Problem of the Week #22: Monday June 5th, 2023 Today's topic suggestion comes from Fraser Ridge. Please let me know if you have topic suggestions as well. 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.
The standard number system we use today is called decimal numbers, or base 10, meaning at every digit, we count to 9 before carrying one to the next smallest digit. It’s of course possible to use other numbers as bases as well. For example, if we are using base 6, then we would only be using the digits 0 to 5 and carry to the next smallest digit when we reach 6.
a). Count from 1 to the largest 2 digit number in base 6.
b). How much is the base 6 number 123 in base 10?
c). Write the first 10 prime numbers in base 6.
d). Write the base 10 number 123 in base 6.
e). Write the decimal 0.123 in base 6.
e). Of the unit fractions (ie 1/2, 1/3, 1/4, etc), which ones would terminate when written as a heximal (base 6 version of a decimal)? Which ones would result in repeating heximals?
f). How do we predict the number of repeating digits and the number of initial non-repeating digits in the heximal expansion of a fraction?
g). Prove that any rational number would result in terminating or repeating heximals when written in base 6.
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.