Latest news with #Wildberger
Yahoo
15-05-2025
- Science
- Yahoo
Mathematicians May Have Solved Impossible Algebra Problem
Two mathematicians have used a new geometric approach in order to address a very old problem in algebra. In school, we often learn how to multiply out and factor polynomial equations like (x² – 1) or (x² + 2x + 1). In real life, these equations get very messy, very fast. In fact, mathematicians typically only approximate the solutions for ones above a certain value, known as higher degree (or higher order) polynomials. In this paper, however, the authors posit that they can use a metric from geometry called a Catalan number, or Catalan series, to find exact solutions to higher degree polynomials. The Catalan numbers are a natural observed consequence of a bunch of different mathematical scenarios, and can be found by engaging in such efforts as distilling Pascal's triangle of polynomial coefficients. They help graph theorists and computer scientists to plan data structures called trees by showing how many different tree arrangements can be made within certain parameters. In this case, they also quantify how many ways you can divide a polygon of any size into a particular number of triangles. The leading mind behind this work, mathematician Norman 'N.J.' Wildberger, is an honorary professor at the University of New South Wales in Australia—a term likely reflecting the fact that he retired in 2021 after teaching at the university since 1990. Wildberger also self identifies as a 'heretic' of certain mathematical foundations, exemplified in part by his longtime belief that we should stop using infinity or infinite concepts in some parts of math. That opposition to infinite or irrational numbers is key to this research. For many, many years now, people studying algebra have known that we simply 'can't' solve certain polynomials. They can't be taken apart into a mathematical term that fits under a square root (a.k.a. radical) sign at all. But, in Wildberger's view, focusing on this divide and dwelling inside the radical is a hindrance. We should 'sidestep' it altogether. To make this argument, Wildberger teamed up with Dean Rubine, a computer scientist who has worked for Bell Labs and Carnegie Mellon University. However, for decades now, Rubine has helped to lead the number-crunching at a secretive hedge fund that focuses on algorithms (more later on his role in this publication). The paper has a teacherly quality, reading somewhat like a chapter from a good textbook. The authors lay out and define their terms, then build their arguments one by one into a complete picture. What results is the 'hyper-Catalan' array, which contains the classic Catalan numbers as well as an extension that includes other numbers that satisfy the conditions to solve polynomials. (Remember, the hyper-Catalan number series doesn't need to line up with all the other uses of the Catalan numbers—rather, the Catalans are a basis from which to begin building a unique set that solves the polynomial problem.) This all wrapped up in an array called the Geode, which encompasses the entirety of the hyper-Catalan number series. After stepping through the work leading up to and including the Geode array, Wildberger gets in one last jab: [F]ormal power series give algebraic and combinatorially explicit alternatives to functions which cannot actually be concretely evaluated (such as nth root functions). Hence they ought to assume a more central position. This is a solid, logical way of removing many of the infinities which currently abound in our mathematical landscape. Having been authored by an aging iconoclast and a longtime quantitative executive, this work may have more of an uphill climb to be broadly recognized. It's also published in the peer reviewed American Mathematical Monthly—a broad interest journal associated with the Mathematical Association of America. The journal accepts advertisers, offers paid editing services, and offers an option where authors can pay to make their articles open access—often a few thousand dollars or more. (The latter is, unfortunately, the normalized model and cost of open access publishing.) In this case, this less-orthodox approach could be a result of the subject matter simply not being on most people's radar anymore. But it also fits right into Wildberger's lifelong quest to trim the mathematical fat and present clear, simple ideas for as many people as possible. On the tech forum Hacker News (from startup incubator Y Combinator), Rubine explained in a post that he'd closely followed Wildberger's work on this problem since 2021, when Wildberger declared he was going to solve this problem on his YouTube channel. '[H]e was doing a series where he'd teach amateurs how to do math research,' Rubine said. 'For the first problem, he said he'd solve the general polynomial. I thought it was a joke, because everybody 'knows' that we can't go beyond degree four. But no, 41 videos later, he had done it. Two years after that he still hadn't written it up, so I wrote a draft and sent it to him, which evolved into this paper.' With that kind of determination, Wildberger may, after all, be an apt opponent for infinity itself. His democratic, open-door approach to mathematical thinking is really admirable. And in the paper, he and Rubine point out a number of questions that this theory opens up. We'll see if others in the mathematics community pick up some of these questions. I, for one, hope so, because 41 more videos is a long time to wait for the next breakthrough. You Might Also Like Can Apple Cider Vinegar Lead to Weight Loss? Bobbi Brown Shares Her Top Face-Transforming Makeup Tips for Women Over 50
Yahoo
13-05-2025
- Science
- Yahoo
Mathematicians Thought This Algebra Problem Was Impossible. Two Geniuses May Have Found a Solution.
