Math

A new proof significantly strengthens a decades-old result about the ubiquity of ways to represent whole numbers as sums of fractions.

How the modern world arose from imaginary numbers.

Three mathematicians show, for the first time, how to form a square with the same area as a circle by cutting them into interchangeable pieces that can be visualized.

Mathematical tools that proved essential during the pandemic were in many cases invented by mathematicians who had no particular goal in mind.

A pair of mathematicians solved a legendary question about the proportion of vertices in a graph with an odd number of connections.

Originally devised as a rigorous means of counting holes, homology provides a scaffolding for mathematical ideas, allowing for a new way to analyze the shapes within data.

Two teams found different ways for quantum computers to process nonlinear systems by first disguising them as linear ones.

The goal of the “busy beaver” game is to find the longest-running computer program. Its pursuit has surprising connections to some of the most profound questions and concepts in mathematics.

At 21, Ashwin Sah has produced a body of work that senior mathematicians say is nearly unprecedented for a college student.

For almost a century, the anonymous members of Nicolas Bourbaki have written books intended as pure expressions of mathematical thought.

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.

Computer scientists are trying to build an AI system that can win a gold medal at the world’s premier math competition.

Reprinted with permission from Quanta Magazine‘s Abstractions blog. The 61st International Mathematical Olympiad, or IMO, began yesterday. It may go down in history for at least two reasons: Due to the COVID-19 pandemic it’s the first time the event has been held remotely, and it may also be the last time that artificial intelligence doesn’t compete. […]

Two computer scientists found—in the unlikeliest of places—just the idea they needed to make a big leap in graph theory.

A past generation’s glory can be the next generation’s headache.

Three mathematicians have resolved a fundamental question about straight paths on the 12-sided Platonic solid.

A powerful technique called SAT solving could work on the notorious Collatz conjecture. But it’s a long shot.

Emperor penguins display rigorously geometric spacing and mathematical efficiency when they huddle together for warmth, which may reveal secrets to their overall health.

Why do mathematicians enjoy proving the same results in different ways?

His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.

Herd immunity differs from place to place, and many factors influence how it’s calculated.

Representation theory was initially dismissed. Today, it’s central to much of mathematics.

By using this methodology, combined with available information from Google maps to measure store size and popularity, one can model the best time to go shopping in a pandemic.Photograph by Diego Cervo / Shutterstock Now is not the time to go to the grocery store, to restock the pantry, to get fresh milk and eggs. […]

To distinguish between fundamentally different objects, mathematicians turn to invariants that encode the objects’ essential features.

Mathematicians try to figure out when problems can be solved using current knowledge—and when they have to chart a new path instead.

Studies of collective behavior usually focus on how crowds of organisms coordinate their actions. But what if the individuals that don’t participate have just as much to tell us?

Like earthquakes and forest fires, outbreaks have a “heavy tail” of large events.

Modern mathematics relies on collaboration and travel. COVID-19 is making it increasingly difficult.

The legendary mathematician, who died on April 11, 2020 was curious, colorful and one of the greatest problem-solvers of his generation.

A landmark proof in computer science has also solved an important problem called the Connes embedding conjecture. Mathematicians are working to understand it.

“Rainbow colorings” recently led to a new proof. It’s not the first time they’ve come in handy.

How an aerial-combat theory can help fight the outbreak.

Reprinted with permission from Quanta Magazine’s Abstractions blog. One sunny day in the summer of 2019, Mathias Kolle, a professor at the Massachusetts Institute of Technology, took a couple of eminent colleagues out sailing. They talked about their research. They had some drinks. Then Kolle noticed something was off: A rowboat tied to his boat had […]

Mathematicians have studied knots for centuries, but a new material is showing why some knots are better than others.

A data analyst uncovers a timeless message about correlation.

A curious physicist has discovered an unexpected link between theoretical block collisions and a famed quantum search algorithm.

Bringing the Drake Equation up to date.

A game theorist breaks down the effects of inequality.

The revolution of machine learning has been greatly exaggerated.

The goal of a 15 puzzle is to put numbered tiles in order. Now mathematicians have solved the opposite problem — how to scramble one.

A major advance toward solving the 60-year-old sunflower conjecture is shedding light on how order begins to appear as random systems grow in size.

It’s time to change the way uncertainty is quantified.

Randomness would seem to make a mathematical statement harder to prove. In fact, it often does the opposite.

Are you as good at Scrabble as you think?

Over a decade ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn’t a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, […]

The building blocks of understanding are memorization and repetition.

In practice, quantum computers can’t run many programs that classical computers can, because they’re not allowed to selectively forget information. A new algorithm for multiplication shows a way around that problem.

