Mathematician Solves Algebra's Oldest Problem

Newsweek

01-05-2025

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A mathematician has uncovered a way of answering some of algebra's oldest problems.
University of New South Wales Honorary Professor Norman Wildberger, has revealed a potentially game-changing approach to solving higher polynomial equations.
Polynomial equations involve a variable being raised to powers, such as the degree two polynomial: 1+ 4x - 3x2 = 0. Until now, a method for solving "higher order" polynomial equations, where x is raised to the power of five or higher, had proven elusive.
Wildberger has developed a fresh approach to the problem though using novel number sequences. His method was detailed in an article published in The American Mathematical Monthly co-authored with computer scientist Dr. Dean Rubine.
Wildberger's findings could have significant implications. Higher order polynomial equations play a fundamental role in both math and science, assisting in everything from writing computer programs to describing the movement of planets.
"Our solution reopens a previously closed book in mathematics history," Wildberger said in a statement.
A Brief History of Polynomials
Solutions to degree-two polynomials have been around since as far back as 1800 BC, when the Babylonians pioneered their "method of completing the square", which would evolve into the quadratic formula that became familiar to many high school math students.
Then in the 16th century, the approach of using roots of numbers dubbed "radicals" was extended to solve three- and four-degree polynomials.
In 1832, French mathematician Évariste Galois demonstrated how the mathematical symmetry behind the methods used to resolve lower-order polynomials became impossible for degree five and higher polynomials. He concluded that no general formula could solve them.
Though some approximate solutions for higher-degree polynomials have been developed in the years since, Wildberger contends that these don't belong to pure algebra.
Wildberger's 'Radical' Approach
He points to the issue of the classical formula's use of third or fourth roots, which are "radicals." These "radicals" tend to represent irrational numbers, decimals that extend to infinity without repeating and can't be written as simple fractions.
Wildberger said this makes it impossible to calculate the real answer as "you would need an infinite amount of work and a hard drive larger than the universe."
For example, the answer to the cubed root of seven, 3√7 = 1.9129118... extends forever. So although 3√7 "exists" in this formula, this infinite, never-ending decimal is being mischaracterized as a complete object.
Wildberger "doesn't believe in irrational numbers" as they rely on an imprecise concept of infinity and lead to logical problems in mathematics.
This rejection of radicals inspired Wildberger's best-known contributions to mathematics, rational trigonometry and universal hyperbolic geometry. These approaches rely on mathematical functions like squaring, adding, or multiplying, rather than irrational numbers, radicals, or functions like sine and cosine.
Wildberger's new approach to solving polynomials avoids radicals and irrational numbers, relying on special extensions of polynomials called "power series," which can have an infinite number of terms with the powers of x.
Wildberger found that by truncating the power series they were able to extract approximate numerical answers to check that the method worked.
He said: "One of the equations we tested was a famous cubic equation used by Wallis in the 17th century to demonstrate Newton's method. Our solution worked beautifully."
University of New South Wales Honorary Professor Norman Wildberger
University of New South Wales Honorary Professor Norman Wildberger
University of New South Wales
Wildberger's Mathematical Logic
Wildberger uses novel sequences of numbers that represent complex geometric relationships in his approach. These sequences belong to combinatorics, a facet of mathematics dealing with number patterns in sets of elements.
The Catalan numbers are the most famous combinatorics sequence, used to describe the number of ways you can dissect a polygon into triangles. The Catalan numbers have a number of important applications in practical life, whether it be computer algorithms, game theory or structure designs.
The Catalan numbers also play a role in biology, helping count the possible folding patterns of RNA molecules.
Wildberger said: "The Catalan numbers are understood to be intimately connected with the quadratic equation. Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers."
This approach sees the Catalan numbers enhanced from a one-dimensional to multi-dimensional array based on the number of ways a polygon can be divided using non-intersecting lines.
"We've found these extensions, and shown how, logically, they lead to a general solution to polynomial equations," Wildberger said. "This is a dramatic revision of a basic chapter in algebra."
Degree five polynomials, or quintics, also have a solution under Wildberger's approach.
Practical Applications of Wildberger's Method
Theoretical mathematics aside, Wildberger believes this new method could have significant promise when it comes to creating computer programs capable of solving equations using the algebraic series rather than radicals.
"This is a core computation for much of applied mathematics, so this is an opportunity for improving algorithms across a wide range of areas," he said.
The novel array of numbers, dubbed the "Geode" by Wildberger and co-author Rubine, offers significant potential for further research.
"We introduce this fundamentally new array of numbers, the Geode, which extends the classical Catalan numbers and seem to underlie them," Wildberger said.
"We expect that the study of this new Geode array will raise many new questions and keep combinatorialists busy for years. Really, there are so many other possibilities. This is only the start."
Newsweek has reached out to the University of New South Wales for comment.
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Reference
Wildberger, N. J., & and Rubine, D. (2025). A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode. The American Mathematical Monthly. https://doi.org/10.1080/00029890.2025.2460966