October 2000
SIMMER Presentation

The first SIMMER meeting this year was held on Saturday, October 14, 2000, 1:30 pm - 4:30 pm. This was a joint session with this year's SOAR Reunion.

Here is a description of the session followed by some references and questions posed by the two presenters.

Title: “A Tale of Two Series”
Presenters: Dr. Greg Martin and Emmanuel Knafo
                    (Mathematics, University of Toronto)

Part 1: Do you hate fractions?  The way they just sit there so smug over their exactness?  The way the denominator lets the numerator walk all over it?  Well, this SIMMER is for you, because we're going to talk about decimals!  We'll take a look at the series of digits making up the decimal expansions of various fractions, thinking about when they terminate and when they repeat and trying to understand how long it takes the decimal expansions to start repeating. Surprisingly, we'll find connections to topics such as Euler's phi function and factorizations of numbers.

Part 2: Do you hate decimals?  The way they take up so much space?  The way they just go on and on and on, never stopping to let anyone get a digit in edgewise?  Well, this SIMMER is for you, because we're going to talk about fractions!  We'll take a look at the series of numbers making up the continued fraction representation of various numbers, thinking about when they terminate and when they repeat and trying to understand how to calculate the continued fractions in the first place. Surprisingly, we'll find connections to topics such as the Euclidean algorithm and the quadratic formula.

In their usual creative manner Greg and Emmanuel divided the 3 hours into 7 equal parts as follows:

1:30-1:55:42.857142
1:55:42.857142-2:21:25.714285
2:21:25.714285-2:47:08.571428
2:47:08.571428-3:12:51.428571
3:12:51.428571-3:38:34.285714
3:38:34.285714-4:04:17.142857
4:04:17.142857-4:30

The first and last three portions were spent on Parts A and B respectively. The middle portion were for a break, refreshments and for informally discussing the "Reunion Problems" students were given on the last day of this year's SOAR Number Theory 'camp'.

References:

Exploring the Real Numbers", Stevenson (Prentice Hall)
* Part I: Section 3.1
* Part II: Sections 3.3, 4.3

"Number Theory with Computer Applications" by Kumanduri & Romero (Prentice Hall)
* Part I: pp. 192-193, problem #1
* Part II: Sections 11.1, 11.4

"Elementary Number Theory and its applications", Rosen (Addison-Wesley)
* Part I: Section 12.1
* Part II: Section 12.2-12.4

Problems to think about:

Part 1:
* Find a number n such that the decimal expansion of 1/n has period 3.
(That is, 1/n = 0.abcabcabcabcabcabc... for some digits a, b, and c.)
* Find all numbers n with that property.
* Do the same with period 3 replaced by period 4 and period 5.
* In general, what is the relationship between the periods of the decimal expansions of 1/m, 1/n, and 1/mn?
* Find numbers n such that the decimal expansion of 1/n has period 15; period 20; period 60.

Part 2:
* Expand the rational fractions 17/3 , 3/17 , and 8/1 into finite simple continued fractions. From the continued fraction expansions of 17/3 and 3/17, can you generalize and show how the continued fraction expansion for a/b relates to the one for b/a ? What is the continued fraction expansion of an integer ?
* Evaluate the infinite continued fractions of <1,1,1,1,...> ,
<2,1,1,1,1,...> , <2,3,1,1,1,1,...> , <1,2,1,2,1,2,...> , <1,3,1,2,1,2,1,2,...> , <3,6,7,3,6,7,3,6,7,...> , and <5,6,9,3,3,6,7,3,6,7,3,6,7,...> .
* Expand each of the following as an infinite simple continued fraction:
sqrt(2) , sqrt(2) - 1 , sqrt(2) / 2 , sqrt(3) , and 1/sqrt(3) .

            Note sqrt stands for square root.