Nassim Nicholas Taleb has overplayed his hand this time and is left looking, well, klueless.Photograph by Salzburg Global Seminar / Flickr Since the midterm elections, a feud has been raging on Twitter between Nate Silver, founder of FiveThirtyEight, and Nassim Nicholas Taleb, hedge-fund-manager-turned-mathematical-philosopher and author of The Black Swan. It began, late last year, with […]

The sum-of-three-cubes problem solved for “stubborn” number 33.

With a better understanding of probability, we won’t be misled.

Recent progress on the “sum product” problem recalls a celebrated mathematical result that revealed the power of miniature number systems.

How a math professor learned to stop worrying and love algebraic geometry.

Digital security depends on the difficulty of factoring large numbers. A new proof shows why one method for breaking digital encryption won’t work.

An experiment in controlling how much of Scrabble is luck.

Alexander Smith’s work on the Goldfeld conjecture reveals fundamental characteristics of elliptic curves.

In math, sometimes the most common things are the hardest to find.

Reprinted with permission from Quanta Magazine‘s Abstractions blog. The first time I heard a mathematician use the phrase, I was sure he’d misspoken. We were on the phone, talking about the search for shapes with certain properties, and he said, “It’s like looking for hay in a haystack.” “Don’t you mean a needle?” I almost interjected. Then he […]

Reprinted with permission from Quanta Magazine‘s Abstractions blog. Imagine winding the hour hand of a clock back from 3 o’clock to noon. Mathematicians have long known how to describe this rotation as a simple multiplication: A number representing the initial position of the hour hand on the plane is multiplied by another constant number. But is a similar […]

What math can teach us about finding order in our chaotic lives.

What’s easy for a computer to do, and what’s almost impossible? Those questions form the core of computational complexity. We present a map of the landscape.

What this World Cup reveals isn’t that the stats were wrong—far from it, they were insightfully calculated—but rather that we relate to stats and probabilities in strange ways.Photograph by Hey! Play! / Wayfair This year’s World Cup has been full of surprises. Tournament mainstays such as the Netherlands and Italy didn’t even qualify, and Germany, […]

Zeta values seem to connect distant geometric worlds. In a new proof, mathematicians finally explain why.

Code is at once a force, or a means, of liberation and constraint.Photograph by Donnie Ray Jones / Flickr How did we humans manage to build a global civilization on the cusp of colonizing other planets? It seems like such an unlikely outcome. After all, we were prone to cycles of war and famine for millennia, […]

Understanding the odds lets you play with them.

A simple, step-by-step breakdown of two “perfect” math proofs.

Generations of researchers have pursued his “Langlands program,” which seeks to create a grand unified theory of mathematics.

Even with no one in charge, army ants work collectively to build bridges out of their bodies. New research reveals the simple rules that lead to such complex group behavior.

  The late statistics wizard Hans Rosling, who died in February of 2017 at age 68,  brought at least 10 toilet paper rolls to some of his beloved presentations.  He would stack them into a tower on a table, each roll representing one billion people. In a 2012 talk at the Skoll World Forum, he […]

Reprinted with permission from Quanta Magazine‘s Abstractions blog. Physics contains equations that describe everything from the stretching of space-time to the flitter of photons. Yet only one set of equations is considered so mathematically challenging that it’s been chosen as one of seven “Millennium Prize Problems” endowed by the Clay Mathematics Institute with a $1 million reward: […]

The Navier-Stokes equations describe simple, everyday phenomena, like water flowing from a garden hose, yet they provide a million-dollar mathematical challenge.

Anybody you can name from ancient history is in your family tree.

Quantum Lab: Scientists are fabricating quantum photonic circuits—consisting of waveguides and other elements—to manipulate single photons for future quantum communications and processing.Oak Ridge National Laboratory / Flickr Justin Trudeau, the Canadian prime minister, certainly raised the profile of quantum computing a few notches last year, when he gamely—if vaguely1—described it for a press conference. But […]

A famously difficult mathematical problem resisted solution for over 40 years. Mathematicians have finally resolved it by following an intuition that links number theory to physics.

Former correction officer Gary Klivans doesn’t want to be photographed more clearly for fear of gang retaliation.Gary Klivans As a corrections officer at a Westchester County, N.Y., prison in the 1990s, Gary Klivans was a one-man gang unit. Members of The Latin Kings and the Bloods made up a sizable part of the prison population. […]

High school teachers are bringing ice cream into the lab.

Voevodsky’s friends remember him as constitutionally unable to compromise on the truth—a quality that led him to produce some of the most important mathematics of the 20th century.

