Green’s Theorem and Area of Polygons
A common method used to find the area of a polygon is to break the polygon into smaller shapes of known area. For example, one can separate the polygon below into two triangles and a rectangle: By...
View ArticleTwo points determine a line, three a quadratic — what has that got to do with...
In this post I describe how simple facts about polynomials are applied in correcting errors, for example scratches on compact disks. The same technique is used in many other places, e.g. in the...
View ArticleBinary quadratic forms over the rational integers and class numbers of...
I wrote an article with Irving Kaplansky on indefinite binary quadratic forms, integral coefficients. At the time, I believe I used high-precision continued fractions or similar. It took me years to...
View ArticleMatching Theory
Matching theory is an active field in mathematics, economics, and computer science. It ensured a Nobel memorial prize for Alvin E. Roth and Lloyd S. Shapley in 2012. The theory is applied in the real...
View ArticleHomology: counting holes in doughnuts and why balls and disks are radically...
There are some questions that are really easily posed, have an obvious answer, but are in fact really, really hard to answer in a mathematically satisfactory way. Two examples are: How many holes does...
View ArticleDealing with Risk
Dealing With Risk Consider a small company which uses a million dollar machine as an essential part of its operations. Suppose that there is a 10% chance that the machine will break down and need...
View ArticleClimbing the ladder of hyper operators: tetration
Arguably the first math lesson we’ve had – ever – dealt with counting. Soon, we’re exposed to addition, and later, multiplication. Finally, when we’re fresh into middle school, we take on...
View ArticleWhen can we do induction?
Introduction Every so often, the question comes up (either here or elsewhere) of why induction is a valid proof technique. And this is of course a very natural question. Induction is after all rather...
View ArticleOn the Möbius function
The Möbius function is a rather useful one, especially when dealing with multiplicative functions. But first of all, a few definitions are in order. Definition 1: Let \(\omega(n)\) be the number of...
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