Something to fight over, for the mathematically inclined:
Since when is the perimeter (circumference) of a circle equal to π?
When the diameter is 1.
I know you can approximate a circle by adding more and more vertices to a polygon, but then all of the vertices are facing outwards. Here the vertices alternately face inwards and outwards, so all we are doing is drawing a zig-zag line. On whatever microscopic scale you choose, we just end up with a circle made up from an infinitely fine zig-zag line, which will always be longer than a smoothly curved line. I’ve no idea how to put this in robust terms.
No, c = 2πr
If I take the graph above at face value (which I shouldn’t), r would be 1 and dropped, then c = 2π (for r = 1)
If the “perimeter” of the black border is 4 (and supposedly = c):
4 = 2π (/2)
2 = π
The person drawing this graph couldn’t even bother to correctly do some basic algebra.
Yeah, I walked away and reasiled i’d substituted r for d. :-[
I’m sorry to be a stick-in-the-mud spoilsport here, but this type of dull-witted mathematical naïveté does no one any favours, least of all those struggling with calculus. You have to calculate the (limiting) circumference using the red line portions, not the black ones — that is, use Pythagoras instead of just foolishly adding the x- and y components together. The diagram’s treatment effectively says that if you go 50 metres east and then 50 metres north, you have gone 100 metres north-east.
There are far more interesting ways to get π, the ratio of a circle’s circumference to its diameter, to be 4 (or any value you like) using non-Euclidean geometry.
Ah … so the sum of a gazzilion tiny hypotenuses? Clever!
My idiots explanation: The circle has a “smooth” outline, the straight-line configuration has a rough outline. You can make that texture approach smooth, but it remains rougher than the red line forever, so there are forever white gaps between the black line and the red line. iow: The black line’s length will always exceed that of the red one.
My bugbear is that, having taught calculus, problems like these confuse students by apparently validating intuitional misconceptions. It’s the sort of problem you give to your star students with the express instruction to identify the flaw.
This explanation fails because, as you can easily validate for yourself, the apparent ratio remains 4, regardless how finely you zigzag.
I thought it would have the exact opposite and pedagogically meaningful effect. The conclusion that Pi=4 is so obviously wrong that the problem actually invites critical thinking. The student will try and find the flaw in such reasoning, which will hopefully serve to deepen understanding.
Erm … but that’s the point, not so? As I understand it the puzzle presents a clearly incorrect conclusion based on faulty reasoning and invites us to spot the error in the reasoning.
My point is you’re zig-zagging, not that the ratio ever changes. The apparent circle is not, and never will be, a circle.
It’s consistently been my experience that on the whole, such problems only serve to confuse all but the most talented students. I suppose that there would be significant merit if students were presented with such material once they have mastered the basics and are comfortably competent — at which point they wouldn’t need my instruction anymore.
Well, there riemann sum doubt about that … :o
Yes, correct. The same cannot be said about the “gazillion hypotenuses”. In the limit, they are the circle.
Only in a certain ε-δ neighbourhood.
the “circle” is not even a “circle” if you look close enough.
True! So it is actually a pixel perfect approximation.
I found it on Quora, where someone posted it in answer to the question “What are some math problems you can confront your grade 10 teacher with?”
Boy, would it cause a breakdown in classroom discipline…
Okay, in that situation I would spend a few minutes describing to the students in detail how and why the misconception is faulty, and say that it illustrates the great importance of being careful how mathematical constructs are formulated.