Intersection of line and lattice points

Question

Consider the Cartesian plane.

A line originates from origin. How many lattice points would it cross?

A lattice pointis a point with both coordinates integers.

My attempt:

Let the line be $y=mx$ since it passes through origin

For $x$ and $y$ to be both integers $m$ must be rational but we can also choose an irrational value of $m$ and hence a line need not pass through any lattice point exept origin

But in practice (on graph paper) it seems to always pass through a lattice point

What is the flaw here?

Answer 1

"In practice" your line does not have zero width. Since a line with irrational slope comes as close as you wish to many lattice points the fuzziness of the pencil lead (or the pixel width on a screen) will show the line (almost) passing through lattice points.

Answer 2

"In practice" your line does not have zero width. Since a line with irrational slope comes as close as you wish to many lattice points the fuzziness of the pencil lead (or the pixel width on a screen) will show the line (almost) passing through lattice points.

Hence your reasonung is correct