"Parallel lines meet at infinity", my cousin in class 10 told me, and also that his teacher had mentioned that if one cannot understand this apparently simple concept then one can never hope to understand mathematics. I was in class 8 at that time. I wonder why did the teacher say so. The concept is still not obvious to me. By linking an obscure statement to the ability to understand mathematics, the teacher very effectively killed any uncomfortable questions about it from the students.
Suppose I am standing near a set of parallel lines, say a railroad. I look towards positive infinity - the lines meet at some point. We will never reach that point and the point is not a well-defined point, but the lines do meet as per the statement. Now, let me gaze towards minus infinity. Again, the parallel lines meet at minus infinity. Now if two lines meet at two distinct points, then they are nothing but the same line. So, we arrive at a contradiction, and parallel lines do not meet at infinity.
My mathematician friends tell me that this line of reasoning is not quite correct, as one cannot have a well defined point at infinity and that the statement that "parallel lines meet at infinity" is only valid in projective geometry.
Suppose I am standing near a set of parallel lines, say a railroad. I look towards positive infinity - the lines meet at some point. We will never reach that point and the point is not a well-defined point, but the lines do meet as per the statement. Now, let me gaze towards minus infinity. Again, the parallel lines meet at minus infinity. Now if two lines meet at two distinct points, then they are nothing but the same line. So, we arrive at a contradiction, and parallel lines do not meet at infinity.
My mathematician friends tell me that this line of reasoning is not quite correct, as one cannot have a well defined point at infinity and that the statement that "parallel lines meet at infinity" is only valid in projective geometry.
