Archive for May, 2022

Fairy Lights on the Farey Tree

Fairy Lights on the Farey Tree. Parity types are coloured as follows: Even: Blue; Odd: Green; None: Red.

The rational numbers ${\mathbb{Q}}$ are dense in the real numbers ${\mathbb{R}}$. The cardinality of rational numbers in the interval ${(0,1)}$ is ${\boldsymbol{\aleph}_0}$. We cannot list them in ascending order, because there is no least rational number greater than ${0}$.

Image Processing Emerges from the Shadows

Satellite images are of enormous importance in military contexts. A battery of mathematical and image-processing techniques allows us to extract information that can play a critical role in tactical planning and operations. The information in an image may not be immediately evident. For example, an overhead image gives no direct information about the height of buildings or industrial installations, but shadows, together with the time, date and basic trigonometry, enable heights to be determined  [TM233 or search for “thatsmaths” at irishtimes.com].

Parity and Partition of the Rational Numbers. Part II: Density of the Three Parity Classes

In last week’s post, we defined an extension of parity from the integers to the rational numbers. Three parity classes were found — even, odd and none. This week, we show that, with an appropriate ordering or enumeration of the rationals, the three classes are not only equinumerate (having the same cardinality) but of equal density in ${\mathbb{Q}}$. This article is a condensation of part of a paper [Lynch & Mackey, 2022] recently posted on arXiv.

Parity and Partition of the Rational Numbers. Part I: The Three Parity Classes

We define an extension of parity from the integers to the rational numbers. Three parity classes are found — even, odd and none. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure.