Discussion of the Activity
Consider extension questions 3 and 4 on "Iterating to Find the Square Root of 2":
3. How many iterations would be required to get one hundred-decimal-place accuracy in the estimate of sqrt 2?
4. If the process converges to sqrt 2, how "fast" does it converge? That is, how does the error change from iteration to iteration?
To address these questions, students at advanced levels need to explore number and operation in depth. For example, after eight iterations, the exact result of the process is a formidable fraction. Students can use a computer algebra system to get the first 300 decimal places in the expansion of this fraction. To see how close this approximation is, we square the approximation, N, and compare the result with 2. If the approximation is squared, the result is 2.0€, with 155 zeros before the next nonzero digit. Counting the 155 zeros after the 2, we can say that |N2 - 2| < 10-155. Solving this inequality, we get
|(N - sqrt 2)(N + sqrt 2)| < 10-155. Therefore,
Thus, the approximation to sqrt 2 is accurate to at least 154 decimal places. The technology enables advanced students to apply not only their understanding of number, place value, and operation but also their knowledge of factoring, inequalities, absolute value, and mathematical reasoning. See also the discussion of a web plot in figure 3.4.
This method of expanding a fraction to 300 decimal places can be used by students to study the repeating-decimal phenomenon of rational numbers and to explore the irrationality of sqrt 2. This process fits with the recommendation in Principles and Standards (NCTM 2000) for
high school students to explore system properties of numbers. By applying the rational-root theorem found in many algebra 2 textbooks, students discover that the equation x2 - 2 = 0 has no rational roots. Therefore, sqrt 2 must be irrational, since it is a root of that equation. CAS can be used by students in the development of a proof of the rational-root theorem, the factor theorem, the remainder theorem, and other results for polynomials.
Fig. 3.4 The web plot modeling the orbit of f(x)
To address question 4, students can use the technique above to count the number of accurate decimal places in their answers after one, two, three, €, eight iterations and make a table. The discussion of how to measure accuracy is important, but good estimates of the number of accurate decimal places are sufficient. Students can look at various regression options using the values in their tables. An exponential regression fits the data well, indicating that the number of digits of accuracy approximately doubles with each iteration. Translating this result into a statement about the error at each step, students can generate a recursive formula for the error as follows: E(n)=E(n-1)2