Algebra often involves manipulating numbers or other objects using operations like addition and multiplication. Flavio Coelho/Moment via Getty ImagesCourtney Gibbons, Hamilton College You scrambled up a Rubik’s cube, and now you want to put it back in order. What sequence of moves should you make? Surprise: You can answer this question with modern algebra. Most folks who have been through high school mathematics courses will have taken a class called algebra – maybe even a sequence of classes called algebra I and algebra II that asked you to solve for x. The word “algebra” may evoke memories of complicated-looking polynomial equations like ax² + bx + c = 0 or plots of polynomial functions like y = ax² + bx + c. You might remember learning about the quadratic formula to figure out the solutions to these equations and find where the plot crosses the x-axis, too.Graph of a quadratic equation and its roots via the quadratic formula.Jacob Rus, CC BY-SA Equations and plots like these are part of algebra, but they’re not the whole story. What unifies algebra is the practice of studying things – like the moves you can make on a Rubik’s cube or the numbers on a clock face you use to tell time – and the way they behave when you put them together in different ways. What happens when you string together the Rubik’s cube moves or add up numbers on a clock? In my work as a mathematician, I’ve learned that many algebra questions come down to classifying objects by their similarities.
Sets and groups
How did equations like ax² + bx + c = 0 and their solutions lead to abstract algebra? The short version of the story is that mathematicians found formulas that looked a lot like the quadratic formula for polynomial equations where the highest power of x was three or four. But they couldn’t do it for five. It took mathematician Évariste Galois and techniques he developed – now called group theory – to make a convincing argument that no such formula could exist for polynomials with a highest power of five or more. So what is a group, anyway? It starts with a set, which is a collection of things. The fruit bowl in my kitchen is a set, and the collection of things in it are pieces of fruit. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 also form a set. Sets on their own don’t have too many properties – that is, characteristics – but if we start doing things to the numbers 1 through 12, or the fruit in the fruit bowl, it gets more interesting.In clock addition, 3 + 12 = 3.OpenStax, CC BY-SA Let’s call this set of numbers 1 through 12 “clock numbers.” Then, we can define an addition function for the clock numbers using the way we tell time. That is, to say “3 + 11 = 2” is the way we would add 3 and 11. It feels weird, but if you think about it, 11 hours past 3 o’clock is 2 o’clock. Clock addition has some nice properties. It satisfies:
closure, where adding things in the set gives you something else in the set,
identity, where there’s an element that doesn’t change the value of other elements in the set when added – adding 12 to any number will equal that same number,
associativity, where you can add wherever you want in the set,
inverses, where you can undo whatever an element does, and
commutativity, where you can change the order of which clock numbers you add up without changing the outcome: a + b = b + a.
By satisfying all these properties, mathematicians can consider clock numbers with clock addition a group. In short, a group is a set with some way of combining the elements layered on top. The set of fruit in my fruit bowl probably can’t be made into a group easily – what’s a banana plus an apple? But we can make a set of clock numbers into a group by showing that clock addition is a way of taking two clock numbers and getting to a new one that satisfies the rules outlined above.
Rings and fields
Along with groups, the two other fundamental types of algebraic objects you would study in an introduction to modern algebra are rings and fields. We could introduce a second operation for the clock numbers: clock multiplication, where 2 times 7 is 2, because 14 o’clock is the same as 2 o’clock. With clock addition and clock multiplication, the clock numbers meet the criteria for what mathematicians call a ring. This is primarily because clock multiplication and clock addition together satisfy a key component that defines a ring: the distributive property, where a(b + c) = ab + ac. Lastly, fields are rings that satisfy even more conditions. At the turn of the 20th century, mathematicians David Hilbert and Emmy Noether – who were interested in understanding how the principles in Einstein’s relativity worked mathematically – unified algebra and showed the utility of studying groups, rings and fields.
