Is a line the same as distance?

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Discussion Overview

The discussion revolves around the relationship between lines and distance in geometry, exploring whether a line can be equated to the distance between points. Participants delve into definitions, properties of lines, and the nature of distance, with references to geometric concepts and coordinate systems.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants propose that the distance between two points is not the same as a line, asserting that a line is a geometric object while distance is a numerical concept.
  • Others argue that while the distance can be represented by a line segment, the line itself extends infinitely and is distinct from the numerical value of distance.
  • A few participants mention that the shortest distance between two points is represented by a straight line connecting them, emphasizing the difference between various types of lines and distances.
  • There is a discussion about the existence of lines in a coordinate system versus a graph of a function, with some uncertainty about the implications of these terms.
  • Some participants question the abstract nature of lines, suggesting that while geometric lines are theoretical constructs, drawn lines are merely approximations.
  • One participant acknowledges a correction regarding the terminology used to describe distance and lines, indicating a refinement of understanding.

Areas of Agreement / Disagreement

Participants generally disagree on whether a line can be equated with distance, with multiple competing views presented. The discussion remains unresolved regarding the definitions and implications of lines and distance.

Contextual Notes

Limitations include varying interpretations of "graph" and "coordinate system," as well as the abstract nature of geometric concepts that may not align with physical representations.

question99
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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?
 
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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.
 
question99 said:
Would it be correct to say that the distance between particular points is the same as a line?
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.
question99 said:
If so, does every possible line on a graph exist?
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,
question99 said:
Are the number of lines infinite?
Yes.
 
question99 said:
Would it be correct to say that the distance between particular points is the same as a line?
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".
 
If you have two points, any lines connecting them are the distance, but the shortest distance is the straight line connecting them.
 
fian said:
If you have two points, any lines connecting them are the distance, but the shortest distance is the straight line connecting them.
It's the case in Euclide geometry
 
fian said:
If you have two points, any lines connecting them are the distance, but the shortest distance is the straight line connecting them.

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"
 
micromass said:
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"
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.
 
Thank you for correcting me, micromass
 
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  • #10
Mark44 said:
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,

But, isn't line a abstract (something not representing or imitating external reality) idea? :confused: 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
Vinay080 said:
But, isn't line a abstract (something not representing or imitating external reality) idea? :confused: I thought we can't draw a line on the (cartesian) coordinate system (consisting of abstract lines as axis'), though we draw them illegally!
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.
 
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  • #12
I wouldn't say that we are doing this illegally, though.
I might require a new thread to deal with this, but thank you for clarification Mark44.
 

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