Two mathematicians have used a new geometric approach in order to address a very old problem in algebra. In school, we often learn how to multiply out and factor polynomial equations like (x² – 1) or (x² + 2x + 1). In real life, these equations get very messy, very fast. In fact, mathematicians typically only approximate the solutions for ones above a certain value, known as higher degree (or higher order) polynomials. In this paper, however, the authors posit that they can use a metric from geometry called a Catalan number, or Catalan series, to find exact solutions to higher degree polynomials. The Catalan numbers are a natural observed consequence of a bunch of different mathematical scenarios, and can be found by engaging in such efforts as distilling Pascal's triangle of polynomial coefficients. They help graph theorists and computer scientists to plan data structures called trees by showing how many different tree arrangements can be made within certain parameters. In this case, they also quantify how many ways you can divide a polygon of any size into a particular number of triangles. The leading mind behind this work, mathematician Norman 'N.J.' Wildberger, is an honorary professor at the University of New South Wales in Australia—a term likely reflecting the fact that he retired in 2021 after teaching at the university since 1990. Wildberger also self identifies as a 'heretic' of certain mathematical foundations, exemplified in part by his longtime belief that we should stop using infinity or infinite concepts in some parts of math. That opposition to infinite or irrational numbers is key to this research. For many, many years now, people studying algebra have known that we simply 'can't' solve certain polynomials. They can't be taken apart into a mathematical term that fits under a square root (a.k.a. radical) sign at all. But, in Wildberger's view, focusing on this divide and dwelling inside the radical is a hindrance. We should 'sidestep' it altogether. To make this argument, Wildberger teamed up with Dean Rubine, a computer scientist who has worked for Bell Labs and Carnegie Mellon University. However, for decades now, Rubine has helped to lead the number-crunching at a secretive hedge fund that focuses on algorithms (more later on his role in this publication). The paper has a teacherly quality, reading somewhat like a chapter from a good textbook. The authors lay out and define their terms, then build their arguments one by one into a complete picture. What results is the 'hyper-Catalan' array, which contains the classic Catalan numbers as well as an extension that includes other numbers that satisfy the conditions to solve polynomials. (Remember, the hyper-Catalan number series doesn't need to line up with all the other uses of the Catalan numbers—rather, the Catalans are a basis from which to begin building a unique set that solves the polynomial problem.) This all wrapped up in an array called the Geode, which encompasses the entirety of the hyper-Catalan number series. After stepping through the work leading up to and including the Geode array, Wildberger gets in one last jab: [F]ormal power series give algebraic and combinatorially explicit alternatives to functions which cannot actually be concretely evaluated (such as nth root functions). Hence they ought to assume a more central position. This is a solid, logical way of removing many of the infinities which currently abound in our mathematical landscape. Having been authored by an aging iconoclast and a longtime quantitative executive, this work may have more of an uphill climb to be broadly recognized. It's also published in the peer reviewed American Mathematical Monthly—a broad interest journal associated with the Mathematical Association of America. The journal accepts advertisers, offers paid editing services, and offers an option where authors can pay to make their articles open access—often a few thousand dollars or more. (The latter is, unfortunately, the normalized model and cost of open access publishing.) In this case, this less-orthodox approach could be a result of the subject matter simply not being on most people's radar anymore. But it also fits right into Wildberger's lifelong quest to trim the mathematical fat and present clear, simple ideas for as many people as possible. On the tech forum Hacker News (from startup incubator Y Combinator), Rubine explained in a post that he'd closely followed Wildberger's work on this problem since 2021, when Wildberger declared he was going to solve this problem on his YouTube channel. '[H]e was doing a series where he'd teach amateurs how to do math research,' Rubine said. 