Klaus Hoechsmann, Spring 1997.

Computing Roots.

Consider a rectangle with base b, height h, and area a = bh. If b ≠ h, we can create a new, more square-like rectangle with the same area a, by averaging base and height, that is, by taking

b_{∗} = \frac{b + h}{2}  and  h_{∗} = \frac{a}{b_{∗}}

as our new base and height. This process can be repeated until the rectangle is as square-like as we wish. The following list shows how this works for a = 9 and b = 2. The results are given both as common fractions and as decimals. We start with b_{1} = 2/1 = 2.0 and h_{1} = 9/2 = 4.5. Then:

b_{2} = \frac{13}{4} = 3.25  h_{2} = \frac{36}{13} = 2.769230769230769231
b_{3} = \frac{313}{104} = 3.009615384615384615  h_{3} = \frac{936}{313} = 2.990415335463258786
b_{4} = \frac{195313}{65104} = 3.000015360039321701  h_{4} = \frac{585936}{195313} = 2.999984640039321499
b_{5} = \frac{76293945313}{25431315104} = 3.000000000039321600  h_{5} = \frac{228881835936}{76293945313} = 2.999999999960678400

Good — but not perfect. Of course, b = h = 3 would fit perfectly, but our algorithm does not know that. Let us see what it does with a = 10, b = 3. We start with b_{1} = 3/1 = 3.0 and h_{1} = 10/3 = 3.3333333333333333333. Then:

b_{2} = \frac{19}{6} = 3.1666666666666666666  h_{2} = \frac{60}{19} = 3.1578947368421052631
b_{3} = \frac{721}{228} = 3.1622807017543859649  h_{3} = \frac{2280}{721} = 3.1622746185852981969
b_{4} = \frac{1039681}{328776} = 3.1622776601698420808  h_{4} = \frac{3287760}{1039681} = 3.162277660166916583
\frac{2161873163521}{683644320912} = 3.1622776601683793319  \frac{6836443209120}{2161873163521} = 3.1622776601683793319

Better — at least on the decimal side. The common fractions still look different, but are they? Is there a perfect square root of 10 (as 3 is the square root of 9)? Of course it could not be a whole number, but it might be a common fraction.

Exercises.

1. Using the same method, approximate \sqrt{10} (a) by decimals, (b) by common fractions, starting with b_{1} = 5/2 = 2.5. Use a calculator for the computations.

2. Let d_{k} = |b_{k} − h_{k}| be the difference between base and height at the k-th turn of this algorithm. What can you say about the size of d_{k+1}? (Hint: visualize b_{k}, h_{k}, and b_{k+1} on the number line.)

3. Suppose there were a fraction n/m, with n and m whole numbers, such that (n/m)^{2} = 10. Then the whole numbers n^{2} and 10m^{2} would be equal (see?). This is impossible because … Complete the last sentence with an irrefutable reason.

4. Devise a method for approximating cube roots. Starting with a square prism of base b × b and height h, create a more cube-like object (with the same volume) by averaging: b_{∗} = \frac{1}{3}(b + b + h), and so on. Try this out for computing the cube root of 9, using decimals.

5. (“Newton’s Method”) (a) In the x, y-plane, make a sketch of a smooth graph y = f(x), its tangent line at a point (x_{0}, y_{0}), and a point (x_{1}, α) on the tangent. (b) Calculate x_{1} in terms of x_{0}, α, y_{0} and the slope s_{0} of the tangent. (c) Using the fact that s_{0} = 2x_{0} for f(x) = x^{2}, compute x_{1} in terms of x_{0} and α/x_{0}. (d) Try this method on f(x) = x^{2} + x and α = 1. What is the idea behind it?