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.
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