Mathematician solves algebra's oldest problem

Yahoo

01-05-2025

Most people's experiences with polynomial equations don't extend much further than high school algebra and the quadratic formula. Still, these numeric puzzles remain a foundational component of everything from calculating planetary orbits to computer programming. Although solving lower order polynomials—where the x in an equation is raised up to the fourth power—is often a simple task, things get complicated once you start seeing powers of five or greater. For centuries, mathematicians accepted this as simply an inherent challenge to their work, but not Norman Wildberger. According to his new approach detailed in The American Mathematical Monthly, there's a much more elegant approach to high order polynomials—all you need to do is get rid of pesky notions like irrational numbers.
Babylonians first conceived of two-degree polynomials around 1800 BCE, but it took until the 16th century for mathematicians to evolve the concept to incorporate three- and four-degree variables using root numbers, also known as radicals. Polynomials remained there for another two centuries, with larger examples stumping experts until in 1832. That year, French mathematician Évariste Galois finally illustrated why this was such a problem—the underlying mathematical symmetry in the established methods for lower-order polynomials simply became too complicated for degree five or higher. For Galois, this meant there just wasn't a general formula available for them.
Mathematicians have since developed approximate solutions, but they require integrating concepts like irrational numbers into the classical formula.
To calculate such an irrational number, 'you would need an infinite amount of work and a hard drive larger than the universe,' explained Wildberger, a mathematician at the University of New South Wales Sydney in Australia.
This infinite number of possibilities is the fundamental issue, according to Wildberger. The solution? Toss out the entire concept.
'[I don't] believe in irrational numbers,' he said.
Instead, his approach relies on mathematical functions like adding, multiplying, and squaring. Wildberger recently approached this challenge by turning to specific polynomial variants called 'power series,' which possess infinite terms within the powers of x. To test it out, he and computer scientist Dean Rubine used 'a famous cubic equation used by Wallis in the 17th century to demonstrate Newton's method.'
You don't need to try wrapping your head around all that, however. Just trust Wildberger when he said the solution 'worked beautifully.'
The same goes for Catalan numbers, a famous sequence of numbers that describes the number of ways to dissect any given polygon. These also appear in the natural world in areas like biology, where they are employed to analyze possible folding patterns of RNA molecules.
'The Catalan numbers are understood to be intimately connected with the quadratic equation,' explained Wildberger. 'Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers.'
Outside of headspinning concepts on paper, Wildberger believes the new approach to higher power polynomials could soon result in computer programs capable of solving equations without the need for radicals. It may also help improve algorithms across a variety of fields.
'This is a dramatic revision of a basic chapter in algebra,' argued Wildberger.
Luckily, none of this will be your next pop quiz.