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Two More Math Problems: Continued Fractions, Nested Square Roots, Digits of Pi

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 + 4a)) / 2. In order for x to be an integer, 1 + 4a 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.

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