Is a line the same as distance?

1. Apr 20, 2015

question99

Would it be correct to say that the distance between particular points is the same as a line? If so, does every possible line on a graph exist? Are the number of lines infinite?

2. Apr 20, 2015

axmls

The distance between two points is the arc length of a line, sure. But it's necessary to differentiate between "distance" and "displacement."

But I'm not quite sure what you're asking.

3. Apr 20, 2015

Staff: Mentor

No. A line is a geometric object that extends infinitely far in some direction. If you have two points on a line, the distance between them is a number, often with some units attached such as feet, meters, miles, etc. The portion of the line between the two points is a line seqment.
I don't know what you mean. By "graph" I assume you mean the graph of some function. If instead of "graph" you mean "coordinate system", then yes, every possible line could be drawn on the coordinate system,
Yes.

4. Apr 21, 2015

Svein

No. A line is a geometric object. Distance (between two points) is a mathematical concept and depends on the (mathematical) environment.

Example: Take a globe. Pick two cities - Say New York and Cape Town. Take a piece of string, fix the ends at those cities and pull. When the string is taut, the length of the string is the distance between the cities, but I do not think you would call the shape of the string a "line".

5. Apr 21, 2015

fian

If you have two points, any lines connecting them are the distance, but the shortest distance is the straight line connecting them.

6. Apr 21, 2015

fian

It's the case in Euclide geometry

7. Apr 21, 2015

micromass

No, a distance is not a line. A line is a geometrical objects consisting of an infinite number of points. A distance is a number. So you should phrase it has "the shortest path is the straight line connecting them" or "the shortest distance is measured along the straight line connecting them"

8. Apr 21, 2015

fian

Ah, yes. Sorry. I used my daily speaking. You said right about the definition of line.
I am correcting it. It should have been the length or magnitude of the line connecting the points. So the distance is the length of the line.

9. Apr 21, 2015

fian

Thank you for correcting me, micromass

10. Apr 21, 2015

Vinay080

But, isn't line a abstract (something not representing or imitating external reality) idea? I thought we can't draw a line on the (cartesian) coordinate system (consisting of abstract lines as axis'), though we draw them illegally!

11. Apr 21, 2015

Staff: Mentor

All of the concepts of geometry, including lines, points, circles, and so on, are abstract concepts. A geometric line has infinite length but its width is zero. Whenever we use a pen or pencil to draw a line on graph paper, what we make is only an approximation to a geometric line. I wouldn't say that we are doing this illegally, though.

12. Apr 21, 2015

Vinay080

I might require a new thread to deal with this, but thank you for clarification Mark44.