Algebraic Numbers

For my Numberlore book  I've been looking into the subject of Algebraic numbers. These are numbers that are the (real) solutions of algebraic equations with integer or rational coefficients. They include all rational numbers and expressions involving nth roots (square roots, cube roots etc), known as radicals. Thee are sufficient to solve all equations up to the fourth degree.

They do not include transcendental numbers like the circular constant, pi, or the exponential constant, e, which require infinite series

There are, however, further algebraic numbers between radicals and transcendentals. Little seems to have been written about them. They are solutions of quintic or higher equations and are known as ultra-radicals. One such is the solution to x^5 - x = 1. which is 1.1673... to four places of decimals. This and similar numbers were studied by a Swedish mathematician, Erland Bring (1736 - 1798). This information is from Wickipedia. I would be interested to know more.

 

In the News

 I found this Polar map projection among my History files. I copied it from somewhere years ago. It shows the location of Greenland, which is in the news, very clearly.

The quality of videos on You Tube seems to have deteriorated considerably lately, due to inferior productions based on Artificial Intelligence, and the automatic generation of mis-spelt captions, which often translate the same word or name in different ways in the same article. A lot of them also have the same didactic style of presentation.

We had a bit of snow on 5th January and since then it has been very cold here in Crewe. However the two named storms didn't reach this far north. Here's a photo I took from my front porch.