In the realm of geometry and mathematics, a line is generally considered to be infinite in both directions. This concept is fundamental to our understanding of lines in Euclidean geometry. Let's explore this concept in more detail.

Firstly, it's important to define what a line is in mathematical terms. A line is often defined as a one-dimensional figure with no thickness that extends infinitely in both directions. This means that a line has no endpoints; it simply continues on forever in a straight path.

The statement that "Any segment with at least two points has infinitely many points" is based on the idea that between any two points on a line, no matter how close they are, there is always another point. This is a reflection of the density of the real numbers, which are used to represent the coordinates of points on a line. For any two real numbers, there is always another real number between them. This is a key property of the real number system known as the Archimedean property.

When we consider the length of a line, we have to distinguish between the concepts of length and extent. A line, by definition, does not have a finite length because it extends indefinitely. However, a line segment, which is a part of a line that is bounded by two distinct endpoints, can have a finite length. This length can be measured and is determined by the distance between the two endpoints.

The totality of points on a line is indeed infinite. This is because for every point on the line, you can always find another point further along the line, no matter how far you have already extended it. This is a reflection of the unbounded nature of a line.

It's also worth noting that the concept of a line being infinite is not just a theoretical abstraction. It has practical applications in fields such as physics, where it is used to model phenomena that are understood to extend without limit, such as the path of a light beam in a vacuum.

In conclusion, a line in mathematics is infinite in the sense that it has no endpoints and can be extended indefinitely in both directions. The points on the line are also infinite in number, and while a line itself does not have a finite length, a line segment, which is a portion of a line, can have a finite length.

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Any segment with at least two points has infinitely many points, because, intuitively, given any two distinct points, there is a third one, distinct from both of them, say, the middle point. ... It may be finite in length or infinite in length. The totality of the points comprising the line is in any case infinite.Dec 3, 2013read more >>