A mathematician solved a 200-year-old problem that once defeated the greatest mathematical thinking in history, a traditional wisdom about what could happen to algebra.
Norman Wildberger of Sydney, Sydney, in collaboration with computer scientist Dean Rubine, has developed a revolutionary approach to solving polynomial equations of five and above – a legendary number of 1832 French mathematician évariste Galois.
“Our solution reopened a book that was previously closed in the history of mathematics,” Wilderberg explained in a seminal paper published in the American Mathematics Monthly.
The key innovation lies in rejecting traditional methods of relying on irrational numbers – those never-ending, non-repetitive decimals that cannot be accurately calculated. Instead, Wildberger developed a special extension of polynomials called Power Series, and a fascinating new numerical sequence they called “Geode.”
The sequence extends the famous Catalan number that will calculate the multidimensional array based on more complex polygon divisions as a way for triangles to be divided into triangles. This method reveals a surprising connection between geometric shapes, combinations and algebra that have been hidden for centuries.
“We have found these extensions and show how they lead to a general solution to polynomial equations. This is a dramatic revision of the basic chapters in algebra,” Wildberger noted.
When mathematician John Wallis used to test the famous cubic equations used by the Newtonian method in the 17th century, Wilderberg’s solution “works well.” In addition to theoretical interest, the method is expected to make practical progress in computational mathematics in many fields.
Innovation is more than just solving equations. The newly discovered array of numerical “Geode” appears to be the basis of the classic Catalan sequence itself, suggesting that deeper mathematical structures have avoided mathematicians so far.
“We hope that research on this new Geode array will raise many new questions and keep the combinationists busy for years,” Wildberger said. “Indeed, there are many other possibilities. This is just the beginning.”
For a major breakthrough that can separate centuries, Wilderberg’s work represents an amazing achievement-showing that revolutionary discoveries are still possible even in our oldest, most established branches of mathematics.
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