What are some of the more classic and wonderful methods of proof in mathematics?

In mathematics, there is an extremely basic axiom called the axiom of choice, and many mathematical contents must be based on this theorem to be established. In 1924, mathematicians Sturt Barnach and Alfred Tarski came up with a strange corollary, the sphere-splitting theorem, based on the axiom of choice. The theorem states that a three-dimensional solid sphere is divided into finite parts, which can then be formed into two solid spheres that are exactly the same as the original, depending on rotation and translation.

That's right, each one is exactly the same as the original. The split-sphere theorem is too counterintuitive, but it is a strict corollary of the axiom of choice, and it cannot be questioned, unless you abandon the axiom of choice, but mathematicians will pay a much greater price for it.

In 1931, the Austrian mathematician Gödel proposed a theorem that shocked the academic community - Gödel's incompleteness theorem. This theorem states that there must be theorems in our current mathematical system that cannot be proved or falsified. As soon as this theorem came out, it shattered the millennia-old dream of mathematicians - that is, to establish a perfect mathematical system, starting from some basic axioms, to derive theorems and formulas of all mathematics.

But Gödel's incompleteness theorem states that the system does not exist because there must be something in it, and we cannot prove or falsify it, that is, a mathematical system cannot be complete, at least its completeness and compatibility cannot be satisfied at the same time.

The map theorem, the theorem is like this, for example, we are in China, holding a map of China, then on the map, there must be a point, so that the point on the map is exactly the same as the real geographical location where the point is located, such a point we can definitely find. The theorem can also be extended to say that there must be a symmetrical point on the earth, and at any moment their temperature and air pressure must be exactly equal, note here"Definitely"It's not probabilistic"Definitely", but the absoluteness guaranteed by the theorem. Of course, some would say that this theorem cannot be applied in practice.

But using this theorem, we know that if we show a map anywhere in a park, we will find it on the map"Current location"。

There are many counter-intuitive facts in mathematics, the most famous of which is the Schwartz cylinder, that is, the infinitely folded cylinder, which is constructed with infinite folds on the surface, and then it can be proved that this cylinder is finite in volume, but the surface area is infinite, and it is the most important counterexample that the surface integral cannot be defined as simply as the volume integral.

If we can't find a positive way to prove it, then let's assume that there is a maximum prime number, and then reason on this basis to see if we will get a ridiculous result, if we can, then it means that our hypothesis is wrong, that is, the assumption that there is a maximum prime number is wrong, because the answer can only be one of the two, there is no more choice, at this time, the denial of one side is equivalent to the affirmation of the other, because the two must be one of them. Either the largest prime number exists, or it doesn't, there is no third possibility.

There are many formulas for calculating pi, and for a long time, we all thought that to calculate the 1000 digits of the circumference, the first 999 digits had to be calculated. However, in 1995, mathematicians discovered a magical formula that could calculate any digit of pi without having to know the preceding number. For example, to calculate the number of the 1 billionth digit, we don't need to know any digit before the 1 billion, the formula can directly give the number of the 1 billionth digit.

This formula is abbreviated as the BBP formula.

Before the twentieth century, mathematicians avoided infinity when encountering it, believing that either something went wrong or the result was meaningless. It was not until 1895, when Cantor established the theory of super-poor numbers, that people learned that infinity also has a hierarchy, for example, the infinity of a real number is higher than the infinity of an integer number. This is also counterintuitive, we never think of infinity as a number, but infinity has different levels in the theory of super-infinite numbers.

1.Gödel's incompleteness theorem.

Any compatible formal system, insofar as it contains Piano's axioms of arithmetic, can construct in it propositions that can neither be proved nor denied in the system (i.e., the system is incomplete).

Any compatible formal system that contains Piano's axioms of arithmetic cannot be used to prove its own compatibility.

2.Continuum assumptions.

There is no set where the cardinality is absolutely greater than the columnable set and absolutely smaller than the set of real numbers.

3.Banach Tuski's theorem.

This theorem states that if the axiom of choice is true, a three-dimensional solid sphere can be divided into finite (unmeasurable) parts, and then simply recombined by rotation and translation elsewhere to form two complete spheres with the same radius.

I think the clever proof process in mathematics has this.

There are many proofs of mathematical formulas. Here's a clever proof of some common formulas.

If the number of the sum of the cubes of the natural numbers is tiled, the number of cubes where it is exactly the square of the sum of the natural numbers. Therefore, we can prove the above equation.

2) The Pythagorean theorem.

The area of the large square is:

a+b）^2

The area of the large square is also equal to the sum of the area of the four triangles and the area of the small square:

4×（1/2ab）+c^2

From this we get the following formula:

a+b）^2=4×（1/2ab）+c^2

After simplification, the Pythagorean theorem is obtained.

a^2+b^2=c^2

This formula is the famous Euler equation, which is known as the most beautiful mathematical formula. A very simple formula combines the most important constants in mathematics – the natural constant e, the imaginary unit i, , the natural number 1, the natural number 0, and the most important mathematical symbol - plus sign + equal sign =.

Obviously, the sum of cos and sin is equal to e (i), so we can prove the euler formula. In Euler's formula, we can get the following formula:

e^（iπ）=-1+0

By transforming the terms of the above formula, we can finally derive the general form of Euler's identity.

4) Prove that pi is an irrational number.

Although it was used more than 3,000 years ago, it wasn't until more than 200 years ago that mathematicians first proved that pi is an irrational number. There are many ways to justify PI. The following is a counter-proof given by mathematician Ivan M. Niven. This method is simple and ingenious.

If is a rational number, there must be integers a and b in order for the following formula to hold:

a b where n is a positive integer.

