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Weird science: Algebra's oldest problem solved using intriguing new number sequences

By Elva Darnell, University of New South Wales - Sydney

A University of New South Wales (UNSW) mathematician has discovered a new method to tackle algebra's oldest challenge: solving higher polynomial equations.

Polynomials are equations involving a variable raised to powers, such as the degree-two polynomial: 1 + 4x - 3x² = 0. The equations are fundamental to mathematics as well as science, where they have broad applications, such as helping describe the movement of planets or writing computer programs.

However, a general method for solving '"higher order" polynomial equations, where x is raised to the power of five or higher, has historically proven elusive.

Now, UNSW Honorary Professor Norman Wildberger has revealed a new approach using novel number sequences, outlined in a recent publication with computer scientist Dr. Dean Rubine.

"Our solution reopens a previously closed book in mathematics history," Prof. Wildberger says.

Mathematician Norman Wildberger, who is known for his work in rational trigonometry and universal hyperbolic geometry, says the new polynomial method has both theoretical and practical applications. [Credit: Photo by Elva Darnell, UNSW]

 

 

 

 

The polynomial problem
Solutions to degree-two polynomials have been around since 1,800 BC, thanks to the Babylonians' "method of completing the square," which evolved into the quadratic formula familiar to many high school math students. This approach, using roots of numbers called "radicals," was later extended to solve three- and four-degree polynomials in the 16th century.

Then, in 1832, French mathematician Evariste Galois showed how the mathematical symmetry behind the methods used to resolve lower-order polynomials became impossible for degree-five and higher polynomials. Therefore, he figured, no general formula could solve them.

Approximate solutions for higher-degree polynomials have since been developed and are widely used in applications. However, says Prof. Wildberger, these don't belong to pure algebra.

Radical rejection behind new method
The issue, he says, lies in the classical formula's use of third or fourth roots, which are radicals.

The radicals generally represent irrational numbers, which are decimals that extend to infinity without repeating and can't be written as simple fractions. For instance, the answer to the cubed root of seven, 3√7 = 1.9129118 ... extends forever.

Prof. Wildberger says this means that the real answer can never be completely calculated, because "You would need an infinite amount of work and a hard drive larger than the universe."

So, when we assume 3√7 "exists" in a formula, we're assuming that this infinite, never-ending decimal is somehow a complete object.

This is why, Prof. Wildberger says, he "doesn't believe in irrational numbers."

Irrational numbers, he says, rely on an imprecise concept of infinity and lead to logical problems in mathematics.

Prof. Wildberger's rejection of radicals inspired his best-known contributions to mathematics, rational trigonometry, and universal hyperbolic geometry. Both approaches rely on mathematical functions such as squaring, adding, or multiplying, rather than irrational numbers, radicals, or functions like sine and cosine.

His new method to solve polynomials also avoids radicals and irrational numbers, relying instead on special extensions of polynomials called "power series," which can have an infinite number of terms with the powers of x.

By truncating the power series, Prof. Wildberger says, they were able to extract approximate numerical answers to check that the method worked.

VIDEO: Prof. Norman Wildberger explains his latest paper on his popular YouTube channel, Insights into Mathematics. [Credit: Insights into Mathematics/N J Wildberger]

"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," he said.

New geometry for a general solution
However, Prof. Wildberger says, the proof for the method is, ultimately, based on mathematical logic.

His method uses novel sequences of numbers that represent complex geometric relationships. These sequences belong to combinatorics, a branch of mathematics that deals with number patterns in sets of elements.

The most famous combinatorics sequence, called the Catalan numbers, describes the number of ways you can dissect a polygon, which is any shape with three or more sides, into triangles.

The numbers have important practical applications, including in computer algorithms, data structure designs, and game theory. They even appear in biology, where they are used to help count the possible folding patterns of RNA molecules. They can also be calculated using a simple two-degree polynomial.

"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."

Prof. Wildberger's work extends these Catalan numbers 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," he says. "This is a dramatic revision of a basic chapter in algebra."

Even quintics -- a degree-five polynomial -- now have solutions, he says.

Aside from theoretical interest, he says, the method holds practical promise for creating computer programs that can solve 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 says.

Geode's unexplored facets
Prof. Wildberger says the novel array of numbers, which he and Dr. Rubine called the "Geode," also holds vast 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.

"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," he says.

Published May 2025

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