Wait before you dismiss this as a crank question 🙂
A friend of mine teaches school kids, and the book she uses states something to the following effect:
If you divide the circumference of any circle by its diameter, you get the same number, and this number is an irrational number which starts off as $3.14159… .$
One of the smarter kids in class now has the following doubt:
Why is this number equal to $3.14159….$? Why is it not some other irrational number?
My friend is in a fix as to how to answer this in a sensible manner. Could you help us with this?
I have the following idea about how to answer this: Show that the ratio must be greater than $3$. Now show that it must be less than $3.5$. Then show that it must be greater than $3.1$. And so on … .
The trouble with this is that I don’t know of any easy way of doing this, which would also be accessible to a school student.
Could you point me to some approximation argument of this form?
If the kids are not too old you could visually try the following which is very straight forward. Build a few models of a circle of paperboard and then take a wire and put it straigt around it. Mark the point of the line where it is meets itself again and then measure the length of it. You will get something like 3.14..
Now let them measure themselves the diameter and circumference of different circles and let them plot them into a graph. Tadaa they will see the that its proportional and this is something they (hopefully) already know.
Or use the approximation using the archimedes algorithm. Still its not really great as they will have to handle big numbers and the result is rather disappointing as it doesn’t reveal the irrationality of pi and just gives them a more accurate number of $\pi$.