At a Glance
- Mathematicians from UNSW Sydney have developed a novel method to solve higher-degree polynomial equations without relying on irrational numbers or radicals, which traditionally complicate such calculations.
- These equations, essential in fields like physics and computing, have long resisted general solutions beyond degree four due to limitations proven by 19th-century mathematician Évariste Galois.
- The breakthrough uses extended versions of Catalan numbers and introduces power series to find approximate solutions to previously unsolvable polynomial equations.
- This method successfully tackles quintic equations and promises to enhance computational efficiency in solving complex algebraic problems without invoking complex or irrational values.
- The innovation also introduces the “Geode” array, offering a foundation for future discoveries that could transform algebra and its applications in science and technology.
Mathematicians have made a breakthrough in solving higher-degree polynomial equations, a challenge that has puzzled scholars for centuries. In a new study published in Nature Communications, researchers from UNSW Sydney, led by Professor Norman Wildberger, introduced a novel method to solve polynomial equations without using “irrational numbers” or “radicals,” values that cannot be expressed as simple fractions. These radicals have caused many complications in traditional algebraic methods, especially for degree five or higher equations.
Polynomial equations are crucial in many areas of science and technology, as they help describe everything from planet motion to computer algorithms. While mathematicians have known how to solve quadratic equations (those of degree two) for centuries, solving higher-degree equations has long been difficult. For example, in the 19th century, mathematician Évariste Galois proved that there was no general formula for solving polynomials of degree five or higher using traditional methods. Wildberger’s new approach, however, offers a way to solve these complex equations.
The breakthrough relies on new mathematical sequences that extend the well-known Catalan numbers, a sequence used to count the number of ways polygons can be divided into triangles. Wildberger and his team applied these sequences to develop a new method for solving higher-degree polynomials. Unlike previous approaches that used complex numbers and irrational values, their method uses “power series,” consisting of terms that can be added to approximate solutions. The result? A clean, practical solution to equations that were once thought unsolvable.
The new technique solves quintic (degree five) equations and can improve computer programs and algorithms. Wildberger believes that the new “Geode” array, which builds on the Catalan numbers, could lead to even more discoveries in the future. “This is just the beginning,” he says, indicating that this innovation could change how mathematicians approach algebra and revolutionize various fields of applied mathematics.
References
- Wildberger, N. J., & Rubine, D. (2025). A hyper-catalan series solution to polynomial equations, and the geode. The American Mathematical Monthly, 132(5), 383–402. https://doi.org/10.1080/00029890.2025.2460966
- University of New South Wales. (2025, May 1). Mathematician solves algebra’s oldest problem using intriguing new number sequences. Phys.Org; University of New South Wales. https://phys.org/news/2025-05-mathematician-algebra-oldest-problem-intriguing.html
