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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Why people trust influencers more than brands – and what that means for the future of marketing
Why people trust influencers? Discover why people trust influencers over traditional brands and what it means for marketing’s future. Learn about parasocial relationships, the 5 types of value influencers provide, and why microinfluencers often outperform mega-creators.
Why people trust influencers more than brands – and what that means for the future of marketing
Kelley Cours Anderson, College of Charleston Not long ago, the idea of getting paid to share your morning routine online would have sounded absurd. Yet today, influencers are big business: The global market is expected to surpass US$32 billion by the end of 2025. Rooted in celebrity culture but driven by digital platforms, the influencer economy represents a powerful force in both commerce and culture. I’m an expert on digital consumer research, and I see the rise of influencers as an important evolution in the relationship between companies, consumers and creators. Historically, brands leaned on traditional celebrities like musicians, athletes and actors to endorse their products. However, by the late 2000s, social media platforms opened the door for everyday people to build audiences. Initially, influencers were viewed as a low-cost marketing tactic. Soon, however, they became a central part of marketing strategies. In the 2010s, influencer marketing matured into a global industry. Agencies and digital marketplaces emerged to professionalize influencer-brand matchmaking, and regulators like the Federal Trade Commission started paying more attention to sponsored content. The rise of video and short-form content like TikTok and Reels in the mid-2010s and 2020s added authenticity and emotional immediacy. These dynamics deepened influencer-follower relations in ways that brands couldn’t easily replicate. Influencers are now recognized as not only content creators, but also as entrepreneurs and cultural producers.
Why people trust influencers
Social media influencers often foster what researchers call “parasocial relationships” – one-sided bonds where followers feel as if they personally know the influencer. While the concept has roots in traditional celebrity culture, influencers amplify it through consistent, seemingly authentic content. This perceived intimacy helps explain why consumers often trust influencers more than brands. Though the parasocial relationship isn’t mutual, it feels real. That emotional closeness cultivates trust, a scarce but powerful currency in today’s economy. The goal for many influencers may be financial independence, but the path begins with social and cultural capital, acquired through community connection, relatability and niche expertise. As an influencer’s following grows, so does their perceived legitimacy. Brands, in turn, recognize and tap into that legitimacy. Although risks exist, like algorithmic incentives and commercial partnerships that undercut authenticity, many influencers successfully navigate this tension to preserve their community’s trust.
The many ways creators add value
Like any economy, the influencer economy revolves around value exchange. Followers spend their valuable resources – time and attention – in return for something meaningful. Researchers have identified several forms of value that influencers’ content can take:
Connection, or what researchers call “social value”: Influencers often build tight-knit communities around shared interests. Through live chats, comments and relatable storytelling, they offer a sense of belonging.
Fun, or “hedonic value”: Many influencers provide enjoyment using entertainment, humor and a touch of allure in their content. Think cat videos, TikTok dances and random acts of kindness that deliver joy and distraction from the day-to-day.
Knowledge, or “epistemic value”: Creators offer informational or educational content to feed consumer curiosity. This can be through tutorials, product reviews or deep dives into niche topics.
Usefulness, or “utilitarian value”: From life hacks to product roundups, like “Amazon must-haves,” influencers provide utilitarian or practical value to help simplify consumer decisions and solve everyday problems.
Money, or “financial value”: People love finding a bargain. Discounts, affiliate links and deal alerts offer direct economic benefit to followers. Some influencers even launch their own products or digital courses, delivering long-term value through entrepreneurial spinoffs.
These forms of value often overlap, reinforcing trust, and can pay off financially for influencers. In fact, consumers are significantly more likely to trust user-generated content like influencer posts over brand-generated advertising.
