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① Comparative Examples
In fact, in the world of infinite numbers, the part may equal the whole!
② Comparison method
This is Cantor's way of comparing two "infinities": we can pair up two sets of infinities, with an element in each set corresponding to an element in the other set, and if they end up exactly one-to-one, there is no extra element in either set. , then the two numbers are equal in size;
Georg Cantor, the founder of "infinity mathematics," suggested that we could use the Hebrew letter ? (aleph) to describe an infinite number, with the corner symbol at the bottom right of the letter representing the number's position in the infinite series.
To this day, almost every branch of theoretical mathematics has become a tool for scientists to explain the physical world, including theories that were once thought to be too pure to be of any practical use, such as group theory, noncommutative algebra, and nonEuclidean geometry. Even today, however, there is a large body of mathematics that maintains the noble status of being "useless" and whose only function is to help people exercise their intellects, a transcendence worthy of the "king of purity" crown. "This system is the so-called Pure King. This system is the so-called "number theory" ("number" here means integer), which is one of the most ancient and complex theoretical mathematical ideas. It is one of the most ancient and complex theoretical mathematical ideas. Strangely enough, although number theory is indeed mathematics in its purest form, it is, in a way, an empirically and even experimentally based science.
In fact, the vast majority of propositions in number theory come from practice - people trying to do things with numbers, and then getting results from which they can form theories. This process is not unlike physics, except that physicists try things on real objects rather than theorized numbers. There is another similarity between number theory and physics: some of their propositions have been proved "mathematically", but others remain empirical, waiting to be proved by the most brilliant mathematicians.
① Goldbach's Conjecture
So we have not been able to come up with a generalized formula for primes until now. Another interesting problem in number theory that has neither been proved nor disproved is known as the Goldbach conjecture. This conjecture, formulated in 1742, claims that any even number can be expressed as the sum of two primes. [
② The theorem of the mean distribution of primes
The theorem of the mean distribution of primes is one of the most important discoveries in all of mathematics, and can be expressed simply as follows: the percentage of primes in the interval from 1 to any natural number N greater than 1 is approximately equal to the inverse of the natural logarithm of N. The larger N is, the greater the number of primes is. The larger N is, the more precise the result of this equation will be.
③ Fermat's Grand Theorem
Fermat wrote a short note in the margins suggesting that the equation x2 + y2 = z2 has an infinite number of sets of integer solutions, but that for equations such as xn + yn = zn [ 22], the equation has no solution if n is greater than 2.
Ra proved that the equations x3 + y3 = z3 and x4 + y4 = z4 cannot have integer solutions; Dirichlet proved that x5 + y5 = z5 does not have an integer solution, and together with the efforts of a few other mathematicians, we have now established that the equation does not have an integer solution as long as n is less than 269.
④ Imaginary numbers
People picked one of the modifiers used by Caldano to name such a number, so it is now called imaginary numbers. Since the creation of imaginary numbers, mathematicians have begun to use the concept more and more frequently.
Perhaps the only thing we can say about such numbers is that they are not zero, but they are not bigger or smaller than zero, so they are entirely imaginary numbers, or impossible numbers.
By analogy, every real number has a corresponding imaginary number. You can also combine real and imaginary numbers into a single equation, written in the form (slightly). Such mixed expressions invented by Caldano are often called complex numbers .
It was not until two amateur mathematicians gave it a simple geometric meaning that the imaginary numbers were given a proper name.
The three dimensions we take for granted could be combined with time to form a unified coordinate system that conformed to four-dimensional geometry.
A little
1) Introduction
Objects that do not have a plane of symmetry can be categorized into two types - left-handed and right-handed.
The threads on the shell of one type of snail are clockwise, the other counterclockwise. Even the basic particles that make up all matter (the so-called "molecules") often come in left- and right-handed forms. For example, sugar comes in both left- and right-handed forms, and believe it or not, there are two types of sugar-feeding bacteria, each of which can only eat sugar that corresponds to a chiral form.
② How to convert the two
But if you take a donkey out of the plane, flip it 180 degrees in space, and then have it come back into the plane, it becomes exactly the same as another donkey. By analogy, we can say that if we take a right-handed glove out of three-dimensional space, flip it in some suitable way in the fourth dimension, and then have it come back into our space, then it can become a left-handed glove as well.
