One day I saw a blogger posting such an idea: If you take out a ruler and move your finger from 3.1cm to 3.2cm, then your fingertip will be at At a certain moment, it just crossed pi.
So is this true? Today we will study this issue.
What are the characteristics of π
First of all, let’s get to know π. It is pi, an irrational number, and a transcendental number. Irrational numbers are infinite non-repeating decimals, while transcendental numbers mean that they cannot be the solution to an equation composed of rational numbers; for example, √2 is an irrational number but not a transcendental number, because it is the solution to the equation x?=2. Notice! These conclusions have been rigorously proved mathematically, and it is not because humans have not yet calculated the last digit of π that they muddleheadedly believe that it is "infinitely non-cyclic". Although the human method of calculating π has been updated for many generations, it still uses the "infinite series" algorithm. You can understand it as an infinitely long calculation formula arranged according to a certain rule. Every time you calculate a little more accuracy, Add a little more.
The most important piece of information we need to extract about π is its "infinity". This concept means that when you touch the ruler, your hand needs to touch an infinitely accurate location, so as to satisfy the above idea.
Can infinite precision be achieved
In fact, as early as ancient Greece, philosophers had some in-depth thinking on issues related to limits. For example, there was a man named Zeno. , he raised a question called "Achilles chasing the tortoise". He said to everyone: I found that the great hero Achilles can never catch up with a tortoise.
The only weakness is the heel buddy
This Achilles is probably like the little Nezha in our Chinese mythology, a demigod hero with relatively great supernatural powers, then Why did Zeno say that a hero cannot outrun a tortoise? This is his analysis: I assume that Achilles's speed is 10 times that of the turtle (this hero doesn't seem to run very fast). The distance between him and the tortoise is about 100 meters. So when Achilles runs 100 meters, the tortoise will run 1 meter. In order to catch up with the tortoise, Achilles will run 1 meter, but at the same time the tortoise will also run 1 meter. It will run 1cm. When Achilles runs 1cm, the turtle will run 100 microns... In this way, it will never catch up.
This is probably how you chase a turtle.
That’s it...
After listening to this, do you have a strange feeling, that is, you clearly know that this conclusion is wrong? , but I don’t know how to criticize it, or I can’t find an angle to criticize it from. If it were at the scene, you would probably blurt out this sentence: "You are talking nonsense! I can't continue chatting with people like you."
One of the spirits of science is that no matter how unreasonable things are, "reason" and "evidence" must be given, so we must find the root of the problem. One very interesting thing is that Democritus, Zeno’s lifelong rival, was the founder of the classical atomic theory. His thoughts were diametrically opposed to those of Zeno. We can get a glimpse of him from his atomic theory. What answers are given to such questions.
The existence of atoms is a basic knowledge for us modern people, but for the ancient Greeks thousands of years ago, it was a very magical and profound philosophical question, because they did not observe it through To understand the possibility of the microscopic world, we can only use "utopian" methods to find answers (this is one of the great contributions of philosophy in the early days of the world. Now most of the functions of philosophy have been replaced by science). At that time, everyone's common understanding on this issue was that "matter can be infinitely divided." This point was expressed in the "Zhuangzi: Chapter of the World": "If you use a stick with a foot, half of it will be taken every day, and it will be inexhaustible for eternity.
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He is dividing objects with thoughts!
But Democritus obviously did not think so. He said:
If matter is infinite It is divisible, so if we continue to divide a small object, what will we get in the end? Is it a lot of small particles with dimensions (that is, volume)? Obviously as long as it has dimensions, it can continue to be divided. , so we have to keep dividing until there are only a lot of points without dimensions (that is, the concept of points in geometry, without volume and area). Okay, now we have to pinch them together, so how many points. Can it produce dimensions? 2? Thousands? Ten thousand? ...No, no matter how many 0s are added together, they will never have dimensions.
So, Democritus concluded that matter must not be infinitely divisible. It must have a minimum dimension that cannot be divided. He used the Greek word "indivisible" (átomos) to mean The name given to it is what we now translate as "atom".
