Sometimes I go into some subject in the class (high school level) and I have to explain to my students how approximate an irrational number by a sequence of rationals. The problem is that I should explain that in a high school level. What I usually do is take $\pi$ as a example and take the sequence:
\begin{align}
& 3,1=31/10\\
& 3,14=314/100 \\
& 3,141=3141/1000 \\
& 3,1415=31415/10000\\
&\vdots
\end{align}
I think that approach is intuitive and the students feel satisfacted with that. I was trying to figure out another way to explain the rational approximation but I coudn't find any. My questions is, does anyone know another way to explain that approximation in a high school level?
Thanks in advance.
Answer
Newton's method for square roots can be a good candidate due to its simplicity.
For example for approximating $\sqrt2$ in few iterations. You can ask your students where do they think $\sqrt2$ is located. Between $1.41$ and $1.42$, then you can start with $x_0=1.41$
$g(x) = x - \frac{(x^2 - 2)}{2x}$
$g(1.41) = 1.41 - \frac{(1.41^2 - 2)}{2\cdot1.41} = 1.4142198581...$
$g(1.4142198581) = 1.4142135623 \approx \sqrt2$
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