The purpose of Series Math Study (SMS) website is to explore and develop the beauty of series formulas that have connections to special mathematical constants. Some series can be used to calculate ever more accurate values of some special math constants as the desire that has challenged mathematicians for many centuries.
Submitted by admin on Wed, 01/03/2024 - 10:08pm
3/3/2024
We are pleased to inform you that the book titled "Embarking on Adventurous Journeys Through Some Subjects in Pure Mathematics" has been published on Amazon at this link, https://www.amazon.com/Embarking-Adventurous-Journeys-Subjects-Mathematics/dp/B0CWF7FZWH (kindle readers - https://www.amazon.com/dp/B0CW1KVN76 ). The publication focuses on two elementary mathematical subjects: the exploration of nested radicals and algebraic identities, and the art of solving polynomial equations using of trigonometric and hyperbolic trigonometric functions. The first subject unveils beautiful nested radical identities in algebra, along with essential known formulas in mathematics. Meanwhile, the second subject unveils an innovative approach to solving polynomial equations, encompassing quadratic to quartic and other higher order solvable equations, by utilizing trigonometric and hyperbolic trigonometric functions.
On this website, we present a summarized series of elegant algebraic identities and solutions to polynomial equations, each rigorously developed and proven in this book.
A. Elegant Nested Radical Identities
The algebraic identities demonstrate how two or more nested radicals can be combined into a single nested radical, whether they are added or subtracted—a transformation that is both mathematically profound and visually striking. These identities are not merely theoretical; they have practical applications in simplifying expressions and solving equations. The beauty of such identities lies in their elegance, simplicity, and the surprising relationships they reveal among seemingly intricate expressions.
1. Combination of Two Nested Radicals into a Single Nested Radical
,
where a, b and x are real numbers.
2. Complex Fourth Root
,
where a and b are real numbers.
3. Radical expression for the quarter of the lemniscate sine
B. Elegant Solutions to Cubic, Quartic, Special quintic, and Certain Higher-Degree Equations
As introduced in the first section, we have developed a systematic and consistent method for solving cubic, quartic, special quintic, and select higher-degree equations. This rigorously proven approach encompasses several groundbreaking discoveries. It marks a significant advancement, distinguished by its refinement, elegance, and novel insights. Here, we present only the summarized solutions to certain polynomial equations.
1. Solution to the cubic equation
2. Solution to the quartic equation
3. Solution to the special quintic equation
4. Solution to the special sextic equation
5. Solution to the special septic equation
6. Solution to the special octic equation
7. Solution to the special 9th degree equation
for k = 0, 1, 2.
Some formulas and solutions presented above are incomplete. For further details, please refer to the book.
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