Ramsey’s theorem predicts a surprising (and useful) consistency in the organization of graphs. Here’s a simple visual proof of how it works.Image by Lucy Reading-Ikkanda/Quanta Magazine; Source: Jonathan Jedwab, Simon Fraser University Reprinted with permission from Quanta Abstractions A recent advance in geometry makes heavy use of Ramsey’s theorem, an important idea in another field—graph theory. […]

The color of LED lights is controlled by a clumsy process. A new mathematical discovery may make it easier for us to get the hues we want.

It’s “a definitive study for all time, like writing the final book,” says one researcher who’s mapping out new classes of geometric structures.

Reprinted with permission from Quanta Abstractions In an interview with Quanta Magazine last fall, the eminent theoretical physicist Helen Quinn recalled her uncertainty, as a Stanford University undergraduate in the 1960s, about whether to pursue a career in physics or become a high school teacher. “There were no women in the faculty at Stanford at that time in the […]

The father of information theory built a machine to game roulette, then abandoned it.

A California housewife who in the 1970s discovered four new types of tessellating pentagons is dead at 94.

Big advances in math can happen when mathematicians move ideas into areas where they seem like they shouldn’t belong.

Science predicts only the predictable, ignoring most of our chaotic universe.

Paul Erdős placed small bounties on hundreds of unsolved math problems. Over the past 20 years, only a handful have been claimed.

Near-miss math provides exact representations of almost-right answers.

Near-miss math provides exact representations of almost-right answers.

Juggling has advanced enormously in recent decades, thanks in part to the mathematical study of possible patterns.

Ramsey’s theorem predicts a surprising (and useful) consistency in the organization of graphs. Here’s a simple visual proof of how it works.

Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system.

The French mathematician was cited “for his pivotal role in the development of the mathematical theory of wavelets.”

Top women in mathematics and physics discuss how they got to where they are—and why there aren’t more of them.

19th-century mathematicians thought the “roots of unity” were the key to solving Fermat’s Last Theorem. Then they discovered a fatal flaw.

Atlas knew the answer. Straining under the task of holding up the Earth, this Titan god likely got a good idea of how much the Earth weighed. But none of us are so conveniently situated. How could a mere mortal, a tiny person residing on Earth’s surface, carry out their own estimate of Earth’s weight? […]

Harvey Friedman is about to bring incompleteness and infinity out of quarantine.

The late statistics wizard Hans Rosling, who died this month at age 68,  brought at least 10 toilet paper rolls to some of his beloved presentations.  He would stack them into a tower on a table, each roll representing one billion people. In a 2012 talk at the Skoll World Forum, he used the rolls […]

The simple Möbius strip illustrates a deep mathematical challenge that has long tormented the field of symplectic geometry.

The surprising mathematics of the extremely rare.

The ancient study of an object’s curvature is guiding mathematicians toward a new understanding of simple equations.

Nature makes chance, humans make luck.

Matthew Peterson is a pretty inspirational guy. As a dyslexic child he found math class difficult, so as an adult he resolved to totally change the way math is taught. After completing his studies in biology, electrical engineering, and Chinese language and literature at the University of California, Irvine, Peterson co-founded the nonprofit MIND Research […]

The legendary graph isomorphism problem may be harder than a 2015 result seemed to suggest.

Can you turn a two-dimensional fractal into a 3-D object? Break out your scissors and tape for a chance to win a 3-D printed sculpture.

How do physicists reconstruct what really happened in a particle collision? Through calculations that are so challenging that, in some cases, they simply can’t be done. Yet.

The same problem that caused the 2007 financial crisis also tripped up the polling data ahead of this year’s presidential election.

Mathematicians have had a hard time finding commonalities in large groups of random shapes — until recently.

When divvying something up, there's more than one way to define what's fair.

Seek balance in all things—including subterfuge.

Bernard Morin developed glaucoma at an early age and was blind by the time he was six years old. Despite his inability to see, Morin went on to become a master topologist—a mathematician who studies the intrinsic properties of geometric forms in space—and earned renown for his visualization of an inside-out sphere. For sighted people, […]

At the Hotel Parco dei Principi in Rome, in September of 1973, the Hungarian mathematician Paul Erdős approached his friend Richard Guy with a request. He said, “Guy, veel you have a coffee?” It cost a dollar, a small fortune to a professor of mathematics at the hinterland University of Calgary who was not much […]

The early 19th century golfer Harry Vardon was the Tiger Woods of his day, and not just because he had marital difficulties. He even had a biography written about him, which recounted, among other things, how he handled losing his first child and living with tuberculosis. But Vardon’s life would be more useful to sports […]

Shattering preconceptions about players isn’t all about the numbers.

