Many everyday things seem trivial to me, but in reality, I know very little about them, such as 2β.
Proof of the Irrationality of 2β
There are many methods to prove that 2β is irrational. You can directly check wiki for these proofs.
Calculation of 2β
During our school days, we used the approximate value of 2β, which is 1.414, for calculations. However, what interests me now is how to calculate 2β 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:
Here, we have a very important form, which is a continued fraction. Because in this expression, we can infinitely iterate by replacing 2β 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 21β.
When we use 21β 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 2β to a high precision. However, some more interesting questions have also emerged:
Why can an equation be substituted into itself? Is this method correct?
For other square root numbers, like 3β, how can we construct an equation that is similar to the equation for calculating 2β?
The form of the continued fraction seems similar to the iterative method for finding the greatest common divisor. What is their relationship?