Obviously, f k(0), f k( ), f(0), and f( ) are integers. In addition, both f(x) and fk(x) satisfy f(x)=f(-x), which are integrable at x=0 and x=.

Because f(0) and f( ) are integers, f( )f(0) is also an integer. If sin(x) sinf(0) is a positive integer (x), then it is clear that there is a positive integral (x) on (0)sinf.

Obviously, when n + f(x)sinx 0, according to the grip theorem, the integral of f(x)sinx on [0, ] also tends to zero. However, the above derivation shows that the integral is a positive integer, so there is a contradiction between the two. This means that =a b is not true, so pi must be an irrational number.

I have seen a clever proof that proves that there are irrational numbers a and b, so that a b is a rational number.

Let a=b=sqrt(2), then if a b is a rational number, the original name is true.

Otherwise, a b is an irrational number. In this way, a is the value of a b, and b remains the original value, and they are still irrational numbers, but at this time, a b=2 is a rational number, and the original name is true.

Principle: [sqrt(2) sqrt(2)] sqrt(2) == sqrt(2) [sqrt(2)*sqrt(2)] == sqrt(2) 2 == 2

This proof solves the problem without giving a definite answer.

denotes power operations, and sqrt denotes arithmetic square roots.

There are many ingenious proof processes in mathematics, for example, the famous pi is proved to be an irrational number in the continuous verification, and the pi of an irrational number is originally a very ingenious mathematical method.

There is a hook in mathematics, and this proof process is still very wonderful, because it is a proof to be obtained in a graph, and it is later cited as a theorem for later people to continue to use this theorem.

Mathematics seems obscure to many, but it can be understood if it is proved in a reasonable and intuitive way, for example, the Pythagorean theorem is very ingenious.

Mathematics itself is a wonderful subject, and there are many wonderful proof processes, such as the proof of the Pythagorean theorem, and there are many different ways to prove it.

I think it's a proof of an infinite number of prime numbers (Euclid?). ）

Multiply all known prime numbers (2*3*5...).), plus 1 (....)+1), a new prime number is created (dividing all prime numbers by 1), so the number of prime numbers is infinite.

The perfect cubic formula of sum (a+b) 3 (a b) (a b) (a b) (a b) The number of species selected x the type of combination 1aaa+3aab+3abb+1bbb a 3 3a 2b 3ab 2 b 3.

The perfect cubic formula for difference (a-b) 3 (a+(-b))(a+(-b))(a+(-b)) 1aaa+3aa(-b)+3a(-b)(-b)+1(-b)(-b)(-b) a 3 3a 2b 3 b 2 b 3.

The proof of the isosceles triangle isosceles is equicentric, and the triangle ABC is all equal to the triangle ACB, so the two angles are equal. Proof is complete.

Euclid had an infinite number of proofs for prime numbers.

Numbers are graphs, graphs are numbers, charts move.

Helen Qin Jiushao's formula, cosine theorem.

Court. Some quotes about mathematics.

Mathematics is an infinite science. Herman Weyl.

The problem is the heart of mathematics. －－

As long as a branch of science asks a large number of questions, it is full of vitality, and the lack of problems indicates the end or decline of independent development. －－hilbert

Some beautiful theorems in mathematics have the property that they are so easy to generalize from the facts, but the proofs are very hidden. Gauss.

The philosopher also has to study mathematics, because he must jump beyond the vast sea of ever-changing phenomena to grasp the real substance. ......And because it is a shortcut to the transition of the soul to truth and eternal existence. Plato.

Gauss (the prince of mathematics) said: "Mathematics is the king of science".

Russell said, "Mathematics is symbols plus logic."

Pythagoras said, "Number governs the universe."

"Mathematics is an ingenious art," Halmos said

"Mathematics is the highest achievement of human thinking," Mishra said

Bacon (English philosopher) said, "Mathematics is the key that opens the doors of science".

According to the Bourbaki school (the French mathematical research group), "mathematics is the study of theories of abstract structures".

Hegel said, "Mathematics is God's symbol for describing nature".

Wilde (President of the American Mathematical Society) said, "Mathematics is a culture that is constantly evolving."

Plato said, "Mathematics is the highest form of all knowledge."

"Mathematics is the crown jewel of human wisdom," said Court

Descartes said, "Mathematics is the instrument of knowledge, and it is the source of other tools of knowledge." All the sciences that study the order and measurement are related to mathematics. ”

Engels (philosopher of the dialectics of nature) said: "Mathematics is mathematics that studies quantitative relations and spatial forms in real life.

Klein (American mathematician) said, "Mathematics is a rational spirit that enables the human mind to be used to the fullest degree."

Galileo said, "Give me space, time, and logarithms, and I can create a universe" and "The books of nature are written in the language of mathematics" Newton said, "Without bold conjectures, no great discoveries can be made", and Halmos said

The creation of mathematics is by no means something that can be obtained by inferences alone, but usually begins with vague speculations, speculation about possible generalizations, and then comes to conclusions that are not very certain. Then organize your thoughts until you see the clues of the facts, and it often takes a lot of effort to put everything into logical proof. This process is not a one-time process, and it is not uncommon for many years to be wasted in the temptation to go through many failures, setbacks, repeated speculations, speculations, and temptations.

"In mathematics, our main tools for discovering truth are induction and simulation," Laplace said

Wittgenstein said: "Mathematics is a variety of provative techniques".

Hua Luogeng said: "New Changchai mathematical methods and concepts are often more important than solving mathematical problems themselves."

Napier said, "I always try my best and my talents to get rid of that heavy and monotonous calculation".

Kepler said, "I spent the best part of my life pursuing that goal......The book has been written. It doesn't matter if modern people read it or future generations read it, maybe it will take a hundred years to have a reader".