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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PG&E Donates $1 Million to Local Food Banks to Help Feed Families
PG&E donates $1 million to local food banks across Northern and Central California—equivalent to about 3 million meals—supporting 38 food banks serving 47 counties.
Just in time for the holidays, Pacific Gas and Electric Company (PG&E) announced a $1 million donation to local food banks across Northern and Central California—support aimed at meeting a surge in demand as more families and seniors struggle to put food on the table. According to PG&E, the contribution is expected to provide the equivalent of roughly 3 million meals and will support 38 food banks serving 47 counties within PG&E’s service territory.
A third major food-bank contribution since September
The $1 million gift marks the third food-bank-focused contribution since September from PG&E or The PG&E Corporation Foundation (the PG&E Foundation). Combined, those efforts bring PG&E’s total community food support in 2025 to $2.37 million. PG&E emphasized that the funding for these charitable contributions comes from PG&E shareholders—not customers.
Food banks facing record-breaking demand
Food banks across California are reporting pressure levels not seen since the pandemic. Officials with the California Association of Food Banks say demand has reached record highs, driven in part by an unexpected surge during the federal government shutdown this fall. “California food banks experienced an unexpected surge with the [federal government] shutdown this fall. So, we reached out for help on their behalf and PG&E responded,” said Stacia Levenfeld, Chief Executive Officer of the California Association of Food Banks. “Their $1 million gift to food banks throughout Northern and Central California will have a meaningful impact on the lives of millions of people this holiday season and help food banks continue their critical work in our communities.” PG&E leaders framed the donation as an extension of a longstanding partnership with food bank networks. “We are grateful to help local food banks fulfill their mission during this time of increasing demand, especially as more families and seniors are struggling through the holiday season,” said Carla Peterman, Executive Vice President, Corporate Affairs, PG&E Corporation and Chair of The PG&E Corporation Foundation Board. “Our longstanding partnership with the California Association of Food Banks supports the safety net that is our local food banks.”
Where the 2025 food support has gone
PG&E outlined additional contributions made earlier in the year:
September: The PG&E Foundation awarded $1.12 million to support local food banks, tribal food banks, and senior meal programs.
November: The PG&E Foundation donated $250,000 to the California Association of Food Banks’ Emergency Response Fund.
Equity-focused grant distribution
The California Association of Food Banks notes that while California produces nearly half of the nation’s fruits and vegetables, more than one in five residents still don’t know where their next meal will come from. Food insecurity rates are even higher in many communities of color. PG&E said grant amounts awarded to local organizations will account for county poverty and unemployment levels, using a formula from the California Department of Social Services. The goal: promote equity by directing more support to counties with higher need.
About the PG&E Corporation Foundation and PG&E
The PG&E Corporation Foundation is an independent 501(c)(3) nonprofit organization, separate from PG&E and sponsored by PG&E Corporation. PG&E is a combined natural gas and electric utility serving more than 16 million people across 70,000 square miles in Northern and Central California. More information is available at pge.com and pge.com/news.
Why this matters
As food banks brace for sustained demand beyond the holiday season, large-scale donations like PG&E’s can help stabilize local supply—especially when distributed with an equity lens that targets the counties facing the steepest economic pressures. For families, seniors, and individuals navigating rising costs, the impact is immediate: more meals available now, and stronger community support systems heading into the new year. Community links:
In the 1970s, Lynwood, CA, dreamed of a downtown mall anchored by Montgomery Ward. Decades later, the empty lots told a story of ambition, delay, and renewal.
In the early 1970s, Lynwood, California, dreamed big.
City leaders envisioned a new, modern downtown — a sprawling shopping and auto mall that would bring jobs, shoppers, and a sense of pride back to this small but growing city in the southeast corner of Los Angeles County. At the heart of the plan stood a gleaming new Montgomery Ward department store, which opened around 1973 and promised to anchor a larger commercial center that never fully came.
But for those of us who grew up in Lynwood during that time, the promise never quite materialized.
Instead, we remember acres of empty lots, chain-link fences, and faded “Coming Soon” signs that sat for decades — silent witnesses to a dream deferred.
The Vision That Stalled
In 1973, Lynwood’s Redevelopment Agency launched what it called Project Area A — an ambitious plan to clear and rebuild much of the city’s downtown core. Small businesses and homes were bought out, land was assembled, and the city floated bonds to support new construction.
For a brief moment, it looked as if the plan might work. Montgomery Ward opened its doors, serving as a retail beacon for the area. Yet the rest of the mall — the shops, restaurants, and auto dealerships — never came.
By the mid-1970s, much of downtown had been bulldozed, but little replaced it. And by the time Ward closed its Lynwood location in 1986, the vast lots surrounding it had become symbols of frustration and unfulfilled potential.
What Happened?
Some longtime residents whispered about corruption or backroom deals — the kind of speculation that grows when visible progress stalls.
But newspaper archives and redevelopment records tell a more complex story.
Lynwood’s plans collided with a series of hard realities:
The construction of the Century Freeway (I-105) disrupted neighborhoods and depressed land values. Environmental cleanup and ownership disputes slowed development. Economic shifts in retail — as malls in nearby Downey, South Gate, and Paramount attracted anchor stores — drained the local market. And later, political infighting among city officials made sustained redevelopment almost impossible.
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To this day, there’s no public record of proven corruption directly tied to the 1970s mall plan. What did exist was a tangle of bureaucracy, economic change, and missed opportunity — a perfect storm that left Lynwood’s heart half-built and half-forgotten.