Whenever I fly, I like to talk to the person sitting next to me. Once in a while, I find that we know at least one person in common. If you are like me, perhaps coincidences such as this happen in your life as well. The most unusual coincidence in my life took place when […]

Hooked on numbers: Pythagoreans celebrate sunrise (1869), a painting by Fyodor Bronnikov (1827–1902).Wikicommons Like most people, my high-school training in mathematics involved next-to-no history, barely touching on the names of a few mathematicians, like Pythagoras, and their theorems. I graduated only vaguely aware that geometry came from ancient Greece and algebra came from the Babylonians. […]

A bit of education can erase a definitive cognitive gap between men and women.

I would cut off my right hand if you find it.” That was the guarantee retired Columbia history lecturer Jens Ulff-Møller made that there was no word for “million” in Old English, a medieval predecessor of the language you’re currently reading. Some Anglo-Saxon writers understood the idea of a million, and they had a term for it: a “thousand […]

The surprising power of Neil Sloane’s Encyclopedia of Integer Sequences.

We’ve predicted and broken human population limits for centuries.

We’ve predicted and broken human population limits for centuries.

If you think computers have gotten really good at solving problems—like voice recognition, or self-driving cars—then your mind would be blown by what they could do if P = NP. Internet cryptography would come crashing down. Voice and image recognition would become near-perfect. Mathematical proofs would be greatly simplified. The stock market would dramatically change. […]

Andrew Wiles thought he had a solution to an age-old puzzle. Until it began to unravel.

Andrew Wiles thought he had a solution to an age-old puzzle. Until it began to unravel.

Workers count votes at a polling place in Worcester, Mass.SuperStock via Getty Images Last Thursday the UK’s Conservative Party stomped to an electoral victory that fairly shocked the country. The Tories won a comfortable majority of seats in parliament, enabling them to govern the nation without a coalition partner. That result contrasted sharply with the […]

A still from Prudential’s commercial, showing the largest domino to ever be toppled. The point was to make a spectacle of the power of compound interest.Courtesy of Prudential It’s a clear warm day in early August, and the Harvard psychologist Dan Gilbert is lucky it isn’t windy. Towering right behind him stands a 30-foot tall […]

Computers can’t invent, but they’re changing the field anyway.

A simple mathematical model of a sandpile shows remarkably complex behavior.

Rags, riches, and fame in mathematics.

Composer Edward Elgar still has cryptographers playing his tune.

The building blocks of understanding are memorization and repetition.

Sorry, education reformers, it’s still memorization and repetition we need.

We can experience nothingness, but does it actually exist?

How the null hypothesis keeps the hairy hominid alive.

How the null hypothesis keeps the hairy hominid alive.

If you’ve ever tried to give yourself a haircut, you know just how hard it is to make something precisely symmetrical. We value symmetry so highly in part because it’s really hard to achieve. Here are five of the most symmetrical objects humans have ever crafted, and why they were so hard to make. 1. […]

Ecelop via Shutterstock Go ahead, admit it. Like a lot of people, you have a favorite number. Maybe you’re not as extreme as Sheldon Cooper, the arch-nerd character on television’s Big Bang Theory, who loves the number 73: “73 is the 21st prime number, its mirror 37 is the 12th and its mirror 21 is […]

Gryffindor and Slytherin are about to play their annual badminton match. The best players from each house are supposed to face off on court one, the second best on court two, and so on. Slytherin’s coach knows that Gryffindor will put their players on the right courts, in order of their skill, because Gryffindors are […]

Computer security is trying to avoid wasting your time.

Nine years ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn’t a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, MP3 […]

Composer Edward Elgar still has cryptographers playing his tune.

Grigori Perelman became famous, despite his adamant opposition, for proving a conjecture from Henri Poincaré, pictured here. In November 2002, a Russian mathematician named Grigori Perelman posted the first of three short preprints to the arXiv (an online repository for drafts of academic papers in math and science), offering a proof for the famous Poincare […]

Shattering preconceptions about players isn’t all about the numbers.

Researchers’ rituals for assessing probability may mislead as much as they enlighten.

How numbers can tip the scales of justice.

Chance encounters in the life of Andrei Kolmogorov.

Chance encounters in the life of Andrei Kolmogorov.

Why it’s hard to recognize the unlikely.

Those aggravating slowdowns aren’t one driver’s fault. They’re everybody’s fault.

When the biggest problem with traffic is nothing at all.

Never mind tornadoes—companies now seek shelter from daily weather.

The reliability of some technologies depends on just the right amount of randomness.

There are two flavors of uncertainty in our lives. Math helps with both.

When a hypothesis is neither true nor false.

When a hypothesis is neither true nor false.

What looks like the golden ratio is sometimes just fool’s gold.