Fifty years ago, a highly respected American physicist warned: “The greatest shortcoming of the human race is our inability to understand the exponential function.” Today, his caution is more relevant than ever as technology advances, demands on resources mount, and changes that once took centuries or decades can happen in years. Sometimes months… or days. Yet humanity’s blindspot is ignoring the dangers of exponential increase or the power of compounding growth.

Professor Albert Bartlett taught physics at the University of Colorado and spent years explaining a simple mathematical principle that even impressively educated people frequently underestimate. He believed humanity’s failure to understand compounding function was leading us toward serious problems that could’ve been anticipated and possibly/probably prevented.

Today, we’re witnessing extraordinary advances in artificial intelligence, computing, medicine, and scientific discovery. Some of these developments are compounding at remarkable rates, bringing enormous opportunities along with equally significant challenges. The trouble is that we aren’t very good at recognizing where these changes may take us.

But Bartlett’s message wasn’t all doom and gloom, and neither is the future. Understanding exponential growth can help us recognize dangers, anticipate change, and take advantage of opportunities that previous generations could scarcely have imagined. And as we’ll discover, the mathematics concealed in humanity’s dangerous blind spot is surprisingly simple.

The World Is Changing Faster Than We Can Understand It

Think back to when the internet came through a telephone line and the modem sounded like it was trying to contact Mars. In a single working lifetime, we’ve moved from waiting for a webpage to load to carrying much of the world’s accumulated knowledge in our pockets. Now artificial intelligence can analyze information, generate explanations, and write computer code in seconds. That’s astonishing progress, but the real problem isn’t speed alone.

What gets us into trouble is treating tomorrow as a slightly larger version of today. When something grows by a fixed percentage, each increase builds on a bigger base. For a while, the difference hardly seems worth mentioning. Then it becomes impossible to ignore, and by the time the consequences become obvious, opportunities to deal with them may have passed.

This matters far beyond computers. Compound interest can quietly build a retirement fund or bury a household in debt. A contagious disease can spread faster than a healthcare system can respond. Growing demands for energy, water, and other resources can overwhelm infrastructure designed around yesterday’s expectations. Not everything grows exponentially, of course, and nothing physical can continue doing so indefinitely. But failing to recognize compounding growth can come at an enormous cost.

There’s an encouraging side to all this. Understanding exponential growth isn’t particularly difficult, and it doesn’t require an advanced mathematics degree. What it does require is learning to look beyond what’s happening today and recognize what might happen tomorrow. That’s the lesson one remarkably persistent physics professor spent much of his life trying to teach.

The Professor Who Tried to Warn the World

Albert Allen Bartlett wasn’t an alarmist who’d discovered a catchy line. He was a physicist with a Harvard doctorate who’d worked at Los Alamos during the Manhattan Project before joining the University of Colorado in 1950. He taught there for decades, became president of the American Association of Physics Teachers, and earned a reputation as a gifted teacher. But the work he’d become best known for had less to do with advanced physics than elementary arithmetic.

On September 19, 1969, Bartlett gave a lecture to university students called Arithmetic, Population and Energy. His concern was that politicians, business leaders, and ordinary citizens kept talking about steady growth as though it could carry on forever. Population was increasing, energy consumption was rising, and apparently sensible projections were ignoring what happens when one year’s growth gets added to the next. The numbers weren’t complicated. Their consequences were.

Bartlett took that lecture on the road and eventually delivered it 1,742 times, across 49 American states and seven other countries. Think about that. A respected scientist spent much of his life repeating essentially the same lesson because he believed people still weren’t getting it. He wasn’t arguing that every kind of progress was bad. He was warning that growing demands on finite resources couldn’t continue indefinitely at a fixed percentage rate, regardless of how comfortable the early numbers looked.

Bartlett died in 2013 at ninety, but the problem he identified didn’t retire with him. We’re now surrounded by growth figures, projections, and technological forecasts that are easy to repeat and much harder to understand. His lasting contribution wasn’t inventing the exponential function. It was insisting that we learn what it actually means before making decisions based on it. And the place to start is with the deceptively simple mathematics.

The Mathematics That Fools Almost Everyone

The mathematics behind exponential growth isn’t particularly difficult. Most of us understand that adding ten to a number makes it ten larger. That’s linear growth, where the same amount gets added each time. Exponential growth works differently because each increase is calculated on the new, larger total. The percentage stays the same, but the amount being added keeps growing. That’s where our intuition often lets us down.

Let’s say you start with 100 dollars and add five dollars every year. After thirty years, you’ll have 250 dollars. But if that original 100 dollars grows by five percent annually, with each year’s earnings added to the total, you’ll have approximately 432 dollars after thirty years. The difference comes from compounding, where growth earns further growth. It’s the same principle that makes compound interest so valuable to investors and so expensive to people carrying unpaid debt.

Bartlett frequently explained exponential growth using the Rule of 70, a simple mathematical shortcut for estimating doubling time. Divide 70 by the annual percentage growth rate and you’ll get approximately how many years it’ll take something to double. At seven percent growth, that’s about ten years. After twenty years, the original amount has roughly quadrupled, and after thirty it’s approaching eight times its starting value. Nothing magical has happened. The mathematics has simply been doing its work.

Here’s what makes this so deceptive. During the early stages, exponential growth can appear almost insignificant, especially when the starting amount is small. By the time the increases become impossible to ignore, the process may have been underway for years. Of course, nothing in the physical world can grow exponentially forever, but understanding how quickly things can change before reaching their limits is essential. Bartlett had a favourite illustration involving a chessboard and a few grains of wheat. It’s a story worth telling because the numbers become almost unbelievable.

