What is the difference between a postulum and an axiom? Difference Between Axiom and Axiom

Euclid. A small number of propositions adopted without proof are taken as axioms and axioms. Geometry Originals

There are only 5 public assumptions used in the

Public Assumption 1 A straight line must be drawn from one point to another.

The public 2 straight lines can be extended without limit.

Public 3 is centered on any point, and any long line segment can be used as a circle as a radius.

Public 4 All right angles are equal.

Public Hypothesis 5 If two straight lines intersect with the third straight line, and the sum of the two internal angles on one side is less than two right angles, the two straight lines must intersect after being sufficiently extended to that side.

Description This is the famous Fifth Public Hypothesis.

It is equivalent to "a point outside the straight line can lead only one straight line parallel to the known straight line", so there is also a "parallel common postulate".

called. There are also 5 axioms in the Geometric Primitives:

Axiom 1 is equal to the equal quantity of the same amount.

Axiom 2 Equal quantity plus equal quantity, its sum is equal.

Axiom 3 Equal minus equal amount, the difference is equal.

Axiom 4 The amount of overlapping must be equal.

Axiom 5 The whole is greater than the parts.

Euclid distinguishes between axioms and postulates in this way:

First, axioms are applicable to all sciences, and axioms are peculiar to geometry;

Second, the axiom itself is self-evident, and the axiom is not as self-evident as the axioms, but it also admits its its truth without proof.

Nowadays, people no longer distinguish between axioms and axioms, and use the word axiom to indicate it.

The public is to prove that there is no right or wrong, and the axiom is deduced from the public postulate.

Axioms and axioms are two words coined by Euclid in the Geometry Primitive.

Axioms are fundamental principles that apply in any mathematical discipline and do not need to be proved. For example, "Equal plus equal." Its sum is still equal to "a=b, b=c, then a=c"......

The axiom is the basic principle of geometry that does not need to be proved, which is the axiom of modern geometry. The most famous "Fifth Public Postulate" is one of them.

Here are the five major postulates of Euclid:

Public Assumption 1: Any two points must be connected by a straight line.

Public assumption 2: The straight line can be extended arbitrarily.

Public assumption three: any point can be the center of the circle, and any length is the radius to draw a circle, public assumption four: all right angles are the same phase liquid.

Postulate five: A little outside the line, there is a straight line parallel to the known straight line, and the postulate five is also called a parallel postulate, because it is not as concise as the other postulates, and it looks more like a proposition, after Bowyer and Lobachevsky removed the fifth postulate, they discovered non-Euclidean geometry.

All axioms of Euclidean geometry:

The dots are partial.

The line is only the length of the line on the plane, and there is no width of the pre-burial.

A straight line can extend infinitely on both sides.

There is only one straight line after two points.

A circle can be drawn at any radius over a point in the plane.

The two straight lines are parallel and the isotope angles are equal.

Equal + equal and equal.

Equal—The difference is equal.

Congruence of figures that can coincide.

The whole is greater than the parts.

In simple terms, the axioms are in terms of figures, while the axioms are in terms of quantities.

Explanation of the public assumption.

postulate]

Logic or Mathematics].

A statement that is accepted to be true and therefore does not need to be proven to be true Detailed explanation of a hypothesis that can be considered true without proof, e.g., a straight line can be drawn from one point to another.

Formula. High seas. Fierce search metric.

State, Society, Public: Public. the security of society as a whole).

Set up. Tree core calendar setting (a established; b Installation). Banquet.

Planning: Design. Try.

Hypothetical: Hypothetical. Set or.

Place oneself in somebody's shoes. Radical : 讠.

A commune refers to a place where ancient Chinese officials worshiped the gods and ghosts of heaven and earth.

Commune is a Chinese word, pronounced gōngshè, which refers to the place where ancient Chinese officials worshiped the gods and ghosts of heaven and earth, and also refers to a social form or group. In primitive society, the members of society jointly produce and consume together. Such as:

clan communes, etc. Groups that have their own economic or social views and live together.

The term commune first referred to the organization of autonomous towns in medieval Europe, which was characterized by the fact that citizens had certain rights, including property rights, administrative rights, etc. Help each other and help each other. The situation of the communes varied from region to region, and in some areas, such as northern Italy, the power of self-government was very strong.

The communes of the Middle Ages did not form a democratic politics, but generally formed an oligarchy dominated by wealthy citizens.

Sentence formation of commune:

1. Before the earthquake, Baitou Commune in Chongqing County and other places smelled the smell of sulfur one after another, which made people feel dizzy and nauseous, and choked uncomfortably.

2. The members of the commune designed a rice baler.

3. This commune was founded in 2004.

4. I am a new member of the Red Star Commune.

5. The commune members designed a rice baler.

6. The county is divided into communes, and Gangou is the station of Gangou Commune.

7. Rural communes are highly organized.

8. It belongs to the Jia'an management area under the jurisdiction of Kangning Commune, and is divided into Jia'an Commune.

9. The main organizational form of society is the rural commune.

Some properties of Euclidean geometry are equivalent to parallel axioms, i.e., if the parallel postulate is true, these properties can be deduced, and conversely, if one of these properties is an axiom, the parallel postulate can also be derived. One of the most important, and most often used as axioms instead of parallel axioms, is the Playfair axiom proposed by the Scottish mathematician John Playfair:

Given a straight line, there is one and only one straight line parallel to it at any point outside of the line.

Many people have tried to prove the parallel hypothesis with the first four public presumptions without success, and instead created hyperbolic geometry that violates the parallel public postulate. Finally, the Italian mathematician Eugenio Beltrami proved that the parallel axes are independent of the first four axes.

Many propositions that are equivalent to parallel postulates seem to have nothing to do with parallel lines. Some of these properties are more apparently obvious, and are therefore inadvertently used by some who claim to prove the parallel public establishment. Here are some propositions:

The sum of the inner angles of the triangle is two right angles.

The sum of the inner angles of all triangles is equal.

There is a pair of similar, but not congruent, triangles.

All triangles have circumscribed circles...

If the three interior angles of the quadrilateral are right angles, then the fourth interior angle is also a right angle.

There is a pair of equally spaced straight lines.

If both lines are parallel to the third, then the two lines are also parallel.