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Many everyday things seem trivial to me, but in reality, I know very little about them, such as $\sqrt{2}$.

Proof of the Irrationality of $\sqrt{2}$

There are many methods to prove that $\sqrt{2}$ is irrational. You can directly check wiki for these proofs.

Calculation of $\sqrt{2}$

During our school days, we used the approximate value of $\sqrt{2}$, which is 1.414, for calculations. However, what interests me now is how to calculate $\sqrt{2}$ to a higher precision.

Geometric Method

Adding auxiliary lines, we get a smaller triangle that is proportional to the larger right-angled triangle. Based on the proportional relationships of the line segments in the diagram, we construct an equation:

Next, we transform this equation:

Move 1 to the right side of the equation:

Here, we have a very important form, which is a continued fraction. Because in this expression, we can infinitely iterate by replacing $\sqrt{2}$ on the right side of the equation.

We can see that the omitted part becomes less important as the depth of iteration increases, but the 2 that repeatedly appears in the denominator follows a fixed pattern. Whether we assume the value at the deepest iteration is infinitely large or infinitely small, we will ultimately get a value very close to the initial value of $\frac{1}{2}$. When we use $\frac{1}{2}$ in this continued fraction for continuous calculation,

After a few iterations, we see that the decimal part 0.4142 is already quite accurate. For a more precise result, we can continuously iterate the numerator and denominator of the fraction until we reach the desired precision.

Here, we have basically solved how to calculate $\sqrt{2}$ to a high precision. However, some more interesting questions have also emerged:

1. Why can an equation be substituted into itself? Is this method correct?
2. For other square root numbers, like $\sqrt{3}$, how can we construct an equation that is similar to the equation for calculating $\sqrt{2}$?
3. The form of the continued fraction seems similar to the iterative method for finding the greatest common divisor. What is their relationship?

Let's solve these questions next time. ❤️