and you by the height you see.

The main body of this planet is still in a melting state, and the "solid earth" we often casually mention is only a relatively thin crust floating on lava. The simplest way to prove this is to measure the temperature at different depths of the earth; So we found that the temperature will rise by about 30℃ for every kilometer of depth increase.

In the world's deepest mine (Robinson Deep Mine, South Africa's gold mine), the shaft lining is boiling hot. In order to prevent miners from being roasted alive, air conditioning must be installed in the mine.

In fact, the newborn earth is a pure liquid sphere. It has cooled down slowly since then. What we are seeing now is only a specific stage in the life course of this planet, and in the distant future, the earth will be completely solidified one day.