January 31, 2012
Host: Dr. Maria Wesslen
Topic: Welcome to Math Circles!
In the first week we will work on a mix of problems and puzzles. Here are a few questions which we will consider:
1. We will try to find fractions between √56 and √58 .
a) Are there any?
b) How many are there?
c) Out of all the fractions between √56 and √58 , what could the denominator be?
E.g. 188/25 is one example, so 25 is one possible denominator. What other small numbers could the denominators be? Are there any denominators that will never show up in a fraction between √56 and √58 ?
Try to come up with answers to these questions, and try to prove your conjectures.
2. The number 7. 565656565656565656…. is between √56 and √58 . How do we know if it is a fraction or not? Can we write it as a fraction? How can we go about doing that? Or is it impossible?
We’ll consider these and many related questions.