We all know that pi is the ratio of the circumference of a circle divided by its diameter.
Why not circumference/radius? After all, the radius is a more useful identity in geometry.
Maybe because it was (and still is) easier to measure the diameter of a circle than its radius? You can see exactly where the perimeter of the circle is, but not necessarily where the centre is.
My understanding was that one needed to know where the center of the circle was to measure either the radius -or- the diameter.
But I’m not really into math analysis…so…
Nah. A pair of calipers will tell you the widest point on a circle quite easily, which is by definition the diameter. To get the exact centre you need to find the widest point and then mark where the two sides are, then find the widest point somewhere, draw lines across the two diamters and the intersection is the center.
Of course all this is somewhat moot. If you know the diameter and can divide by two you know the radius. If you know the radius and can multiply by two you know the diameter. I don’t think it’s got much to do with ease of measurement, just tradition.
A very related question - in other situations where PI arises (e.g. probability), is there a strange factor of 2 hanging around, that would disappear if pi were defined as circumference/radius ? I.e., forgetting circles, what is a “natural” definition of pi in other areas?
I’m guessing the definition of pi related to circles came first - is it just a lucky coincidence that they came up with the “right” pi that appears in other fields like probability?
For a defenition of pi that does not involve circles, yyou could sum the series:
4/(2n+1)*(-1)^(n+1)
that is:
4 - 4/3 + 4/5 - 4/7 + 4/9 … to 4/infinity
Just in case you wanted to know.
For a defenition of radius-pi (i.e., 6.28…) that does not involve circles, yyou could sum the series:
8/(2n+1)*(-1)^(n+1)
that is:
8 - 8/3 + 8/5 - 8/7 + 8/9 … to 8/infinity
Just in case you wanted to know.
Well, the area of the circle is pi times radius squared.
If you used a pi’ that was defined as the circumference divided by the radius, then the formula for area would be pi’ times radius squared, divided by four.
The famous relationship, e^(i * pi) + 1 = 0 would have to be changed to e^(i * pi’/2) + 1 = 0
So, you get the extra factor of two in some places, you lose it in others.
I wouldn’t call pi appearing in other places a “coincidence” exactly, it’s more that the analysis is related.
Just a note… if you remove the “circle” part of it, you have a definition of pi that only applies in euclidian geometry. The ratio gives a definition that can be used in other, non-euclidian, geometries.