Polynomials are equations in which a variable is raised to different powers, like 1 + 4x—3x² = 0 (a degree-two polynomial). These equations are essential in math and science, helping explain things like planetary motion and computer algorithms.
However, solving more complex higher-order polynomials (where the variable is raised to the fifth power or higher) has always been difficult.
A UNSW mathematician recently developed a new method using unique number sequences to tackle this long-standing algebra problem.
The Babylonians developed a method for solving degree-two polynomials around 1800 BC, which later evolved into today’s quadratic formula. In the 16th century, mathematicians extended this approach to solve third- and fourth-degree polynomials using radicals (roots of numbers).
However, in 1832, Évariste Galois discovered that the symmetry used to solve lower-order polynomials doesn’t work for degree five or higher polynomials, proving that no general formula could solve them.
While approximate solutions are widely used, Prof. Wildberger argues that they don’t belong to pure algebra. His main issue is that traditional formulas rely on third or fourth roots (radicals), which often represent never-ending irrational numbers. He believes these numbers are problematic because they can never be fully computed.
Radicals often represent irrational numbers, meaning their decimal form never ends and never repeats. For example, ³√7 ≈ 1.9129118…, the exact value keeps going forever.
Prof. Wildberger argues that since these numbers can’t be fully computed, they aren’t actual mathematical objects. He claims that assuming ³√7 exists in a formula implies treating an infinite, never-ending number as something complete, which he rejects.
This is why he doesn’t believe in irrational numbers. Instead, he proposes alternative methods that avoid relying on them entirely.
Prof. Wildberger argues that irrational numbers rely on a flawed concept of infinity, creating logical mathematics problems. His skepticism led him to develop rational trigonometry and universal hyperbolic geometry, both of which avoid irrational numbers and traditional functions like sine and cosine.
Instead, they rely on basic mathematical operations such as squaring, adding, and multiplying. His new method for solving polynomials also avoids radicals and irrational numbers.
Instead, it uses power series, which are extended polynomials with infinite terms involving powers of x. To make calculations manageable, he truncates the power series, allowing for approximate numerical answers, confirming the method’s validity.
That’s a strong endorsement of Prof. Wildberger’s approach! Testing it on a historic cubic equation from the 17th century, particularly one used to showcase Newton’s method, adds credibility to his claim. The fact that his solution worked flawlessly suggests that his rejection of irrational numbers in algebra might be more practical than many had assumed.
“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.
Journal Reference:
- N. J. Wildberger et al, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, The American Mathematical Monthly (2025). DOI: 10.1080/00029890.2025.2460966
