Los Angeles Math Circle

LAMC Calendar // 2018-2019 Academic Year. Jump to:

For meetings prior to Fall 2018, visit the Circle Archive.

AdvancedBeginnersBreaking Numbers into PartsEarly Elementary IEarly Elementary IIHigh School IHigh School IIJunior CircleOlympiad Training
We will introduce Gaussian integers in order to decide which integers can be written as a sum of two squares.
We will wrap up the discussion of Gaussian integers and prove which integers are the sum of two squares.
We introduce the Gini index, an economic metric to measure wealth inequality.
Handouts: Gini Index
We introduce Legendre symbols and quadratic reciprocity to study residues modulo primes.
We explore applications of quadratic reciprocity.
Instead of our typical definition of addition and multiplication, tropical arithmetic looks at minimum and addition operations. We will graph and find roots of tropical polynomials.
We will continue our study of tropical arithmetic by proving a version of the Fundamental Theorem of Algebra for tropical quadratic polynomials.
Given a ruler, how many inch markings can you remove and still measure each increment between 1 and 12 inches? Is there some way to construct a 12-inch ruler such that each increment from 1 to 12 can be measured in a unique way?
Handouts: Golomb Ruler
We will introduce continued fractions and learn how to calculate them. We will also investigate the relationship between the irrationality of a number and properties of its continued fraction expansion.
We will continue our study of continued fractions with an imporant application in number theory: Given an irrational number, how efficiently can it be approximated by rational numbers? Continued fraction expansions play an important role in solving this problem.

In this power-point presentation, we will address the following questions: Why do some musical intervals sound pleasant, while others do not? Why do we have exactly 12 notes in an octave of a piano? Why aren't distances between frets on a flute or a guitar equal to each other? The answers, surprisingly, involve deep mathematical analysis involving continued fractions, the problem of doubling the cube, and rational approximations.

We will introduce the formal defnition of a limit of a sequence and develop basic properties.
We will continue our practice with formally proving limits of sequences and we will prove some additional properties of sequence limits.
We characterize all polynomials that have integer outputs for integer inputs.