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
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
Students will be able to:
- Investigate how technology makes an old algorithm easy to use.
- Use different modes to see how effective an iterative algorithm is.
NCTM Standards and Expectations
- Generalize patterns using explicitly defined and recursively defined functions.
- Understand relations and functions and select, convert flexibly among, and use various representations for them.
- Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.
Common Core State Standards – Practice
Attend to precision.
Look for and make use of structure.