One of the unsolved problems in the history of algebra for two centuries may have been resolved with a new method developed by a mathematician from the University of New South Wales (UNSW) in Sydney.
The method developed by Professor Norman Wildberger challenges the traditional approach to solving fifth-degree and higher polynomial equations.
Polynomials are equations where variables (such as x) are raised to different powers. These equations are used in various fields, not just theoretical mathematics but also in calculating planetary movements and software coding. However, a universal solution formula for high-degree polynomials, such as those involving the fifth power of x and beyond, had not been developed until now.
In a paper co-published by UNSW Honorary Professor Norman Wildberger and computer scientist Dr. Dean Rubine, an innovative approach is proposed that avoids "radical" numbers, that is, irrational numbers.
Quadratic equations began being solved with the "completing the square" method in ancient Babylon around 1800 BCE, and this method eventually evolved into the classic quadratic formula taught at the high school level. In the 16th century, this technique was applied to cubic and quartic equations. However, in 1832, French mathematician Évariste Galois proved that a general solution formula for equations of degree five and higher was impossible. After this, approximate solutions were developed, but according to Wildberger, these solutions were outside of pure algebraic methods.
Professor Wildberger argues that the solution should be developed without using radicals—i.e., expressions involving roots. According to him, expressions like the cube root of seven are based on irrational numbers, which can never be exactly calculated due to infinite decimals. Wildberger notes that irrational numbers lead to logical issues in mathematics.
This perspective was foundational in his previously developed fields such as "rational trigonometry" and "universal hyperbolic geometry." In the new method, "power series," which are infinite extensions of polynomials, are used. These series are truncated at a specific point, allowing the solution's accuracy to be checked with approximate numbers.
Wildberger's method, unlike classical solution methods, operates through combinatorial number sequences. Especially using "Catalan numbers," which explain how polygons are divided into triangles, multidimensional extensions of these numbers were developed.
This new sequence of numbers, called "Geode," is said to offer a general solution to high-degree polynomials, including fifth-degree equations.
Wildberger emphasized that the method they developed could have extensive applications in both theoretical and applied mathematics. The new method may allow for the solution of equations in computer programs using power series instead of radicals.
The Geode sequence is expected to open the door to many new areas of research in the field of mathematical combinatorics.
Professor Wildberger stated, "By introducing the Geode sequence, we have extended the classical Catalan numbers. We believe this discovery represents a fundamental revision in a key area of algebra. This is just the beginning—there's much more to explore.