Lessons for brands
First, there’s evidence that smaller is often stronger. Marketing researchers categorize influencers based on how many followers they have, and nano- and microinfluencers – defined as those with fewer than 10,000 and 100,000 followers, respectively – often generate stronger engagement than mega-influencers with more than 1 million. Influencers with smaller followings can interact with their communities more closely, making their endorsements feel more credible. This has driven brands to focus on mid-tier and microinfluencers, where return on investment is often stronger. As a result, influencer agencies, brokers, platforms and trade associations have sprung up to facilitate these partnerships. Second, brands should remember that influencers’ role in the market comes with new challenges. As the field continues to become more professionalized, it’s also become more complex. Like other entrepreneurs, influencers must keep up with shifting regulations – namely, FTC sponsorship guidelines – which can lead to hefty fines if violated. Many struggle to identify how to best file their taxes when they receive freebies they are expected to build content around. It can also be a challenge for influencers to keep up with continued algorithm tweaks from the multiple social media platforms where they publish. Influencers manage more than content creation. Their role includes quickly responding to followers’ comments and managing communities, as well as handling trolls, all of which is stressful. Personal brand management adds another layer of pressure. As influencers gain more brand partnerships, they run the risk of being seen as “selling out.” Because parasocial trust depends on being viewed as authentic, aligning with the wrong brand or being too promotional can damage the very connection that built an influencer’s following. A single misstep can trigger public backlash. While growing a following can bring brand recognition and financial independence, some influencers even fear that they will lose their own identity. Influencers can struggle with work-life balance, as this is not a nine-to-five job. It requires being “always on” and the constant blurred lines. Their lives become their livelihoods, with little separation between personal and professional identity. In short, when engaging with influencers, strategic brands will recognize that they operate within an intense, high-pressure environment. Organizations such as the American Influencer Council offer support and advocacy, but industry-wide protections are lacking. Influencers have earned a central place in consumer culture not just by selling products, but by offering emotional proximity, cultural relevance and value. They’re not just marketers – they’re creators, community leaders and entrepreneurs. As the creator economy continues to grow, trust will remain its cornerstone. However, the next chapter will require thoughtful navigation of issues like regulation, platform ethics and creator well-being. Understanding influencers means recognizing both their creative work and the evolving market that now depends on them. Kelley Cours Anderson, Assistant Professor of Marketing, College of Charleston This article is republished from The Conversation under a Creative Commons license. Read the original article.
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Glad and Oscar the Grouch Team Up for a Trashy, Toe-Tapping Campaign
Glad teams up with Oscar the Grouch for a playful revival of the “Don’t Get Mad. Get Glad.” campaign, featuring a musical number, limited-edition Oscar-inspired trash bags, and a fresh take on making trash day fun for all ages.
Glad revives its most popular, decades-long, star-studded ad campaign, “Don’t Get Mad. Get Glad.”
What happens when the world’s most iconic grouch meets the nation’s go-to name in trash bags? You get a campaign that’s equal parts nostalgia, Broadway-style fun, and a reminder that even trash can bring a little joy to your day.
A Classic Campaign Gets a Grouchy Remix
Glad has officially revived its legendary “Don’t Get Mad. Get Glad.” campaign, but this time, they’re ditching the usual celebrity faces for a true original: Oscar the Grouch. For the first time, the campaign’s star is none other than Sesame Street’s resident trash enthusiast himself, and he’s bringing his signature tune “I Love Trash” back with a contemporary twist.
The musical number, directed by the award-winning duo Will Speck and Josh Gordon, opens with Oscar in his element—surrounded by trash and a little bit of grumpiness. But the real magic happens when Oscar imagines a world where everyone else shares his passion for trash. The result? A joyful, Broadway-inspired remix that transforms everyday frustration into a celebration of Glad’s dependable trash solutions.
Why Oscar? Why Now?
According to Glad’s Marketing Director, Kellie Li, the choice was simple: “No one feels more strongly about trash than Oscar the Grouch.” The campaign aims to flip the script on how we think about trash—turning a dreaded chore into something a little more lighthearted. With Glad’s reliable bags, there’s less to get mad about, and maybe, just maybe, a little more to sing about.
Nostalgia Meets New Audiences
If “Don’t Get Mad. Get Glad.” sounds familiar, you’re not imagining things. The campaign has been a staple since 1987, featuring everyone from TV stars to athletes. But this new chapter, featuring Oscar and a cast of trash-loving co-stars, is designed to connect with both longtime fans and a new generation discovering Sesame Street on Netflix and PBS KIDS.
Limited-Edition Oscar Goodies and Where to Find Them
To celebrate the campaign, Glad is releasing limited-edition Oscar-inspired trash bag totes—complete with green fur, of course. Fans can snag these playful bags through a social media giveaway this December (follow @gladproducts on Instagram and TikTok for details). And if you miss out, don’t worry: special Oscar-branded Glad ForceFlex with Gain bags will hit Walmart shelves this April, just in time for spring cleaning.
Where to Watch
The campaign is rolling out across the U.S. and Canada, with full-length videos, bite-sized social teasers, and everything in between. Look for it on TikTok, Instagram, Facebook, and Reddit (for our friends up north). Featured products include Glad ForceFlex with Gain and Glad Cherry Blossom.