It’s all fun and games until you do the math
Groups, rings and fields are abstract, but they have many useful applications. For example, the symmetries of molecular structures are categorized by different point groups. A point group describes ways to move a molecule in space so that even if you move the individual atoms, the end result is indistinguishable from the molecule you started with.The water molecule H₂O can be flipped horizontally and the end result is indistinguishable from the original position.Courtney Gibbons, CC BY-SA But let’s take a different example that uses rings instead of groups. You can set up a pretty complicated set of equations to describe a Sudoku puzzle: You need 81 variables to represent each place you can put a number in the grid, polynomial expressions to encode the rules of the game, and polynomial expressions that take into account the clues already on the board. To get the spaces on the game board and the 81 variables to correspond nicely, you can use two subscripts to associate the variable with a specific place on the board, like using x₃₅ to represent the cell in the third row and fifth column. The first entry must be one of the numbers 1 through 9, and we represent that relationship with (x₁₁ – 1)(x₁₁ – 2)(x₁₁ – 3) ⋅⋅⋅ (x₁₁ – 9). This expression is equal to zero if and only if you followed the rules of the game. Since every space on the board follows this rule, that’s already 81 equations just to say, “Don’t plug in anything other than 1 through 9.” The rule “1 through 9 each appear exactly once in the top row” can be captured with some sneaky pieces of algebraic thinking. The sum of the top row is going to add up to 45, which is to say x₁₁ + x₁₂ + ⋅⋅⋅ + x₁₉ – 45 will be zero, and the product of the top row is going to be the product of 1 through 9, which is to say x₁₁ x₁₂ ⋅⋅⋅ x₁₉ – 9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 will be zero. If you’re thinking that it takes more time to set up all these rules than it does to solve the puzzle, you’re not wrong.Turning Sudoku into algebra takes a fair bit of work.Courtney Gibbons What do we get by doing this complicated translation into algebra? Well, we get to use late-20th century algorithms to figure out what numbers you can plug into the board that satisfy all the rules and all the clues. These algorithms are based on describing the structure of the special ring – called an ideal – these game board clues make within the larger ring. The algorithms will tell you if there’s no solution to the puzzle. If there are multiple solutions, the algorithms will find them all. This is a small example where setting up the algebra is harder than just doing the puzzle. But the techniques generalize widely. You can use algebra to tackle problems in artificial intelligence, robotics, cryptography, quantum computing and so much more – all with the same bag of tricks you’d use to solve the Sudoku puzzle or Rubik’s cube.Courtney Gibbons, Associate Professor of Mathematics, Hamilton College This article is republished from The Conversation under a Creative Commons license. Read the original article.
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A young George Washington was thrust into the dense, contested wilderness of the Ohio River Valley as a land surveyor for real estate development companies in Virginia. Henry Hintermeister/Public domain via Wikimedia Commons
This Presidents Day, I’ve been thinking about George Washington − not at his finest hour, but possibly at his worst.
In 1754, a 22-year-old Washington marched into the wilderness surrounding Pittsburgh with more ambition than sense. He volunteered to travel to the Ohio Valley on a mission to deliver a letter from Robert Dinwiddie, governor of Virginia, to the commander of French troops in the Ohio territory. This military mission sparked an international war, cost him his first command and taught him lessons that would shape the American Revolution.
As a professor of early American history who has written two books on the American Revolution, I’ve learned that Washington’s time spent in the Fort Duquesne area taught him valuable lessons about frontier warfare, international diplomacy and personal resilience.
The mission to expel the French
In 1753, Dinwiddie decided to expel French fur trappers and military forces from the strategic confluence of three mighty waterways that crisscrossed the interior of the continent: the Allegheny, Monongahela and Ohio rivers. This confluence is where downtown Pittsburgh now stands, but at the time it was wilderness.
King George II authorized Dinwiddie to use force, if necessary, to secure lands that Virginia was claiming as its own.
As a major in the Virginia provincial militia, Washington wanted the assignment to deliver Dinwiddie’s demand that the French retreat. He believe the assignment would secure him a British army commission.