Rather, it is the so-called " M?bius face ". This surface gets its name from a German mathematician who first studied it more than a hundred years ago. Making a M?bius noodle is very simple: take a long strip of paper, coil it into a ring; twist one end of the strip 180 degrees and finally glue the ends together. Take a look at Figure 23 to see what you can do. The M?bius surface has many strange properties, one of which is easy to discover: take a pair of scissors and make a complete circle parallel to the edge of the M?bius surface (shown by the arrow in Fig. 23). Of course, as you would expect, we should end up with two separate rings. But when you actually do it, you realize you've got it wrong: instead of two rings, we've got one big ring that's twice as long as the original one, but only 1/2 as wide! This donkey finds itself in a predicament where it somehow gets on all fours! Of course, it can flip over and get itself back on its feet, but if it does, it becomes a right-side donkey. In short, our "left-side" donkey becomes a "right-side" donkey after walking around on the M?bius plane.
On a twisted surface, a right-handed object can be converted into a left-handed object simply by passing through the twist, and vice versa. The M?bius ring actually represents part of another, more universal, surface, the Klein bottle.
But just think about it a bit more, and you'll realize that the fourth dimension isn't really that mysterious. In fact, there's a word that most of us use every day that can be thought of as, or actually is, the fourth dimension of the physical world, and that word is "time.
In the terminology of four-dimensional space-time geometry, the line representing the life history of each individual particle of matter is called a "world line". Similarly, the bundle of worldlines that make up a composite object is called a "worldband".
Thus, if we could find an accepted standard speed, we would be able to describe the span of time in units of length.
With the term "light year", we have turned time into a practical dimension, and the unit of time has thus become a unit that can be used to measure space. Conversely, we can also coin another term " light mile" and use it to describe the time it takes for light to travel a distance of 1 mile. Using the values for the speed of light presented above, we can calculate that 1 light mile equals 0. 0000054 seconds.
We can calculate the four-dimensional distance by simply generalizing the Pythagorean Theorem; the four-dimensional distance is a much more fundamental value than the separate space and time intervals to study the physical relationships between events.
The difference between space and time is then completely obliterated, which means we recognize that space can be transformed into time, and vice versa.
We can define the fourth coordinate as a purely imaginary number.
Since we believe that spatial distances are always real, and temporal distances are always purely imaginary, it might be fair to say that real four-dimensional distances are more closely related to ordinary spatial distances, while imaginary four-dimensional distances are more closely related to time intervals. In Minkowski's terminology, the first kind of four-dimensional distance is called spatial, and the second temporal.
Spatial distances translate into ordinary spatial distances, and temporal distances translate into ordinary time intervals. But there's an impenetrable barrier between these two kinds of distances, real and imaginary, so they can't be converted into each other, and for that reason we can't turn a ruler into a clock, and vice versa.
A little
Same:
This is the Rutherford model.
Difference:
According to existing physics, if the structure inside an atom is really the same as that of a planetary system, then it can only last for one billionth of a second, in other words, such an atom can't exist for a long time as it is spinning around and around and around and around and around and around and around and around and around and around and around and around and around and around and around and around. But despite this pessimistic theoretical outlook, reality tells us that the structure of the atom is so stable that the electrons inside the atom happily and tirelessly circle around the central nucleus, never losing any energy and showing no signs of falling!
The electrons do not revolve around the nucleus, and the Rutherford model is incorrect.
① Nuclei and electrons
Although there are thousands of known forms of matter, there are many different kinds, but to trace their roots, they are in fact different combinations of two fundamental particles: 1. nuclei, the fundamental particles of matter, which may be electrically neutral (neutron), or may carry a positive charge (proton); 2. electrons, free negative charge.
Actually, positrons do exist in nature, and they are very similar to negatively charged ordinary electrons, except that they are electrically opposite. Negatively charged protons may also exist, but physicists have not yet detected such particles. The reason that positrons and negative protons (if they exist) are not as common as negative electrons and positive protons in our physical world is that these two groups of particles are "antagonistic" to each other. As you know, if two charges have opposite electrical properties, they will cancel each other out if they come into contact. Therefore, since positive and negative electrons represent positive and negative free charges respectively, they cannot **** exist in the same region of space. Such annihilation will produce strong electromagnetic radiation (gamma rays) at the location where they meet, and the process of annihilation of two electrically opposite electrons is a mirror image of the process of "creating" a pair of electrons seemingly out of thin air by strong gamma rays. The process of "annihilation" of two electrically opposite electrons is mirrored by the seeming "creation" of a pair of electrons out of thin air by strong gamma rays.
For all we know, there could be planetary systems of antimatter in the universe, and if an ordinary rock from the solar system were thrown into one, or vice versa, it would become an atomic bomb as soon as it hit the ground.
② Neutrinos
The existence of neutrinos was deduced by the mathematical reductio ad absurdum. This exciting achievement did not begin with the discovery of something, but with the realization that something was missing from some physical process. That "something missing" is energy.
It was once believed that this was the first experimental evidence that the law of conservation of energy was not working, but Pauli suggested that the "Baghdad thief" of nuclear energy might be a hypothetical particle called a neutrino, which carries no charge and has a mass smaller than that of an ordinary electron. .
