Author: Vincent Granville

These problems are for college undergrads after a first course in calculus. They are provided with solutions, and could be used by college professors as exercises or exam questions.

**1. Digits of Pi/4**

Prove that in base *b*, if *b* is an even integer, *n* > 3, and *x* = Pi/4, then the *n*-th digit of *x,* denoted as *a*(*n*), is given by the formula below. We start with *n* = 1 after the decimal point, for the first digit. Also show that the formula below is not valid if the base *b* is an odd integer, or if *x* is different from Pi/4.

where the brackets represent the integer part (also called floor) function.

**Solution**

Regardless of the number *x* in [0, 1] and the base *b*, the *n*-th digit *a*(*n*) of *x* can be computed as follows:

See here for details. Thus we have

Using the angle difference formula for sinus, the fact that *n* > 3, *b* is an even integer, and *x* = Pi/4, it simplifies to

The result for *a*(*n*) follows immediately.

**2. Continued Fractions and Nested Square Roots**

Let us consider the two following expressions, assuming *a* is a strictly positive real number:

Prove that *x* is an integer if and only if *a* is the product of two consecutive integers. Prove that the same is true for *y*.

**Solution**

Let’s focus on the first case. The second case is almost identical. The strictly positive number *x* must satisfy *x*^2 = *a* + *x*, thus *x* = (1 + SQRT(1 + 4*a*)) / 2. In order for *x* to be an integer, 1 + 4*a* must be a perfect odd square, which is possible only if *a* is the product of two consecutive integers. For instance,

- If
*a*= 1 * 2, then*x*= 2 - If
*a*= 2 * 3, then*x*= 3 - If
*a*= 3 * 4, then*x*= 4 - If
*a*= 4 * 5, then*x*= 5 - and so on.

Note that the expansion of the number *x* = 2 in the nested square root numeration system, when *x* tends to 2, has all its “digits” equal to *a* = 1 * 2. See this spreadsheet for details. More on this here.

*For related articles from the same author, click here or visit www.VincentGranville.com. Follow me on on LinkedIn.*

**Related article**

**DSC Resources**

- Free Book: Applied Stochastic Processes
- Comprehensive Repository of Data Science and ML Resources
- Advanced Machine Learning with Basic Excel
- Difference between ML, Data Science, AI, Deep Learning, and Statistics
- Selected Business Analytics, Data Science and ML articles
- Hire a Data Scientist | Search DSC | Classifieds | Find a Job
- Post a Blog | Forum Questions