Washington received his marching orders on Oct. 31, 1753. He traveled to Fort Le Boeuf in northwestern Pennsylvania and returned a month later with a polite but firm “no” from the French.George Washington held an honorary commission as a major in the British army prior to the French and Indian War. Dea/M. Seemuller/De Agostini collection/Getty Images
Dinwiddie promoted Washington from major to lieutenant colonel and ordered him to return to the Ohio River Valley in April 1754 with 160 men. Washington quickly learned that French forces of about 500 men had already constructed the formidable Fort Duquesne at the forks of the Ohio. It was at this point that he faced his first major test as a military leader. Instead of falling back to gather more substantial reinforcements, he pushed forward. This decision reflected an aggressive, perhaps naive, brand of leadership characterized by a desire for action over caution.
Washington’s initial confidence was high. He famously wrote to his brother that there was “something charming” in the sound of whistling bullets.
The Jumonville affair and an international crisis
Perhaps the most controversial moment of Washington’s early leadership occurred on May 28, 1754, about 40 miles south of Fort Duquesne. Guided by the Seneca leader Tanacharison – known as the “Half King” – and 12 Seneca warriors, Washington and his detachment of 40 militiamen ambushed a party of 35 French Canadian militiamen led by Ensign Joseph Coulon de Jumonville. The Jumonville affair lasted only 15 minutes, but its repercussions were global.The Jumonville affair became the opening battle of the French and Indian War. Interim Archives/Archive Collection/Getty Images
Ten of the French, including Jumonville, were killed. Washington’s inability to control his Native American allies – the Seneca warriors executed Jumonville – exposed a critical gap in his early leadership. He lacked the ability to manage the volatile intercultural alliances necessary for frontier warfare.
Washington also allowed one enemy soldier to escape to warn Fort Duquesne. This skirmish effectively ignited the French and Indian War, and Washington found himself at the center of a burgeoning international crisis.
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Defeat at Fort Necessity
Washington then made the fateful decision to dig in and call for reinforcements instead of retreating in the face of inevitable French retaliation. Reinforcements arrived: 200 Virginia militiamen and 100 British regulars. They brought news from Dinwiddie: congratulations on Washington’s victory and his promotion to colonel.
His inexperience showed in his design of Fort Necessity. He positioned the small, circular palisade in a meadow depression, where surrounding wooded high ground allowed enemy marksmen to fire down with impunity. Worse still, Tanacharison, disillusioned with Washington’s leadership and the British failure to follow through with promised support, had already departed with his warriors weeks earlier. When the French and their Native American allies finally attacked on July 3, heavy rains flooded the shallow trenches, soaking gunpowder and leaving Washington’s men vulnerable inside their poorly designed fortification.Washington was outnumbered and outmaneuvered at Fort Necessity. Interim Archives/Archive Collection/Getty Images
The battle of Fort Necessity was a grueling, daylong engagement in the mud and rain. Approximately 700 French and Native American allies surrounded the combined force of 460 Virginian militiamen and British regulars. Despite being outnumbered and outmaneuvered, Washington maintained order among his demoralized troops. When French commander Louis Coulon de Villiers – Jumonville’s brother – offered a truce, Washington faced the most humbling moment of his young life: the necessity of surrender. His decision to capitulate was a pragmatic act of leadership that prioritized the survival of his men over personal honor.
The surrender also included a stinging lesson in the nuances of diplomacy. Because Washington could not read French, he signed a document that used the word “l’assassinat,” which translates to “assassination,” to describe Jumonville’s death. This inadvertent admission that he had ordered the assassination of a French diplomat became propaganda for the French, teaching Washington the vital importance of optics in international relations.A log cabin used to protect the perishable supplies still stands at Fort Necessity today. MyLoupe/Universal Images Group/Getty Images
Lessons that forged a leader
The 1754 campaign ended in a full retreat to Virginia, and Washington resigned his commission shortly thereafter. Yet, this period was essential in transforming Washington from a man seeking personal glory into one who understood the weight of responsibility.