This light, uncharged particle is undetectable by any existing physical device, and it can easily penetrate any matter. A thin film of metal is enough to block visible light; a few inches of lead significantly reduces the intensity of the more penetrating X-rays and gamma rays; but a neutrino beam can easily pass through a layer of lead several light-years thick! No wonder we can't observe neutrinos anyway.
③ Summary - Transitions Between Particles
Neutrinos can combine with electrons to form the unstable mesons we observe in cosmic rays, which also have the not-so-appropriately-named "heavy electrons":
④ More
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① Temperature and Thermal Motion. p> ① Temperature and Thermal Motion
Brownian motion is actually the result of the invisible thermal motion of matter, and what we usually call temperature is really just a measure of how intense the thermal motion of a molecule is.
When the temperature reaches ?273°C (or ?459°F), absolute zero, the molecules of matter stop moving at all.
And if the temperature continues to rise, even the molecules themselves are in jeopardy, as increasingly violent collisions tear them apart into atoms. This process of thermal dissociation depends on the strength of the molecule itself. Some organic molecules break down into individual atoms or groups of atoms at a "low" temperature of a few hundred degrees, but other, more stable molecules (such as water) require temperatures of more than a thousand degrees to break apart. But no molecule can survive thousands of degrees, and at such high temperatures, matter becomes a gaseous mixture of pure chemical elements.
This process of thermal ionization becomes more and more pronounced if the temperature rises to hundreds of thousands or even millions of degrees. Such extreme high temperatures exceed the upper limit of what we can achieve in the lab, but are common inside stars, especially the Sun. Not even an atom can survive such intense heat; it is stripped of all its outer electrons, and matter ends up as a naked mixture of nuclei and free electrons, with the electrons moving through space at high speeds, colliding with each other with extreme force .
To use heat to completely break down matter, splitting the nucleus into separate nuclei (protons and neutrons), we need at least a few billion degrees. While we don't find temperatures this high even inside the hottest stars, it is likely to exist in a young universe billions of years old.
② Thermal motion and the law of disorder
The complete irregularity of thermal motion can be described by a new law we call the law of disorder, or the law of statistical behavior. To understand this tongue-in-cheek description, let's look at the famous " drunkard's walk" problem.
This equation implies that after the drunkard has made numerous random turns, the most likely distance between him and the lamppost is equal to the average length of each straight line journey he has traveled multiplied by the square root of the number of line segments.
But if there are a large number of drunks moving randomly from the same lamppost position, and they don't interfere with each other, then you'll find that after a long enough period of time all the drunks will be spread out in a certain area around the lamppost, and we can use the method we just described to figure out the average distance between them and the lamppost.
① Introduction
Any spontaneous process in a physical system is bound to move in the direction of increasing entropy until it finally reaches an equilibrium state where entropy is maximized. This is the famous law of increasing entropy, also known as the second law of thermodynamics (the first law is the law of conservation of energy), the law of increasing entropy is also called the law of increasing disorder.
② Misconceptions
1. The existence of living organisms seems to completely violate the law of increasing entropy.
Plants use the negative entropy (order) from sunlight to build their bodies from inorganic compounds, while animals can only eat plants (or other animals) and in this way gain negative entropy.
2.
But how is it that an ordinary steam engine can convert heat into motion without violating the law of increasing entropy? The mystery lies in the fact that the steam engine utilizes only a portion of the energy generated by the combustion of the fuel, and more energy is exhausted in the form of exhaust gases or absorbed by specially installed cooling equipment. In this case, the entropy within the whole system undergoes two opposite changes: 1. part of the heat is converted into mechanical energy of the piston, which is a process of entropy decrease; 2. another part of the heat from the boiler flows into the cooling equipment, which is a process of entropy increase. The law of entropy increase requires only an increase in the total entropy of the system, as long as the increase in entropy in the latter part exceeds the decrease in entropy in the earlier part .
3.
Another example can help us better understand the law of entropy increase. Suppose a 5-pound weight is placed on a shelf 6 feet above the ground. According to the principle of conservation of energy, it is impossible for this weight to run up to the ceiling by itself without an external force. On the other hand, it is possible for it to throw part of its weight at the floor, thereby gaining energy to fly the rest of the way up. Similarly, we can allow a localized region within the system to experience a decrease in entropy, as long as the rest of it increases entropy enough to compensate for the difference. In other words, we can indeed allow the disordered motion of molecules in some regions within the system to become more ordered, as long as we don't care that such an operation makes the motion of molecules in other regions more disordered.
