Parallel lines intersect at infinity. How should this sentence be understood

It's an infinite idea that two parallel lines never intersect because you extend the straight line forever and ever, but you just can't see them intersecting; But you can't say that they never intersect if you don't see it, and your eyes can't look at it all the way with the extension of the straight line, so think that the straight lines intersect in this limit of infinity ... Again, if you think about it the other way around, you draw two rays on the paper at a very small angle, and then extend them all the time, and when you draw a very small angle, you will find that at a certain point the spacing between these two rays changes very slowly and very slowly, and we know that the spacing between the parallel lines is equal everywhere, and if I extend the rays indefinitely, the distance between them changes smaller and smaller, so small that it is almost equal to zero, that is, it is close to no change at all. We can assume that the two rays are parallel at that limit. If so, it can be considered that infinite places intersect.

"Parallel lines intersect at infinity. is a geometric hypothesis, which is explained as follows:

In Euclidean geometry, parallel lines are straight lines that never intersect. However, if two parallel lines are extended infinitely, they will intersect at infinity. This assumption is based on the infinite points of Cantor's point set theory, but in practice, with any measurement of finite length, it will be found that these two lines will never intersect.

The essence of this assumption is that it goes against our perception and common sense, and it is reasonable from a mathematical and philosophical point of view.

When the lines are very long, the distance between them is basically ignored, so they intersect, I don't know if it's right or not.

No, the so-called parallel lines are in the same plane, two straight lines that do not intersect, and the two ends of the straight line extend infinitely, and the parallel lines belong to the idealized model, just like the smooth inclined plane is often used in physics, and the smooth plane is not resisted.

Euclid's five axioms are:

1. Any two points determine a straight line.

2. Any line segment can be extended into a straight line.

3. A circle can be made with a point as the center of the circle and a line segment as the radius.

4. All right angles are equal.

5. There is only one straight line parallel to the known straight line at the point outside the straight line.

Axioms are assumptions that are not provable. From these five axioms, Euclid derives a series of theorems. But it was found that the formulation of the fifth axiom was more complex (the original formulation was:

If both lines intersect the third line, and the sum of the interior angles on the same side is less than the sum of the two right angles.

Then the two lines must intersect on this side. As a result, it is suspected that this is not an axiom, but a theorem that can be deduced from the first four axioms. As a result, for a long time, many mathematicians tried to solve this problem, but most of them were in vain.

Riemannian geometry has a large role in Einstein's general theory of relativity. Legend has it that Albert Einstein encountered great mathematical difficulties in studying the general theory of relativity, and it was not until he discovered the powerful tool of Riemannian geometry that he successfully expressed his thoughts in mathematics.

To sum up, whether parallel lines exist or not, and how many exist, is essentially an assumption, and it doesn't matter whether it's right or wrong. Mathematics is the study of hypotheses and logical reasoning, which is different from the experiment-based sciences of physics, chemistry and biology in the natural sciences. The assumptions of mathematics are not provable, so mathematics should be relegated to the philosophical category.

OK. 1. The general concept is that two straight lines that cannot intersect are parallel lines. It is the most common geometric principle. It is also in line with formal logic.

2. However, from the perspective of the large scale of the universe, a line is curved even though it is a straight line, and it is impossible to bend the same as other lines (including "straight lines") in the infinite distance of the large scale, so there is an intersection.

3. We can also extrapolate the following: (1) Two intersecting straight lines, when they leave the intersection point for a distance, and at the same time intercept a small segment, and the small scale measurement within this small segment can be regarded as parallel. (2) These small parallel lines, on a small scale, do not intersect.

But when we go back to where we left, we find that point of intersection. Parallel lines also intersected.

4. Two parallel lines do not intersect, which is in line with formal logical reasoning; The fact that two parallel lines can intersect at infinity is from a large-scale point of view, which is also in line with dialectical logic.

Analysis: Two straight lines that do not intersect in the same plane are called parallel lines, and it can also be said that these two straight lines are parallel to each other According to this solution, two straight lines that do not intersect in the same plane are called parallel lines, and it can also be said that these two straight lines are flat with each other; So the answer is: in the same plane, parallel to each other Comments:

The key to this question is to understand the meaning of parallel lines, and it should be noted that the premise is "in the same plane".

Summary. Two straight lines in the same plane (when two straight lines never intersect) are called parallel lines; When two straight lines intersect at a (right) angle, one of the lines is called the perpendicular of the other line, and their intersection is called the perpendicular point

Within the same plane, two straight lines that do not intersect are called parallel lines, and these two are parallel to each other. One of the lines is called the () of the other line.

Hello. Two straight lines in the same plane (when two straight lines never intersect) are called parallel lines; When two straight rocks intersect at a (right) angle, one of the straight lines is called the vertical of the other straight line of the rough mountain, and their intersection is called (vertical foot).

From the analysis, it can be seen that two straight lines may intersect, and the wheels may or may not intersect or be parallel;

Therefore, the answer is: neither intersecting nor parallel