Just how did those old timers compute, say, square roots? Or logarithms? Or trigonometric functions? After all, on the mathematical time scale, computers were invented just yesterday -- and electronic calculators only the day before.
Is there an effective way to compute square roots, or any of the so-called transcendental functions, by hand?
The answer is: Yes! One technique for computing square roots, which dates back to ancient Babylon (roughly 4000 years ago), is easy to describe (and even easier to implement).
Let's suppose that we're interested in finding the square root of a positive number . That is, we're looking for a positive number satisfying .
To begin, let's suppose that is a "good guess" for ; in other words, suppose that is approximately equal to . Then, we can say that / is also a "good guess" for , since
Notice, too, that if is smaller than , then / is larger than , and conversely. For example,
In short, given one estimate for , we automatically have a second estimate, and the two estimates lie on opposite sides of the actual value of .
But now there's an easy way to improve on both estimates. Take the average!
In a moment, we'll check the details behind this claim. For now, let's turn this observation into an algorithm for computing .
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