① Introduction
The distribution of air molecules at the microscopic scale is actually not uniform. If you zoom in enough, you'll see that the molecules within a gas keep clumping together into small clusters, which then quickly disperse, only to have similar clusters of molecules appear at other locations. This effect is called density rise and fall. Ordinary liquids also have density and pressure rise and fall effects, they just don't seem to be as pronounced;
② Case 1 - Why the sky is blue
Part of the reason that the sky is blue is that the atmosphere scatters molecules, partly from suspended dust and mostly from density rise.
A pure sky is supposed to be extremely homogeneous, and no amount of molecules can make it "sky blue". It is like an extremely flat mirror, with only refraction or reflection, and very little scattering. In a uniform environment, the scattering of different molecules cancel each other out. But it is the density rise and fall effect that causes "the air to have ineliminable 'impurities', i.e., the rise and fall of the air itself. The scattering of sunlight by density fluctuations and so on creates a blue sky.
③ Case 2 - Why water boils to a milky white color
So we can describe Brownian motion in a different way: a suspended particle in water is pushed around because the pressure on it in different directions is always changing rapidly. As the liquid is heated near the boiling point, the density rise and fall becomes more pronounced, giving the liquid a slightly milky look.
Life is complex, but in essence it is no different from ordinary physical and chemical phenomena, so it is difficult to draw a clear line between life and non-life.
Raw materials are picked up from the surrounding medium to generate structural units similar to themselves. These viral particles are both ordinary chemical molecules and living organisms, so they are the "missing link" between life and non-life.
Genes are indeed the smallest biological units (each individual gene consists of about a million atoms). Genes seem to be the missing link between life and non-life.
1) Genetic traits
Genetic traits such as color blindness require both chromosomes to be affected in order to be visible, hence the term "recessive traits".
"Dominant inheritance" is the opposite of recessive inheritance, where only one chromosome needs to be affected for the trait to manifest.
In addition to dominant and recessive inheritance, there is also a " neutral " genetic trait.
Of course, even under the most advanced microscopes, all genes still look pretty much the same, with their different functions hidden deep inside the molecular structure.
② Others
But before division begins, pairs of chromosomes are often entangled, so it's possible for them to produce partial swaps. Such cross-mixing (as shown in Figures 99a and b) can lead to confusion of gene sequences from both parents, resulting in a mixture of inherited traits.
Traits that are independent of each other and do not affect each other are necessarily far apart on the chromosomes.
If you use only one eye, it is difficult to judge the distance between the nose of the needle and the thread; but if both eyes were open, you could easily pass the thread through the nose of the needle, or at least learn to do so easily. When you look at an object with both eyes, you unconsciously allow both eyes to focus on an object at the same time.
Try closing one eye and then switching to the other, and you'll notice that the position of the object (in this case, the needle) changes relative to a distant background (say, a window across the room). This effect is parallax shift .
The farther away an object is, the less parallax shift there is, so we can use this to judge distance .
1. We don't have to actually build a device that pulls your eyes so far apart that, say, your left eye is in Washington, D.C., and your right eye is in New York, just take a picture of the moon on a starry background from both cities at the same time. Put those two photos in a stereoscope.
2. Use the size of the Earth itself to measure the size of the Earth's rotational orbit
3. Use the size of the rotational orbit to measure the distances to the stars (of course, this means that we'll have to wait six months for the two observations to be completed, but what's the harm in that?)
What if it's further away?
1. Ranging based on pulsating stars
Harlow Shapley, an astronomer at Harvard University, has found a new "ruler" capable of measuring the distance to distant stars: the so-called pulsating star, or parent variable star. or parent variable stars.
If you find a parent variable star that is farther away than the upper limit of the parallax shift method, all you have to do is look through a telescope, note its pulsation period, and figure out its actual brightness; compare your observed brightness to its actual brightness, and you'll immediately know how far away it is. Using this ingenious method, Shapley succeeded in measuring those very long distances within the Milky Way; it is also particularly useful when estimating the general size of the galaxy.
2. Other
By this stage, we can only tell how far away a galaxy is based on its visible size; in line with previous 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, and that there are neither tall people nor short people, then you can judge the distance between someone and you by how tall you see them.
The main body of the planet is still molten, and the "solid earth" that we often refer to inadvertently is a relatively thin crust floating on top of the lava. The easiest way to prove this is to measure the temperature at different depths in the interior of the Earth; we found that for every kilometer of depth, the temperature rises by about 30°C.
What's more, the temperature of the Earth's interior rises by about 30°C for every kilometer of depth.
In the world's deepest mine (the Robinson shaft in the South African gold mine), the walls are so hot that air conditioning had to be installed to prevent the miners from being cooked alive.
In fact, the newly born Earth was a pure liquid sphere, and since then, it has been slowly cooling down, and what we see now is just one particular stage in the life of the planet, and in the distant future, the Earth will one day be completely solidified.