Latest news with #NormanWildberger
Yahoo
03-05-2025
- Science
- Yahoo
Mathematicians devise new way to solve devilishly difficult algebra equations
When you buy through links on our articles, Future and its syndication partners may earn a commission. Polynomial equations are a cornerstone of modern science, providing a mathematical basis for celestial mechanics, computer graphics, market growth predictions and much more. But although most high schoolers know how to solve simple polynomial equations, the solutions to higher-order polynomials have eluded even seasoned mathematicians. Now, University of New South Wales mathematician Norman Wildberger and independent computer scientist Dean Rubine have found the first general method for solving these devilishly difficult equations. They detailed their approach April 8 in the journal The American Mathematical Monthly. A polynomial is a type of algebraic equation that involves variables raised to a non-negative power — for example, x² + 5x + 6 = 0. It is among the oldest mathematical concepts, tracing its roots back to ancient Egypt and Babylon. Mathematicians have long known how to solve simple polynomials. However, higher-order polynomials, where x is raised to a power greater than four, have proved trickier. The approach most often used to solve two-, three- and four-degree polynomials relies on using the roots of exponential numbers, called radicals. The problem is that radicals often represent irrational numbers — decimals that keep going to infinity, like pi. Related: Mathematicians just solved a 125-year-old problem, uniting 3 theories in physics Although mathematicians can use radicals to find approximate solutions to individual higher-order polynomials, they have struggled to find a general formula that works for all of them. That's because irrational numbers can never fully resolve. "You would need an infinite amount of work and a hard drive larger than the universe," Wildberger said in a statement. In their new method, Wildberger and his colleagues avoided radicals and irrational numbers entirely. Instead, they employed polynomial extensions known as power series. These are hypothetically infinite strings of terms with the powers of x, commonly used to solve geometric problems. They belong to a sub branch of mathematics known as combinatorics. RELATED STORIES —Mathematicians solve vexing 'crowd problem' that explains why public spaces devolve into chaos —14-year-old known as 'the human calculator' breaks 6 math world records in 1 day —High school students who came up with 'impossible' proof of Pythagorean theorem discover 9 more solutions to the problem The mathematicians based their approach on the Catalan numbers, a sequence that can be used to describe the number of ways to break down a polygon into triangles. This sequence was first delineated by Mongolian mathematician Mingantu around 1730 and was independently discovered by Leonhard Euler in 1751. Wildberger and Rubine realized that they could look to higher analogues of the Catalan numbers to solve higher-order polynomial equations. They called this extension "the Geode." The Geode has numerous potential applications for future research, especially in computer science and graphics. "This is a dramatic revision of a basic chapter in algebra," Wildberger said.
Yahoo
01-05-2025
- Science
- Yahoo
Mathematician solves algebra's oldest problem
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.
Newsweek
01-05-2025
- Science
- Newsweek
Mathematician Solves Algebra's Oldest Problem
Based on facts, either observed and verified firsthand by the reporter, or reported and verified from knowledgeable sources. Newsweek AI is in beta. Translations may contain inaccuracies—please refer to the original content. A mathematician has uncovered a way of answering some of algebra's oldest problems. University of New South Wales Honorary Professor Norman Wildberger, has revealed a potentially game-changing approach to solving higher polynomial equations. Polynomial equations involve a variable being raised to powers, such as the degree two polynomial: 1+ 4x - 3x2 = 0. Until now, a method for solving "higher order" polynomial equations, where x is raised to the power of five or higher, had proven elusive. Wildberger has developed a fresh approach to the problem though using novel number sequences. His method was detailed in an article published in The American Mathematical Monthly co-authored with computer scientist Dr. Dean Rubine. Wildberger's findings could have significant implications. Higher order polynomial equations play a fundamental role in both math and science, assisting in everything from writing computer programs to describing the movement of planets. "Our solution reopens a previously closed book in mathematics history," Wildberger said in a statement. A Brief History of Polynomials Solutions to degree-two polynomials have been around since as far back as 1800 BC, when the Babylonians pioneered their "method of completing the square", which would evolve into the quadratic formula that became familiar to many high school math students. Then in the 16th century, the approach of using roots of numbers dubbed "radicals" was extended to solve three- and four-degree polynomials. In 1832, French mathematician Évariste Galois demonstrated how the mathematical symmetry behind the methods used to resolve lower-order polynomials became impossible for degree five and higher polynomials. He concluded that no general formula could solve them. Though some approximate solutions for higher-degree polynomials have been developed in the years since, Wildberger contends that these don't belong to pure algebra. Wildberger's 'Radical' Approach He points to the issue of the classical formula's use of third or fourth roots, which are "radicals." These "radicals" tend to represent irrational numbers, decimals that extend to infinity without repeating and can't be written as simple fractions. Wildberger said this makes it impossible to calculate the real answer as "you would need an infinite amount of work and a hard drive larger than the universe." For example, the answer to the cubed root of seven, 3√7 = 1.9129118... extends forever. So although 3√7 "exists" in this formula, this infinite, never-ending decimal is being mischaracterized as a complete object. Wildberger "doesn't believe in irrational numbers" as they rely on an imprecise concept of infinity and lead to logical problems in mathematics. This rejection of radicals inspired Wildberger's best-known contributions to mathematics, rational trigonometry and universal hyperbolic geometry. These approaches rely on mathematical functions like squaring, adding, or multiplying, rather than irrational numbers, radicals, or functions like sine and cosine. Wildberger's new approach to solving polynomials avoids radicals and irrational numbers, relying on special extensions of polynomials called "power series," which can have an infinite number of terms with the powers of x. Wildberger found that by truncating the power series they were able to extract approximate numerical answers to check that the method worked. He said: "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." University of New South Wales Honorary Professor Norman Wildberger University of New South Wales Honorary Professor Norman Wildberger University of New South Wales Wildberger's Mathematical Logic Wildberger uses novel sequences of numbers that represent complex geometric relationships in his approach. These sequences belong to combinatorics, a facet of mathematics dealing with number patterns in sets of elements. The Catalan numbers are the most famous combinatorics sequence, used to describe the number of ways you can dissect a polygon into triangles. The Catalan numbers have a number of important applications in practical life, whether it be computer algorithms, game theory or structure designs. The Catalan numbers also play a role in biology, helping count the possible folding patterns of RNA molecules. Wildberger said: "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." This approach sees the Catalan numbers enhanced 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," Wildberger said. "This is a dramatic revision of a basic chapter in algebra." Degree five polynomials, or quintics, also have a solution under Wildberger's approach. Practical Applications of Wildberger's Method Theoretical mathematics aside, Wildberger believes this new method could have significant promise when it comes to creating computer programs capable of solving 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 said. The novel array of numbers, dubbed the "Geode" by Wildberger and co-author Rubine, offers significant 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," Wildberger said. "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." Newsweek has reached out to the University of New South Wales for comment. Do you have a tip on a science story that Newsweek should be covering? Do you have a question about algebra? Let us know via science@ Reference Wildberger, N. J., & and Rubine, D. (2025). A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode. The American Mathematical Monthly.
