Ramanujan (1887-1920) discovered some formulas on algebraic nested radicals. This article is based on one of those formulas. The main aim of this article is to discuss and derive them intuitively. Nested radicals have many applications in Number Theory as well as in Numerical Methods.
Tag Archives: Nested Radicals
A Problem on Ordinary Nested Radicals
ProblemProve or disprove that |
Solution
In order to simplify the radicals, the radicands should be forced to equal square numbers (e.g., should be a square of some number). Numbers whose squares have a rational and radical part are usually in the form
.
So let
and set
and
Thus
which on simplification yields
And also
Thus,
Using the same process for other radicals:
Thus, now we can easily prove (by addition) that