Chapter 7 Approximation Methods
Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man. Every careful measurement in science is always given with the probable error . . . every observer admits that he is likely wrong, and knows about how much wrong he is likely to be.―Bertrand Russell(1872–1970)1
https://mathshistory.st-andrews.ac.uk/Biographies/Russell/
I think that it is a relatively good approximation to truth—which is much too complicated to allow anything but approximations—that mathematical ideas originate in empirics.―John von Neumann(1903—1957)2
https://mathshistory.st-andrews.ac.uk/Biographies/Von_Neumann/
In a world where you can take your phone out of your pocket, ask it for the square root of two (\(\approx 1.414213562\)) or the fifth root of seven (\(\approx 1.475773162\)), and instantly obtain those numbers, accurate to nine decimal places, it is difficult to convey the profound importance of having good methods of approximation. With this level of accuracy available to us there appears to be no need for approximations.
But stop and think about this for a moment. Both \(\sqrt{2}\) and \(\sqrt[5]{7}\) are irrational numbers so neither can be completely represented by a terminating decimal. That is, the decimal form of both numbers is infinitely long. So if all we have is a value accurate to only nine decimal places what we really have is an approximation, right? It’s not even a particularly good approximation in the sense that most of the information we need to completely specify \(\sqrt{2}\) or \(\sqrt[5]{7}\) in decimal form is missing.
The fact is that the modern world could not exist without good approximation methods because very little of the information necessary to functioning in the modern world can be computed precisely. Moreover in those cases where it can be computed precisely the exact number is often less useful than the approximation. For example, if you are driving to Cincinnati your GPS will tell you that you are \(2\) hours and \(25\) minutes away, not \(145.22434554656546456\) minutes away, even if the latter number is exactly correct.
Based on the very noisy audio signal it receives your phone constantly approximates what signal to send to the speaker for you to hear. Because signal processing is such a ubiquitous problem, many very sophisticated approximation techniques have been developed and they are used all of the time. We don’t see them because they are usually embedded in software on our many electronic devices. Because they are so accurate, we tend not to see them as approximations.
Yet, every scientific, engineering, or financial computation involves approximations because it is almost always impossible to get perfect information. We must approximate and we do it all of the time. Well, actually most of the time our technology does it for us. But our technology is simply the realization of ideas that begin with paper, pencil, and thought. Without these no new technology is possible.
After the invention of Calculus — and especially in the twentieth century — the number of very good approximation techniques ballooned. We will look, very briefly, at two pre–Calculus methods of approximation. Then we will consider two early approximation methods that came from Calculus: Newton’s Method and Euler’s Method.