He learned that leadership required more than courage – it demanded understanding of terrain, cultural awareness of allies and enemies, and political acumen. The strategic importance of the Ohio River Valley, a gateway to the continental interior and vast fur-trading networks, made these lessons all the more significant.
Ultimately, the hard lessons Washington learned at the threshold of Fort Duquesne in 1754 provided the foundational experience for his later role as commander in chief of the Continental Army. The decisions he made in Pennsylvania and the Ohio wilderness, including the impulsive attack, the poor choice of defensive ground and the diplomatic oversight, were the very errors he would spend the rest of his military career correcting.
Though he did not capture Fort Duquesne in 1754, the young George Washington left the woods of Pennsylvania with a far more valuable prize: the tempered, resilient spirit of a leader who had learned from his mistakes.
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The Building That Proved Los Angeles Could Go Vertical
Los Angeles once banned skyscrapers, yet City Hall broke the height limit and proved high-rise buildings could be engineered safely in an earthquake zone.
How City Hall Quietly Undermined LA’s Own Height Limits
The Knowledge Series | STM Daily News
For more than half a century, Los Angeles enforced one of the strictest building height limits in the United States. Beginning in 1905, most buildings were capped at 150 feet, shaping a city that grew outward rather than upward.
The goal was clear: avoid the congestion, shadows, and fire dangers associated with dense Eastern cities. Los Angeles sold itself as open, sunlit, and horizontal — a place where growth spread across land, not into the sky.
And yet, in 1928, Los Angeles City Hall rose to 454 feet, towering over the city like a contradiction in concrete.
It wasn’t built to spark a commercial skyscraper boom. But it ended up proving that Los Angeles could safely build one.
A Rule Designed to Prevent a Manhattan-Style City
The original height restriction was rooted in early 20th-century fears:
Limited firefighting capabilities
Concerns over blocked sunlight and airflow
Anxiety about congestion and overcrowding
A strong desire not to resemble New York or Chicago
Los Angeles wanted prosperity — just not vertical density.
The height cap reinforced a development model where:
Office districts stayed low-rise
Growth moved outward
Automobiles became essential
Downtown never consolidated into a dense core
This philosophy held firm even as other American cities raced upward.
Los Angeles banned skyscrapers for decades — except one. 🏛️ While most buildings were capped at 150 feet, LA City Hall rose three times higher. This wasn’t a loophole — it was power, symbolism, and city planning shaping the skyline we know today. Why was City Hall the exception? And how did this one decision change Los Angeles forever? 📍 Forgotten LA 🧠 The Knowledge Series 📰 STM Daily News LosAngelesHistory LACityHall ForgottenLA UrbanPlanning CityPlanning LASkyline DidYouKnow HistoryTok TheKnowledge STMDailyNews ♬ original sound – STMDailyNews – STMDailyNews
Why City Hall Was Never Meant to Change the Rules
City Hall was intentionally exempt from the height limit because the law applied primarily to private commercial buildings, not civic monuments.
But city leaders were explicit about one thing: City Hall was not a precedent.
It was designed to:
Serve as a symbolic seat of government
Stand alone as a civic landmark
Represent stability, authority, and modern governance
Avoid competing with private office buildings
In effect, Los Angeles wanted a skyline icon — without a skyline.
Innovation Hidden in Plain Sight
What made City Hall truly significant wasn’t just its height — it was how it was built.
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At a time when seismic science was still developing, City Hall incorporated advanced structural ideas for its era:
A steel-frame skeleton designed for flexibility
Reinforced concrete shear walls for lateral strength
A tapered tower to reduce wind and seismic stress
Thick structural cores that distributed force instead of resisting it rigidly
These choices weren’t about aesthetics — they were about survival.