Glad revives its most popular, decades-long, star-studded ad campaign, “Don’t Get Mad. Get Glad.”
Glad revives its most popular, decades-long, star-studded ad campaign, “Don’t Get Mad. Get Glad.”
Bringing the Campaign Home: Phoenix Community Clean-Up
Here in Phoenix, we know the value of coming together to keep our neighborhoods clean and vibrant. Glad’s collaboration with Oscar the Grouch isn’t just a fun national campaign—it’s a reminder that tackling trash can be a community effort, too.
With spring cleaning right around the corner and special Oscar-branded Glad bags hitting Walmart shelves this April, it’s the perfect time for local groups, schools, and neighbors to organize clean-up events across the Valley. Whether you’re sprucing up a park, refreshing a neighborhood, or just making your own block a little brighter, every bag makes a difference.
Ready to join the movement? Rally your friends, family, or local organization and plan a Phoenix clean-up day this spring. Snap a photo of your crew with your Glad or Oscar-inspired trash bags and share it on social media using #GladToCleanPHX and #OscarLovesTrash. Let’s show how Phoenix turns trash day into a reason to celebrate!
“Phoenix, let’s get grouchy about litter and Glad about clean streets! Join our community clean-up and share your photos with #GladToCleanPHX.”
“Spotted: Oscar the Grouch in Phoenix! Grab your Glad bags, clean up your neighborhood, and tag #OscarLovesTrash for a chance to be featured.”
“Spring cleaning in Phoenix just got a lot more fun—thanks to Glad and Oscar! Who’s joining our next clean-up day? #GladToCleanPHX”
About the Brands
Glad, a member of The Clorox Company, has long been a leader in household waste solutions, while Sesame Workshop continues to inspire and educate families worldwide. This collaboration is a perfect blend of dependable products and beloved characters—reminding us all that even the messiest moments can spark a little joy.
The collaboration between Glad and Sesame Workshop for the “Don’t Get Mad. Get Glad.” campaign marks a creative partnership that blends household dependability with beloved children’s entertainment. By bringing Oscar the Grouch into the spotlight, Glad not only revives a classic campaign but also highlights the importance of making everyday chores more enjoyable for families. This partnership leverages Glad’s reputation as the nation’s leading provider of kitchen and outdoor trash bags and food protection products—trusted solutions designed to handle life’s messes with ease (Glad.com). Sesame Workshop, the nonprofit behind Sesame Street, has spent over 50 years enriching families worldwide through educational media and community outreach, helping children grow smarter, stronger, and kinder (Sesame.org). Together, their collaboration aims to inspire a new generation to see the positive side of cleaning up, all while celebrating the joy of community and play.
More than half of new articles on the internet are being written by AI – is human writing headed for extinction?
A new study finds over 50% of online articles are now AI-generated, raising questions about the future of human writing. Discover why formulaic content is most at risk, and why authentic, creative voices may become more valuable than ever.
Preserving the value of real human voices will likely depend on how people adapt to artificial intelligence and collaborate with it. BlackJack3D/E+ via Getty Images
More than half of new articles on the internet are being written by AI – is human writing headed for extinction?
Francesco Agnellini, Binghamton University, State University of New York The line between human and machine authorship is blurring, particularly as it’s become increasingly difficult to tell whether something was written by a person or AI. Now, in what may seem like a tipping point, the digital marketing firm Graphite recently published a study showing that more than 50% of articles on the web are being generated by artificial intelligence. As a scholar who explores how AI is built, how people are using it in their everyday lives, and how it’s affecting culture, I’ve thought a lot about what this technology can do and where it falls short. If you’re more likely to read something written by AI than by a human on the internet, is it only a matter of time before human writing becomes obsolete? Or is this simply another technological development that humans will adapt to?
