A University of New South Wales (UNSW) Sydney mathematician has revealed the first successful solution of an ‘impossible’ equation once considered unsolvable.
Described as algebra’s oldest problem, previous efforts to solve ‘higher order’ polynomial equations have consistently failed, leaving mathematicians without a critical tool. The new method solves that problem, potentially changing mathematics forever.
“Our solution reopens a previously closed book in mathematics history,” said UNSW Honorary Professor Norman Wildberger, who led the research.
How the Impossible Equation Was Finally Solved
According to a statement from the University announcing the solution to algebra’s oldest impossible equation, polynomials are represented by an established equation. For example, the degree two polynomial would be written as 1+ 4x – 3x2 = 0, where the variable “x” is raised to the second degree. However, the statement notes that whenever the variable is raised to five or higher, solving the equation has “historically proven elusive.”
While solutions for two-degree polynomials have existed since the ancient Babylonians discovered them in 1800 BCE, the equation and its limits were not discovered until 1832 by French mathematician Évariste Galois. According to the release, Galois determined that the impossible equation was the limit and that “no general formula could solve them.”
Since then, several “approximate” solutions for higher-order polynomials have been found. Still, according to Wildberger, these solutions do not belong to pure algebra.
In his newly published study outlining the novel solution, the professor explains that the limit to the traditional equation exists because the formula uses third and fourth roots, which are radical numbers. Since radicals generally represent irrational numbers, such as Pi, which extend to infinity without repeating, they cannot be represented as simple fractions. Wildberger says this makes it impossible to calculate higher-order polynomials in a traditional way, since “you would need an infinite amount of work and a hard drive larger than the universe”.
Fortunately for the mathematics community, the professor says he doesn’t believe in irrational numbers. Wildberger’s most successful contributions to mathematics, rational trigonometry and universal hyperbolic geometry, function without radicals. Instead, his work, including the new solution to an impossible equation, taps into special extensions of polynomials known as the “power series,” which can have an infinite number of terms for the variable x without using radicals.
To test whether or not his new solution to an impossible equation worked, Wildberger says he looked at a problem that has remained unsolved for centuries: a famous cubic equation used by Wallis in the 17th century to demonstrate Sir Isaac Newton’s method. When the centuries-old equation was tested using the new method, Wildberger could extract “approximate numerical answers,” proving that it works.
“Our solution worked beautifully,” the professor said.
Combinatorics to the Rescue
Although the first algebraic equation to solve the problem, the new method is based on a branch of mathematics called Combinatorics that represents sequences of numbers. The most famous of these sequences is called the “Catalan” numbers and is used to represent the number of ways you can dissect any polygon, or any shape with three or more sides.
“The Catalan numbers are understood to be intimately connected with the quadratic equation,” Wildberger explained. “Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers.”
Extending these numbers from one-dimensional to a multi-dimensional array, the mathematician found the elusive solution to a previously impossible equation.
“We’ve found these extensions and shown how, logically, they lead to a general solution to polynomial equations,” he explained. “This is a dramatic revision of a basic chapter in algebra.”
Promise for a Wide Range of Applications
Although the new solution offers theoretical interest to mathematicians, Wildberger says his approach could lead to computer programs designed to solve higher-order polynomials using an algebraic series instead of 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.”
The new study, co-authored by computer scientist Dr. Dean Rubine, names the newly discovered number array the “Geode.” The co-authors say this array could hold “vast potential” for further research.
“We expect that the study of this new Geode array will raise many new questions and keep combinatorialists busy for years,” Wildberger said. “Really, there are so many other possibilities. This is only the start.”
Christopher Plain is a Science Fiction and Fantasy novelist and Head Science Writer at The Debrief. Follow and connect with him on X, learn about his books at plainfiction.com, or email him directly at christopher@thedebrief.org.