Recently, I have come across this very neat poem titled A Line in the blog Making Sense of Complications. Even though today’s discussion has little to do with that post, I can’t help but feel inclined to delve into a related topic it touches on. This has been a problem of interest to me ever since I began learning more advanced maths, but it is an interesting bit of philosophical mathematics that I’d like to share with you all!
But first I would like to give a bit of quick background and context to the idea and how it originally cropped up.
It starts with the age old problem of finding the area under a curve.
The first guess is to use a technique which involves drawing rectangles under the curve and counting the total size of the rectangles as a way to approximate its true area. The illustration below shows this process and how the rectangles both overshoot and undershoot the true area of the curve.
The green area is indicating bits that were overestimated, while the red area is indicating bits that were underestimated.
The point is to realize that as we make the width of the rectangles smaller, we can fit in more rectangles with less overshooting and undershooting. This reduces the error and thus obtains a tighter fit to our curve using this approximation method.
Unfortunately, there is always going to be some ‘small error’ associated with this method that we can never get rid of no matter how small we try to make the width of the rectangles. That is, until we force the widths of the rectangles to approach 0 meaning we allow the widths to get closer and closer to 0 without end!
Using this idea we can get an EXACT VALUE for the area under ANY CURVE without any of the previous error we were talking about. Quite remarkable!
At the time, my initial thoughts to this were the following:
“Hold on, how is this possible when the rectangles have a width of 0? How can you approximate the area under a curve using rectangles with no area (lines)?”
Almost 4 years passed till I finally resolved this question with some proper mathematical foundations…but the seed had been sown for a deeper problem that simply could not go away no matter how much Mathematics was learnt. I shall be giving an easier example of this question for the sake of clarity.
Suppose you have a line composed of all the numbers between 0 and 1.
What is the length of this line?
Ignoring real life scales, the line is already a type of ruler and it clearly indicates that its length is 1 (in whatever unit the line is measuring). Suppose for a sake of argument that you pick a single point from this line, representing a number say 0.6, and you remove it.
What is the length of this new line?
To do this, we need to figure out what the length of the section we just removed is. So the question then becomes: What is the length of a single point?
It is important to clarify that while my diagrams have an actual length and width to them, they are supposed to represent objects which do not. In the case of a line, when I draw a line it is supposed to represent an object with just length and no width. In the case of a point, even though it has a physical length and width, it is supposed to represent a singular point with no length nor width.
Due to the point’s lack of length, we say that the point has 0 length.
Okay, let us bring everything all together:
- We started with a line with length 1
- We removed a point of length 0
- The length of our new line is ?
How strange…we removed a point from the line but it does not appear to have changed its length. The stranger bit is…we can keep on doing this again and again. Continually removing points even INFINITELY, and despite that, it will remain the same length of 1.
To top it all off, we have previously noted that our line of length 1 is solely composed of points (each of length 0). So the question remains:
How can it possibly be that points of length 0 can come together to form a line length 1?
The idea seems almost contradictory but it’s true. What is going on here? Is there even a reasonable explanation to this?
This is a fun little conundrum that I’m sure all of you will enjoy trying to wrap your head around, and it’s for this reason that I’ll be posting a conclusion to this post in a week’s time so make sure to check in again to find out!
Here are some alternative resources on the topic in case you want to investigate this question a bit further :)!
Feel free to discuss any ideas you might have in the comments below!
Some great Mathematics Stack Exchange posts regarding this question:
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Thanks for reading!