CHAPTER 3

Approximation

When Mathematics, Computer Science, and Regulations Converge

It is the mark of an instructed mind to rest satisfied with the degree of precision to which the nature of the subject admits and not to seek exactness when only an approximation of the truth is possible.^{1}

—Aristotle

The Greek mathematician and inventor Archimedes, who lived circa 290–211 B.C.E., determined that the ratio of the circumference of a circle to its diameter was somewhere between and .^{2} That ratio is better known as π, or pi. In decimal format pi equals approximately 3.1415927.

Johann Lambert, a Swiss mathematician, created a proof showing that pi was an irrational number.^{3} That means that pi cannot be written as a fraction by using two integers. By contrast, for example, the fraction 5/2 uses the integers 5 and 2, and the resulting number is 2.5. Therefore it is a rational number.^{4} There are no digits beyond the decimal value of 5 in the number 2.5 other than zeros; in other words, there is no remainder. The number need only be written as 2.5.

Pi, on the other hand, has an infinite number of decimals to the right of the decimal point, and the decimals do not repeat. In September 1999 at the University of Tokyo, a computer calculated pi to 200 billion digits (206,158,430,000 digits to be ...