The Earthquake That Changed the Conversation
In 1933, the Long Beach earthquake struck Southern California, causing widespread damage and reshaping building codes statewide.
Los Angeles City Hall survived with minimal structural damage.
This moment quietly reshaped the debate:
A tall building had endured a major earthquake
Structural engineering had proven effective
Height alone was no longer the enemy — poor design was
City Hall didn’t just survive — it validated a new approach to vertical construction in seismic regions.
Proof Without Permission
Despite this success, Los Angeles did not rush to repeal its height limits.
Cultural resistance to density remained strong, and developers continued to build outward rather than upward. But the technical argument had already been settled.
City Hall stood as living proof that:
High-rise buildings could be engineered safely in Los Angeles
Earthquakes were a challenge, not a barrier
Fire, structural, and seismic risks could be managed
The height restriction was no longer about safety — it was about philosophy.
The Ironic Legacy
When Los Angeles finally lifted its height limit in 1957, the city did not suddenly erupt into skyscrapers. The habit of building outward was already deeply entrenched.
The result:
A skyline that arrived decades late
Uneven density across the region
Multiple business centers instead of one core
Housing and transit challenges baked into the city’s growth pattern
City Hall never triggered a skyscraper boom — but it quietly made one possible.
Why This Still Matters
Today, Los Angeles continues to wrestle with:
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Housing shortages
Transit-oriented development debates
Height and zoning battles near rail corridors
Resistance to density in a growing city
These debates didn’t begin recently.
They trace back to a single contradiction: a city that banned tall buildings — while proving they could be built safely all along.
Los Angeles City Hall wasn’t just a monument. It was a test case — and it passed.
Doing things alone is on the rise, and businesses should pay more attention to that – even on Valentine’s Day
Peter McGraw discusses the increasing prevalence of solo living and its implications for businesses, particularly during Valentine’s Day, which typically emphasizes couples. Despite many individuals enjoying activities alone, the marketplace often neglects this growing demographic. Recognizing and catering to solo consumers can yield significant opportunities for businesses.
Doing things alone is on the rise, and businesses should pay more attention to that – even on Valentine’s Day
Every February, Valentine’s Day amplifies what single people already know – that public life is built for two. Restaurants roll out prix fixe menus for couples. Hotels promote “romantic getaway” packages designed for double occupancy. A table for one still invites the question, “Just you?”
Yet there’s irony that’s hard to miss. While Valentine’s Day doubles down on togetherness, more adults are living – and moving through the world – alone.
As a behavioral economist, I study what I call the “solo economy.” A growing share of economic life today is organized around people who live, spend and make decisions on their own.
1-person households aren’t outliers
Half of U.S. adults are unmarried, and one-person households are now the nation’s most common living arrangement. This isn’t a temporary phase confined to young adults waiting to settle down. It includes never-married professionals, divorced empty nesters, widows and widowers, and people who simply prefer to live independently.
It’s a slow-moving demographic shift away from long-term partnership as the dominant adult life path, but a consequential one – reshaping everything from housing and travel to social policy and commerce. One of its clearest expressions is the number of people doing things alone in public.
The rise of public solo life
It would be one thing if the economy were built for two and solos stayed home. But they are going to museums, traveling and, of course, dining alone in restaurants. To assess this behavior, I surveyed single and married Americans about their participation in 25 activities that occur in public – from shopping and dining to attending movies and concerts.
The pattern was striking. Overall, singles were much more likely to do things alone in public than their married counterparts – 56% versus 39%. The difference held across every activity I measured.
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The biggest gaps weren’t for practical tasks like grocery shopping. They were for leisure experiences like going to the movies, dining out and attending concerts. In fact, seven of the 10 largest differences involved retail or entertainment settings – the very places most designed and marketed with couples in mind.
Why hasn’t the business world paid more attention to the singles market?
The answer lies in psychology. Some reluctance stems from the belief that other customers will perceive solo diners or moviegoers as sad or lonely. These fears are amplified by what psychologists call the spotlight effect – our tendency to overestimate how much other people notice and judge us.
