- risky, but we proceed nonetheless. First, let's factor 5280. Its factors are
- 2, 2, 2, 2, 2, 3, 5, and 11
- divisible by
2 3 5
4 = 2 * 2 6 = 2 * 3 10 = 2 * 5 15 = 3 * 5
8 = 2 * 2 * 2 12 = 2 * 2 * 3 20 = 2 * 2 * 5 30 = 2 * 3 * 5
and so on.
We have not divided by 11, and it seems like a curious factor. Why is it in there?
The key to this may be in the well-known approximation to pi of 22/7.
Suppose that the definers of the mile wished to define the unit so that if a circle has a integral number of feet in its diameter, then it will have an integral number of feet in its circumference. This might be desirable if building units (bricks) come in standard lengths, like feet. Since pi is irrational this cannot be done with absolute precision, but nothing in nature is absolutely precise. The definers only needed to work within a certain accuracy.
Here is a table of the various integral fractions of a mile and the diameter that corresponds to a circle with circumference equal to that fraction, using 22/7 as the ratio of circumference to diameter. Because of the factor of 11 in 5280, each of these diameters is a integer.
Because 22/7 is such a good approximation to pi, the relative error of the integral diameter to the true diameter is 0.04%. This corresponds to less than one hundredth of an inch over one foot. Even today it probably is not feasible to mass produce bricks with that degree of accuracy. It certainly was not possible at the time that the mile was defined.
- numbers from 100 to 10,000
// // What number from 100 to 10000 has the property that it is evenly divisible // by all the factors in array d, and if a circle is built with // (number / factor) units in it, then the diameter of that circle // will be an integral number of units within better than one part in 1000? // Answer: 5280 //