Community Service Mathematics is supposed to be a process of discovery. Definitions, propositions, and methods of proof don’t come from nowhere, although after the fact (when presented in a textbook) they often seem to. As opposed to a textbook, real maths is highly non-linear.

Problem: What is the area of the triangle within the rectangle? Solution: In a moment of inspiration, we draw the following additional line: Now the answer is obvious. Once we split the rectangle into two smaller rectangles, the sides of the triangle become diagonals of their respective rectangles. The diagonals obviously split each of the two smaller rectangles into halves, where one half lies inside our original triangle.

Problem: Find the sum of the first 1000 natural numbers. Solution: Write the numbers twice as follows: $ \begin{matrix} 1 & + & 2 & + & \dots & + & 999 & + & 1000 \\\ 1000 & + & 999 & + & \dots & + & 2 & + &

Problem: At any party of 1000 people, must there always exist two people at the party who have the same number of friends at the party? For the sake of this problem, one cannot be friends with oneself, and friendship is bidirectional. Solution: This must always happen. Suppose to the contrary, that every person at the party has a different number of friends at the party. The minimum number of friends one could have is 0, while 999 is the maximum.

Problem: 1000 players compete in a tournament. In each round, players are matched with opponents, and the winner proceeds to the next round. If there are an odd number of players in a round, one player chosen at random sits out of that round. What is the total number of games are played in the tournament? Solution: 999. Each player loses exactly one game, except for the winner of the tournament.

A Bully By Any Other Name From the New York Times: “Shopping online in late July, Clarabelle Rodriguez typed the name of her favorite eyeglass brand into Google’s search bar. In moments, she found the perfect frames — made by a French company called Lafont — on a Web site that looked snazzy and stood at the top of the search results.

Dangling Nodes and Non-Uniqueness Recall where we left off last time. Given a web $ W$ with no dangling nodes, the link matrix for $ W$ has 1 as an eigenvalue, and if the corresponding eigenspace has dimension 1, then any associated eigenvector gives a ranking of the pages in $ W$ which is consistent with our goals.

My next book will be Practical Math for Programmers A High-Level Overview of Fully Homomorphic Encryption Searching for Riemann Hypothesis Counterexamples Linear Programming and Healthy Diets Hybrid Images Bezier Curves and Picasso

Story Time Linear algebra was founded around the same time as Calculus (think Leibniz, circa 1700) solely for the purpose of solving general systems of linear equations. The coefficients of a system were written in a grid form, with rows corresponding to equations and columns to the unknown variables.

The Web as a Graph The goal of this post is to assign an “importance score” $ x_i \in [0,1]$ to each of a set of web pages indexed $ v_i$ in a way that consistently captures our idea of which websites are likely to be important. But before we can extract information from the structure of the internet, we need to have a mathematical description of that structure. Enter graph theory.

The Quest to Capture Speed Companies and researchers spend hundreds of millions of dollars for the fruits of their algorithms. Whether one is indexing websites on the internet for search, folding proteins, or figuring out which warehouse is the most cost-effective to ship a product from, improvements in algorithm speed save immense amounts of money.