From the basic elementary level when we start reading geometry we get this idea developed in us that a straight line is the conjuction of infinite points.but how to prove this? I mean is this an axiom or its provable?
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$\begingroup$There are many different foundations of geometry. Regarding the infinitude of the points of a straight line, I think, the following axioms of Ordered Geometry are the best to shape intuition:
- We do not define what a point is. But we will use capital letters to denote these undefined objects. We do not define what intermediacy is. We just say that triplets of our undefined points may be in intermediacy relation.
- Definition: If $[APB]$ then we say that $P$ is in between $A$ and $B$.
- Axiom 1: There are at least two distinct points.
- Axiom 2: If $A$ and $B$ are two distinct points, there is at least one point $P$ for which $[ABP]$.
- Axiom 3: If [ABC], then $A$ and $C$ are distinct.
- Axiom 4: If $[ABC]$ then $[CBA]$ an not $[CAB]$.
- Theorem: (1) If $[ABC]$ then not $[CAB]$. (2) If $[ABC]$ then $A$, $B$, $C$ are distinct.
- Definition: If $A$ and $B$ are two distinct points then all the points $P$ for which either $[APB]$ or $[PAB]$ or $[ABP]$ form a line.
- Theorem: A line contains infinitely many points. Proof: There is at least one line because there are at least two points. The second axiom says that besides the two points defining the line the line has further points. Repeated application of of Axiom 2 shows that a line has infinitely many points.
- The axioms I use are the first two axioms of the Pasch-Veblen foundation of ordered geometry.
- Reference: Coxeter: Introduction to geometry.
Take your endpoints, call them $S$ and $E$. Pick some point between $S$ and $E$, let us call it $M$. Then, pick a point between $S$ and $M$. Call it $M_2$. Pick a point between $S$ and $M_2$ ....
If you can keep going like that, it is easy to see that the line must have infinite points... and you can keep going on like that. The numbers don't just end.
$\endgroup$ 4 $\begingroup$any point, y, on line in 2d cartesian space can satisfy following equation:
y = a*x + b, where a and b are constant real numbers, and x is real number with range (-inf, +inf)
because x has infinite range, so y has inf range.
this prove line has infinite number of points
$\endgroup$ 2 $\begingroup$You might know the equation of a straight line through 2 points which is: $$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$$ From that we can see that there are infinite solutions to satisfy this equation which proves that a straight line contains infinite points.
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