I'm looking for some intuition for why does pi equal 3.1416... and not some other number.
I started with the first digit 3. What I've done so far is:
Assume a circle of diameter 1. Assume a square of side 1 touching the circle at four points. The perimeter of the square is 4. The circle lies inside the square and both the circle and the square are convex figures, so the length of the circle must be less than 4. Does this seem right?
Also, the circumference must be greater than 2 because 2 is the shortest journey (forward and return) along the diameter of the circle, so the circle must be longer than that.
Now I can't think of anything to show that the length of the circle must be greater than 3.
Also, it'd help if you could give intuition for the next few digits of pi.