Number of points in line segment
What is the number of points in a line segment?
In schools we are taught that a line segment is made up of infinitely many points.
Let us suppose that there is a line segment $AB$ of length $a$ units and another line segment $CD$ of length $b$ units where $a>b$. According to the schools, both the line segments are made up of $\infty$ points but since $a>b$ won't it mean that $AB$ contains more points than $CD$?
Or both have infinite points but some one has larger infinity than other?
I have taken reference from here: How many points in a line segment? but I couldn't get a satisfying answer.
Any comment will be greatly appreciated.
Solution 1:
Suppose that you have a coordinate system where the line segments $AB$ and $CD$ are located in. Just to simplify things, suppose that $AB=[a,b],$ where $[a,b]$ is the interval of real numbers satisfying $a\leq{x}\leq{b}.$ Similarly, suppose $CD=[c,d].$ The length of each line segment is given $b-a$ and $d-c,$ respectively. These lengths could be different, but even if they are different, these two line segments contain the same number of points, or to put it in more formal mathematical words, the same cardinality. In other words, the "infinity" of points for both intervals is the same.
These may seem like a contradiction, but it is not. In higher-level mathematics, we have a rigorous definition for determining the cardinality of two sets, and this definition has very little to do with any notion of length. For two sets $X$ and $Y,$ we say that they have the same cardinality, size, or number of elements, whenever there exists some function $f:X\rightarrow{Y}$ such that $f$ is an invertible function. I assume that, since you said you are a high school student, you likely know what an invertible function is. In the specific case of $X=[a,b]$ and $Y=[c,d],$ such an invertible function f exists, even if the lengths of $X$ and $Y$ are different. To convince you of this, I will explicitly find this function. First of all, notice that there is an invertible function between $[a,b]$ and $[0,b-a]$: this function is $f_0(x)=x-a.$ Second of all, there is an invertible function between $[0,b-a]$ and $[0,d-c]$, given by $f_1(x)=\frac{d-c}{b-a}x.$ Finally, there is an invertible function between $[0,d-c]$ and $[c,d],$ given by $f_2(x)=x+c.$ Combining all three functions to obtain $f,$ we have that $f_2\circ{f_1}\circ{f_0}=f,$ implying $f(x)=\frac{d-c}{b-a}(x-a)+d.$ You can prove that $f,$ given like this, is an invertible function between $[a,b]$ and $[c,d].$ As such, these two intervals, surprisingly, have the same cardinality, the same number of elements.
This may seem counterintuitive, because they have different lengths. But it turns out that length is not a measurement of how many points an interval has. Length is a more abstract concept that involves a discipline of mathematical study called "measure theory." The idea, to put it in simplified terms, is that there exists some function $\lambda$ that takes certain sets from within a special family of sets, and it, so to speak, "assigns" a length to those sets. There are certain restrictions that $\lambda$ must satisfy in order to be a length-assigning function. I will not get into the technical details, since that will be beyond of what you want to know, but if you ever feel like you are ready to tackle the concepts, search the keywords "Jordan content," "measure (mathematics)" and "Lebesgue measure." The idea is that these length-assigning functions do not preserve the cardinality of a set. In other words, if $|X|=|Y|,$ it does not follow that $\lambda(X)=\lambda(Y).$ But what you will see, when you do get to study measure theory eventually, is that the number of elements, or cardinality, is, in some rigorous sense, the 0-dimensional length of a set, while the ordinary concept of length is the 1-dimensional length of a set. The thing you call "area" is like the 2-dimensional length of a set, and so on. This is what will ultimately give you the satisfaction of understanding why there is no contradiction here: they involve different dimensionalities.