Rational approximation for irrational number

Question

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 1

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 $x_0=1.41$

$g(x) = x - \frac{(x^2 - 2)}{2x}$

$g(1.41) = 1.41 - \frac{(1.41^2 - 2)}{2\cdot1.41} = 1.4142198581...$

$g(1.4142198581) = 1.4142135623 \approx \sqrt2$

Answer 2

I think high schoolers can understand Farey sequences. The $n$th Farey sequence is all the fractions between $0$ and $1$, inclusive, with denominators less than or equal to $n$. $F_5 = \{0,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,1\}.$ If your irrational number is between $0$ and $1$, then it lies between two of those fractions. By increasing $n$, there are more and more fractions and some of them get closer to your number. Wiki has a page:

https://en.wikipedia.org/wiki/Farey_sequence

I don't think this is more intuitive, but it has the advantage of leading to a number of elementary exercises.