Growing Up Among the Vacant Lots
For those of us who were kids in Lynwood during that era, the story is more personal.
We remember the sight of the Montgomery Ward building — modern and hopeful at first, then shuttered and fading by the mid-1980s.
We remember riding bikes past the empty dirt fields that were supposed to become shopping plazas. And we remember the quiet frustration of adults who had believed the city’s promises.
Those empty blocks became our playgrounds — but they also became symbols of the gap between what Lynwood was and what it wanted to be.
A New Chapter: Plaza México and Beyond
By the late 1990s and early 2000s, the dream finally resurfaced in a new form.
Developers transformed the long-idle site into Plaza México, a vibrant commercial and cultural hub that celebrates Mexican and Latin American heritage.
It took nearly 30 years for Lynwood’s downtown to come alive again.
The result is beautiful — but it’s also bittersweet for those who remember how long the land sat empty, and how many local businesses and residents were displaced in pursuit of a dream that took a generation to fulfill.
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Looking Back
The story of Lynwood’s lost mall isn’t just about urban planning.
It’s about hope, change, and resilience. It’s about how a community tried to reinvent itself — and how the children who grew up watching that effort still carry its memory.
Sometimes, when I drive through that stretch of Imperial Highway and Long Beach Boulevard, I still imagine what might have been: the bustling mall that never was, and the voices of a neighborhood caught between ambition and uncertainty.
📚 Further Reading
Montgomery Ward will close its Lynwood store. (Jan 3 1986) — Los Angeles Times.
Dive into “The Knowledge,” where curiosity meets clarity. This playlist, in collaboration with STMDailyNews.com, is designed for viewers who value historical accuracy and insightful learning. Our short videos, ranging from 30 seconds to a minute and a half, make complex subjects easy to grasp in no time. Covering everything from historical events to contemporary processes and entertainment, “The Knowledge” bridges the past with the present. In a world where information is abundant yet often misused, our series aims to guide you through the noise, preserving vital knowledge and truths that shape our lives today. Perfect for curious minds eager to discover the ‘why’ and ‘how’ of everything around us. Subscribe and join in as we explore the facts that matter. https://stmdailynews.com/the-knowledge/
Population Density: How Los Angeles Compares to New York and Chicago
How dense are America’s biggest cities? A clear breakdown of population density in Los Angeles, New York City, and Chicago—city limits vs metro areas—and why it matters.
Population Density: How Los Angeles Compares to New York and Chicago
When people think of crowded American cities, New York City usually comes to mind first. Los Angeles, by contrast, is often labeled as “sprawling,” while Chicago is seen as a middle ground. But population density tells a more nuanced story—especially when comparing city proper numbers versus metro-area density.
City Proper: How Dense Are the Cities Themselves?
Looking only at official city boundaries, the differences are stark:
New York City averages about 27,000–28,000 people per square mile, making it by far the most densely populated major city in the United States.
Chicago comes in at roughly 12,000 people per square mile, dense but far more spread out than New York.
Los Angeles, despite being the nation’s second-largest city by population, averages just 8,400–8,500 people per square mile.
This gap reflects development patterns. New York grew upward with dense apartment buildings and extensive transit. Los Angeles expanded outward with single-family neighborhoods and car-oriented planning.
Metro Areas Tell a Different Story
When the lens widens to include surrounding suburbs and commuter communities, the rankings shift:
Los Angeles Metro Area: ~7,000 people per square mile
New York Metro Area: ~5,300 people per square mile
Chicago Metro Area: ~3,500 people per square mile
This surprises many readers. While New York’s core is extremely dense, its metro region stretches across a vast, lower-density area spanning parts of New York, New Jersey, and Pennsylvania. Los Angeles, on the other hand, has a metro region that is more consistently built-up, with fewer truly rural gaps.
Why Density Feels Different in Each City
Population density doesn’t always match perception:
New York feels crowded because density is concentrated vertically and transit funnels millions into compact areas.
Los Angeles feels congested not because of extreme density, but because people are spread out and heavily reliant on cars.
Chicago balances both, with dense neighborhoods near the core and more traditional suburban sprawl outward.
For cities like Los Angeles—now reinvesting in rail, buses, and transit-oriented development—understanding density is critical. As coverage on LA Metro and urban revival continues, these numbers explain why transit challenges in Southern California differ so sharply from those in New York or Chicago.
The Big Picture
Most dense city: New York City
Most dense metro area: Los Angeles
Most balanced: Chicago
Density isn’t just about how many people live in a place—it’s about how they live, move, and interact with the city around them.
Further Reading: Population Density & Urban Development
Dive into “The Knowledge,” where curiosity meets clarity. This playlist, in collaboration with STMDailyNews.com, is designed for viewers who value historical accuracy and insightful learning. Our short videos, ranging from 30 seconds to a minute and a half, make complex subjects easy to grasp in no time. Covering everything from historical events to contemporary processes and entertainment, “The Knowledge” bridges the past with the present. In a world where information is abundant yet often misused, our series aims to guide you through the noise, preserving vital knowledge and truths that shape our lives today. Perfect for curious minds eager to discover the ‘why’ and ‘how’ of everything around us. Subscribe and join in as we explore the facts that matter. https://stmdailynews.com/the-knowledge/
Courtney Gibbons, Associate Professor of Mathematics, Hamilton College This article is republished from The Conversation under a Creative Commons license. Read the original article.
