Archimedes recognized the power of the infinite, and in the process laid the groundwork for calculus.
Archimedes used a similar strategy to approximate pi. He replaced a circle by a polygon with many straight sides, and then kept doubling the number of sides to get closer to perfect roundness. But rather than settling for an approximation of uncertain accuracy, he methodically bounded pi by sandwiching the circle between “inscribed” and “circumscribed” polygons, as shown below for 6-, 12- and 24-sided figures.
Then he used the Pythagorean theorem to work out the perimeters of these inner and outer polygons, starting with the hexagon and bootstrapping his way up to 12, 24, 48 and ultimately 96 sides. The results for the 96-gons enabled him to prove that
In decimal notation (which Archimedes didn’t have), this means pi is between 3.1408 and 3.1429.
This approach is known as the “method of exhaustion” because of the way it traps the unknown number pi between two known numbers that squeeze it from either side. The bounds tighten with each doubling, thus exhausting the wiggle room for pi.
In the limit of infinitely many sides, both the upper and lower bounds would converge to pi. We can discover more and more of its digits — the current record is over 2.7 trillion decimal places — but we will never know it completely.
Aside from laying the groundwork for calculus, Archimedes taught us the power of approximation and iteration. He bootstrapped a good estimate into a better one, using more and more straight pieces to approximate a curved object with increasing accuracy.
More than two millennia later, this strategy matured into the modern field of “numerical analysis.” When engineers use computers to design cars to be optimally streamlined, or when biophysicists simulate how a new chemotherapy drug latches onto a cancer cell, they are using numerical analysis. The mathematicians and computer scientists who pioneered this field have created highly efficient, repetitive algorithms, running billions of times per second, that enable computers to solve problems in every aspect of modern life, from biotech to Wall Street to the Internet. In each case, the strategy is to find a series of approximations that converge to the correct answer as a limit.
There is no limit to where that will take us.
Source: Take It to the Limit