Problem of the Week #21: Monday May 29th, 2023 This is the last problem on the May Number Theory topic. As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
A continued fraction is a way to represent any real number as a sequence of integers by writing down the integer part of the real number and taking the reciprocal of the remaining part at each iteration. For example, here is the process of writing the real number 2.315 as a continued fraction:
Separate the integer part of the number: 2.315 = 2 + 0.315
Take the reciprocal of the remaining part: 1/0.315 = 3.174603174603…
Separate the integer part: 3.174603174603… = 3 + 0.174603174603…
Take the reciprocal of the remaining part: 1/0.174603174603… = 5.727272…
Separate the integer part: 5.727272… = 5 + 0.727272…
Take the reciprocal of the remaining part: 1/0.727272… = 1.375
Separate the integer part: 1.375 = 1 + 0.375
Take the reciprocal of the remaining part: 1/0.375 = 2.666666…
Separate the integer part: 2.666666… = 2 + 0.666666…
Take the reciprocal of the remaining part: 1/0.6666666… = 1.5
Separate the integer part: 1.5 = 1 + 0.5
Take the reciprocal of the remaining part: 1/0.5 = 2
Separate the integer part: 2
The process is terminated when there is no remaining part to use to take the reciprocal, and the result is the list of integer parts we have separated from the process. This list of integers can be reconstructed to be the original number this way:
Or it can be written as the list of coefficients of this continued fraction:
2.315 = [2, 3, 5, 1, 2, 1, 2]
Notice this continued fraction has 7 terms.
a). Write the number 3.21 in the form of a continued fraction.
b). Write the continued fraction [1, 2, 3, 4, 5] in the form of a decimal.
c). Find two real numbers who have exactly 4 terms in their continued fraction representations.
d). Using your answer in part a), write 1/3.21 as a continued fraction.
e). Using your answer in part a), write -3.21 as a continued fraction.
f). Without going through the process of conversion, can you predict how many terms are in the continued fraction representation of the sum of the numbers in part a) and b)?
g). Given two natural numbers m and n, how many terms are there in the continued fraction representation of the quotient m/n?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.