Findings by consumer researchers Rebecca Hamilton and Rebecca Ratner can help explain why this bias is so persistent. Across studies conducted in the U.S., China and India, people consistently predicted they would enjoy activities less if they did them alone – even though they’d be seeing the same movie or visiting the same museum.
But when people actually went alone, they enjoyed the experience just as much as those who went with others. The fear, it turns out, is largely imagined.
Another problem is that solo consumers don’t always feel welcome.
While behavior is changing, markets have been slower to adapt. Most businesses still design experiences around pairs, families or groups. Consider restaurants that seat solo diners at the bar or near the kitchen or bathrooms, or ticketing systems that require purchasing in pairs. The result is friction for solo consumers – and missed opportunities for companies.
Valentine’s Day promotions make that mismatch especially visible. In 2024, IKEA Canada offered a Valentine’s Day dining experience in its showroom priced and designed for two – and only two – people.
After backlash, the company revised the promotion the following year to be more inclusive: “Bring a loved one, a good friend, or the whole family.” It was a small change, but a revealing one.
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Why solo shoppers have outsized influence
Solo consumers represent a large, growing and profitable market segment, yet they’re navigating a marketplace that still treats them as edge cases.
Another study that Ratner conducted with business school professor Yuechen Wu adds an important twist.
Analyses of more than 14,000 Tripadvisor reviews of restaurants and museums show that reviews written by solo diners and solo museumgoers are rated as more helpful – and receive more positive feedback – than reviews written by people who went with others.
Follow-up experiments showed that when otherwise identical recommendations differed only in whether the reviewer experienced the activity alone or with others, respondents were more likely to rely on the solo reviewer when deciding what to do.
Why? Observers infer that people who go alone are more genuinely interested in the experience and more focused on its quality, rather than simply going along with someone else’s preferences.
Being alone, it turns out, functions as a credibility cue. For businesses, that means solo customers aren’t just customers − they can be very influential customers.
Designing for 1 in Asia
Asian businesses are far ahead of the West in recognizing the buying power of people doing things alone.
In South Korea, for example, “honjok,” which translates as “alone tribe,” culture has fueled products and services designed explicitly for solo living. Think single-serve meals at convenience stores, one-person karaoke booths, and restaurants that promise judgment-free service.
Similarly, in Japan, the ramen chain Ichiran built its brand around the idea of “flavor concentration,” which encourages diners to eat alone in private booths.
Officially, the design is meant to eliminate distractions and heighten the dining experience. In practice, it does something more important: It legitimizes solo dining.
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Progress in the US
In the U.S., Disney theme parks and some of the company’s competitors have long used single-rider lines that reward solo visitors with shorter waits, turning independence into operational efficiency – a logic ski resorts adopted decades ago to fill empty seats on chairlifts.
And solo tourism has become a major trend. Demand is growing, and tour operators are adapting offerings to meet it, including specialized tours for singles and adjustments to historically prohibitive pricing practices.
Industry analysis also shows the global solo travel market expanding rapidly, with tailored products and experiences emerging worldwide. Some companies now offer dedicated solo travel collections with no single supplement − the extra fee traditionally charged to travelers who occupy a room alone − and tours designed specifically for independent travelers.
Doing things alone is an opportunity
Valentine’s Day offers a chance to see how outdated many widespread assumptions still are.
It treats solitude as a problem to be solved, even as people’s behavior tells a different story. Yet businesses, policymakers and U.S. culture more broadly have not designed a world that fully acknowledges that about 42% of American adults are single.
In the meantime, singles aren’t waiting at home. They’re out there – at the movies, on planes, in museums and restaurants – moving through public life on their own terms.
Valentine’s Day may always be built for two. But the economy won’t be.
Courtney Gibbons, Associate Professor of Mathematics, Hamilton College This article is republished from The Conversation under a Creative Commons license. Read the original article.
