Suppose there is an object in front of a convex lens and we know that the light rays from each point on the surface of object will converge at a different point and form an image. So that means that the no. of points in the object and the image are EQUAL because of one to one correspondence so does that mean the size of the two MUST be same? But that just doesn't happen. Where am I wrong. Basically I am comparing infinity to infinity and I know that's not an easy task. But I want to know where my logic is wrong?
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1$\begingroup$ Are there more, less, or equally many points on the interval $[0,2]$ compared to $[0,1]$? $\endgroup$– Marius Ladegård MeyerCommented Jun 30 at 10:04
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1$\begingroup$ This is an extremely poor question. What do you mean by the "number of points" in an image and why do you think it is something that can be defined? Why do you think that such a concept has anything to do with the size of the image? Note that even if you define size to be the cardinality, you can have an uncountable set with zero total physical length. $\endgroup$– Vincent ThackerCommented Jun 30 at 10:30
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1$\begingroup$ @MariusLadegårdMeyer Hmm your comment does make sense. We shouldn't related the size of something to the points because the points are dimensionless so every object will have the same no. Of points but that doesn't mean the size are equal $\endgroup$– Virender BhardwajCommented Jun 30 at 13:21
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1$\begingroup$ @VirenderBhardwaj yes you are understanding well. Regarding your earlier question why do we talk in terms of them [points] while discussing image formation and things like that in optics? This is because distances can be found between points! For instance, the height of an object is the distance from the "top point" to the "bottom point". Note that we are not counting points, we only need the two at the ends. $\endgroup$– Marius Ladegård MeyerCommented Jun 30 at 13:28
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1$\begingroup$ @MariusLadegårdMeyer I get what you are trying to say. Thanks for the insight $\endgroup$– Virender BhardwajCommented Jun 30 at 13:46
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1 Answer
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Dicing with infinities is usually dangerous.
Start with the idea that a small part of the object, size o$\times$o, produces an image of size i$\times$i.
Now let o become smaller and smaller.
What would happen?
i would become smaller and smaller in proportion and there is your one to one correspondence.
The problem with the word point is that mathematically a point has no spatial dimension and so a magnified point is still a point.
Thus your statement $\dots$ we know that the light rays from each point on the surface of object will converge at a different point and form an image is not really referring to a mathematical point but rather a very small area.
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$\begingroup$ Can u help me understand this even with considering small areas as points. Like if we consider that a small area converge onto a different small area in the image then still the argument will be valid right? $\endgroup$ Commented Jun 30 at 12:42
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$\begingroup$ When the areas are finite in size the idea of magnification is valid. $\endgroup$– FarcherCommented Jun 30 at 12:52