It isn’t all or nothing
Thinking about these questions reminded me of Umberto Eco’s essay “Apocalyptic and Integrated,” which was originally written in the early 1960s. Parts of it were later included in an anthology titled “Apocalypse Postponed,” which I first read as a college student in Italy. In it, Eco draws a contrast between two attitudes toward mass media. There are the “apocalyptics” who fear cultural degradation and moral collapse. Then there are the “integrated” who champion new media technologies as a democratizing force for culture.Italian philosopher, cultural critic and novelist Umberto Eco cautioned against overreacting to the impact of new technologies.Leonardo Cendamo/Getty Images Back then, Eco was writing about the proliferation of TV and radio. Today, you’ll often see similar reactions to AI. Yet Eco argued that both positions were too extreme. It isn’t helpful, he wrote, to see new media as either a dire threat or a miracle. Instead, he urged readers to look at how people and communities use these new tools, what risks and opportunities they create, and how they shape – and sometimes reinforce – power structures. While I was teaching a course on deepfakes during the 2024 election, Eco’s lesson also came back to me. Those were days when some scholars and media outlets were regularly warning of an imminent “deepfake apocalypse.” Would deepfakes be used to mimic major political figures and push targeted disinformation? What if, on the eve of an election, generative AI was used to mimic the voice of a candidate on a robocall telling voters to stay home? Those fears weren’t groundless: Research shows that people aren’t especially good at identifying deepfakes. At the same time, they consistently overestimate their ability to do so. In the end, though, the apocalypse was postponed. Post-election analyses found that deepfakes did seem to intensify some ongoing political trends, such as the erosion of trust and polarization, but there’s no evidence that they affected the final outcome of the election.
Listicles, news updates and how-to guides
Of course, the fears that AI raises for supporters of democracy are not the same as those it creates for writers and artists. For them, the core concerns are about authorship: How can one person compete with a system trained on millions of voices that can produce text at hyper-speed? And if this becomes the norm, what will it do to creative work, both as an occupation and as a source of meaning? It’s important to clarify what’s meant by “online content,” the phrase used in the Graphite study, which analyzed over 65,000 randomly selected articles of at least 100 words on the web. These can include anything from peer-reviewed research to promotional copy for miracle supplements. A closer reading of the Graphite study shows that the AI-generated articles consist largely of general-interest writing: news updates, how-to guides, lifestyle posts, reviews and product explainers. https://stmdailynews.com/wp-admin/post-new.php#visibility The primary economic purpose of this content is to persuade or inform, not to express originality or creativity. Put differently, AI appears to be most useful when the writing in question is low-stakes and formulaic: the weekend-in-Rome listicle, the standard cover letter, the text produced to market a business. A whole industry of writers – mostly freelance, including many translators – has relied on precisely this kind of work, producing blog posts, how-to material, search engine optimization text and social media copy. The rapid adoption of large language models has already displaced many of the gigs that once sustained them.
Collaborating with AI
The dramatic loss of this work points toward another issue raised by the Graphite study: the question of authenticity, not only in identifying who or what produced a text, but also in understanding the value that humans attach to creative activity. How can you distinguish a human-written article from a machine-generated one? And does that ability even matter? Over time, that distinction is likely to grow less significant, particularly as more writing emerges from interactions between humans and AI. A writer might draft a few lines, let an AI expand them and then reshape that output into the final text. This article is no exception. As a non-native English speaker, I often rely on AI to refine my language before sending drafts to an editor. At times the system attempts to reshape what I mean. But once its stylistic tendencies become familiar, it becomes possible to avoid them and maintain a personal tone. Also, artificial intelligence is not entirely artificial, since it is trained on human-made material. It’s worth noting that even before AI, human writing has never been entirely human, either. Every technology, from parchment and stylus paper to the typewriter and now AI, has shaped how people write and how readers make sense of it. Another important point: AI models are increasingly trained on datasets that include not only human writing but also AI-generated and human–AI co-produced text. This has raised concerns about their ability to continue improving over time. Some commentators have already described a sense of disillusionment following the release of newer large models, with companies struggling to deliver on their promises.
Human voices may matter even more
But what happens when people become overly reliant on AI in their writing? Some studies show that writers may feel more creative when they use artificial intelligence for brainstorming, yet the range of ideas often becomes narrower. This uniformity affects style as well: These systems tend to pull users toward similar patterns of wording, which reduces the differences that usually mark an individual voice. Researchers also note a shift toward Western – and especially English-speaking – norms in the writing of people from other cultures, raising concerns about a new form of AI colonialism. In this context, texts that display originality, voice and stylistic intention are likely to become even more meaningful within the media landscape, and they may play a crucial role in training the next generations of models. If you set aside the more apocalyptic scenarios and assume that AI will continue to advance – perhaps at a slower pace than in the recent past – it’s quite possible that thoughtful, original, human-generated writing will become even more valuable. Put another way: The work of writers, journalists and intellectuals will not become superfluous simply because much of the web is no longer written by humans. Francesco Agnellini, Lecturer in Digital and Data Studies, Binghamton University, State University of New York This article is republished from The Conversation under a Creative Commons license. Read the original article.
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