'For the first problem, he said he'd solve the general polynomial. I thought it was a joke, because everybody 'knows' that we can't go beyond degree four. But no, 41 videos later, he had done it. Two years after that he still hadn't written it up, so I wrote a draft and sent it to him, which evolved into this paper.' With that kind of determination, Wildberger may, after all, be an apt opponent for infinity itself. His democratic, open-door approach to mathematical thinking is really admirable. And in the paper, he and Rubine point out a number of questions that this theory opens up. We'll see if others in the mathematics community pick up some of these questions. I, for one, hope so, because 41 more videos is a long time to wait for the next breakthrough. You Might Also Like The Do's and Don'ts of Using Painter's Tape The Best Portable BBQ Grills for Cooking Anywhere Can a Smart Watch Prolong Your Life?
Yahoo
05-05-2025
- Science
- Yahoo
Mathematician Finds Solution to One of The Oldest Problems in Algebra
Solving one of the oldest algebra problems isn't a bad claim to fame, and it's a claim Norman Wildberger can now make: The mathematician has solved what are known as higher-degree polynomial equations, which have been puzzling experts for nearly 200 years. Wildberger, from the University of New South Wales (UNSW) in Australia, worked with computer scientist Dean Rubine on a paper that details how these incredibly complex calculations could be worked out. "This is a dramatic revision of a basic chapter in algebra," says Wildberger. "Our solution reopens a previously closed book in mathematics history." As you might expect, understanding how this works isn't easy for the non-algebra geniuses amongst us. Essentially, polynomials are equations that include variables raised to non-negative powers (e.g. x3). When those powers are five or above, that's a higher-degree polynomial. Mathematicians have figured out how to solve lower-degree versions, but it was thought that properly calculating the higher-degree ones was impossible. Before this new research, we've been relying on approximations. Wildberger and Rubine took a new approach to the problem, which is based on Catalan numbers. These numbers are used in advanced number counting and arrangements, including counting how many ways polygons can be subdivided into triangles. By extending the idea of Catalan numbers, the researchers were able to demonstrate that they could be used as a basis for solving polynomial equations of any degree. Part of the clever method involved extending polygon counts to other shapes besides triangles. It's a departure from the traditional method of using radical expressions (like square roots and cube roots) to solve equations like this, instead relying on combinatorics – counting numbers, fundamentally, but in increasingly advanced ways. "The Catalan numbers are understood to be intimately connected with the quadratic equation," says Wildberger. "Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogs of the Catalan numbers." The researchers put their new algebra up against some well-known polynomial equations of the past, including a famous cubic equation studied by John Wallis. The numbers checked out, validating the new work. Wildberger and Rubine didn't stop there. They also discovered a new mathematical structure called the Geode, which ties in with Catalan numbers and seems to act as a foundation for them. This Geode could form the basis of many future studies and discoveries, the researchers say. As the approach taken here is so different to what's gone before, there's the potential to rethink many key ideas that mathematicians have long relied on for computer algorithms, the way data is structured, and game theory. It might even have applications in biology – for counting RNA molecule folding, for example. "This is a core computation for much of applied mathematics, so this is an opportunity for improving algorithms across a wide range of areas," says Wildberger. The research has been published in The American Mathematical Monthly. Scientists Build First-Ever 'Black Hole Bomb' Analog Gravity May Be a Clue That The Universe Is a Giant Computer Scientists Just Confirmed a 67-Year-Old Hypothesis About Vitamin B1
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.