The answer of quantum mechanics
So when we use this idea to look at "Achilles Chase" "Turtle" problem, you will find that if space is not infinitely divisible like matter, then there will finally be a moment when Achilles and the turtle will run through this "minimum length" at the same time, and then Achilles will Easily leave the turtle behind, and Zeno's problem will be solved.
This picture is actually misleading. Electrons are extremely small and do not move like planets.
So is there such a minimum scale? The answer is yes. This is the scope of the most advanced and cutting-edge research field in physics, "quantum mechanics". The core is the law of the smallest unit of all physical phenomena. The so-called "quantum" refers to the most basic particle that can only be measured in "units", such as the electron, which is the smallest unit of charge. There is no half electron in the world. There are so many charges, so the electron is a kind of quantum.
In this way, if we look at Democritus's classical atomic theory, we will find that the "atom" he said should actually be a "quantum". Only quantum is truly "indivisible". Similarly, if there is the smallest unit of particles, then there is no reason why time and space are infinitely divisible. Planck, one of the founders of quantum mechanics, gave the answer: Planck. Planck time and Planck length. They are the shortest time and the shortest length in the physical sense, which are 10-43 seconds and 10-35 meters respectively.
Planck’s fascination with appearance
The real world is inseparable from counting
So how did Planck calculate such a value? This requires us to think about it? The boat sailed back to the real world, and let me think about a question: How do you know information in the real world, such as how many melon seeds are on the table? The answer is actually very simple, just count it. Is there any other way for us to know the number of melon seeds? Actually no, even if someone tells you, someone (or something else that can be counted) must have counted them. So we get a seemingly nonsense fact: if something wants to prove its existence, it must be "countable".
No one can tell what the shape of photons is. We only know that they are not connected to each other
Why should we emphasize this point? That is naturally because "countability" does not necessarily exist. On a very small scale, if we want to recognize a particle, we must hit it with a photon. However, photons have a characteristic, that is, the higher the energy, the higher the accuracy. However, the energy cannot be infinitely high. When high When it reaches a certain maximum value, it will immediately become a very small black hole and destroy the space there. The accuracy of this extreme energy photon before becoming a black hole is the Planck length, and the time it takes for light to travel through the Planck length is the Planck time.
Everything is limited
Does it mean that it cannot be measured and does not exist? There is no evidence to prove that space is discontinuous, but all physical laws of any real world (including the amount of information) point to "everything is finite." Let’s imagine that there are many strange creatures living on a different world without air. If these creatures want to know how high the highest mountain is, they can only climb mountains. However, the gravity of the earth determines that the highest mountain is not high. It can exceed 10,000 meters, so this 10,000 meters is not only the highest measured height, but also the highest "actually existing" height in another world.
So the "measurement" in the above content is not a "measuring behavior" that is limited by human capabilities in the true sense, but a calculated theoretical upper limit. In fact, humans have not been able to create one that is sufficient. Photons that destroyed space on the spot. Just like the highest peak of the earth cannot exceed 10,000 meters, this is determined by the earth's gravitational acceleration. We do not need to actually pile the mountains to 10,000 meters to draw the conclusion that "the height of mountains has a limit."
Standing on the highest peak, the earth will also look round.
The conclusion that "everything is finite" solves all paradoxes about the universe in philosophy. People used to think that the universe was infinite space and infinite time. And now we already know that the size of the universe is limited (finite but unbounded), and the lifespan of the universe is not several times longer than the sun. At the same time, we also know that everything is not infinitely divisible. You may think that time and space have infinite precision (that is, the so-called continuity, of course, there is no evidence to support this), but they are definitely not infinitely divisible. The diameter of an electron is one billion times the Planck length. We have not found any evidence that electrons can be divided. Humanity's quantum research may be about to hit the ceiling once it enters.
The concept of infinity can only exist in the human mind. In some sense, it is greater than the entire universe
You cannot touch π
Now let's go back to the original question, can your fingertips touch π? Of course not, because at least your finger does not have infinite precision. It will only touch a point a little smaller than π and smaller than π. π "jumps" through a slightly larger interval. The infinite number π can only exist forever in the imagination of humans (or any intelligent individual).
That’s it for the inside of this article. If you like my article, please give it a like. You are welcome to comment below. I am Qizhiyu. See you in the next article!