What is the exponential function, and how does exponential growth work? The exponential function describes a quantity that changes by a constant multiplying factor over equal intervals. Unlike linear growth, which adds the same amount each time, exponential growth compounds on an increasingly larger base. For example, $100 growing at seven percent annually becomes approximately $197 after ten years. The Rule of 70 provides a simple way to estimate doubling time: divide 70 by the annual percentage growth rate.

The King, the Chessboard, and the Grain of Rice

Bartlett had a favourite story that illustrated exponential growth better than any mathematical formula. According to an ancient legend, a king was so pleased with the invention of chess that he offered its creator whatever reward he wanted. The inventor made what seemed like a ridiculously modest request. He wanted one grain of rice placed on the first square of a chessboard, two grains on the second, four on the third, eight on the fourth, and so on, doubling the amount on each successive square until all sixty-four were covered.

The king must’ve thought he’d struck a pretty good bargain. After all, how much could a few grains of rice possibly cost? But the inventor understood something the king didn’t. The first ten squares required only 1,023 grains in total. By the twentieth square, the total exceeded a million, and by the thirtieth, it had passed a billion. Things were getting expensive.

Now comes the astonishing part. By the sixty-fourth square, the king would’ve needed 9.2 quintillion grains for that square alone. The total for the entire chessboard would’ve been approximately 18.4 quintillion grains, or 18,446,744,073,709,551,615 to be precise. Depending on the size of the grains, that’s hundreds of billions of tonnes of rice. The royal treasury wasn’t going to cover that bill.

Bartlett used wheat rather than rice in his telling, but the lesson was identical. Each new square required one grain more than the combined total of every preceding square. Think about that. After sixty-three doublings, the final increase was greater than everything that had accumulated before it. That’s the extraordinary mathematical power of compounding, and it explains why apparently manageable growth can eventually produce overwhelming consequences.

The chessboard is only a legend, but the arithmetic is indisputable. The unsettling question is why something so straightforward remains so difficult for intelligent people to appreciate when the numbers involve real money, resources, populations, or technological change.

Why Our Brains Keep Getting the Future Wrong

The king’s mistake wasn’t that he couldn’t do arithmetic. He simply couldn’t imagine where all that doubling would lead, and he’s hardly alone. Most of us make judgments about the future based on what we’ve experienced in the past. If something’s been changing slowly, we tend to expect more of the same. That works reasonably well for many everyday situations, but it can get us into serious trouble when growth starts compounding.

Psychologists have studied this tendency for decades and given it a name: exponential-growth bias. Research consistently shows that people underestimate how quickly quantities increase when they’re growing by a fixed percentage. In a 2022 study involving 521 participants, researchers found that people were reasonably accurate at predicting straight-line growth but regularly underestimated exponential trends. Mathematical knowledge helped, although even understanding the principle doesn’t always prevent us from making the mistake.

There’s another problem. We’re often more confident in our predictions than the evidence warrants. During the COVID-19 pandemic, for example, many people had difficulty appreciating how quickly infections could multiply before hospitals became overwhelmed. Researchers found that helping people understand exponential growth changed their perception of the danger. What seemed manageable one week could become a public health emergency shortly afterward.

None of this means humans are incapable of understanding exponential change. It means our first impressions aren’t always reliable, particularly when we’re dealing with unfamiliar numbers and long-term projections. The sensible approach is to question our assumptions, examine the actual growth rates, and do the arithmetic rather than trust our instincts. That’s especially important when the consequences involve something more substantial than a king’s imaginary rice bill.

Why do humans have difficulty understanding exponential growth? Humans tend to judge the future by extending familiar patterns from the past, which makes the accelerating effects of compounding difficult to appreciate. Physicist Albert Bartlett warned that this misunderstanding leads societies to underestimate the consequences of population growth and increasing resource consumption. The classic chessboard illustration demonstrates the problem: doubling one grain on each successive square produces more grains on the sixty-fourth square than on all sixty-three preceding squares combined.

Exponential Growth Isn’t Just a Mathematical Curiosity

Bartlett wasn’t particularly concerned about chessboards or imaginary rice bills. He was worried about what happens when people fail to recognize exponential growth in the real world, where the consequences can affect their finances, health, resources, and future. Consider compound interest. Money invested at a reasonable rate of return can grow substantially over several decades because the earnings themselves generate further earnings. But the same principle works against anyone carrying unpaid credit card debt. Compounding doesn’t care whether you’re getting richer or poorer.

Infectious diseases provide another sobering example. During the early stages of an epidemic, when most people remain susceptible, infections can multiply exponentially as each infected person passes the disease to others. That’s why apparently small outbreaks can develop into major public health emergencies before authorities have time to respond. The good news is that infections don’t keep multiplying indefinitely. Immunity, changing behaviour, medical interventions, and other factors can slow or reverse the spread.

Population growth was Bartlett’s principal concern. The world’s population reached approximately one billion around 1800 and passed eight billion in 2022. But here’s an important distinction. Global population isn’t currently growing exponentially at a constant rate. Growth rates have been declining for decades, and United Nations projections suggest the population could peak at approximately 10.3 billion during the 2080s. That’s a reminder that growth rates change, and projections aren’t guarantees.

Nevertheless, Bartlett’s warning about finite resources remains relevant. Increasing populations and rising consumption can place enormous demands on energy, water, food production, and infrastructure. Technological improvements can expand supplies and increase efficiency, but they can’t abolish physical limits. The challenge is understanding how demand, resources, innovation, and time interact before shortages or other consequences become serious.

And there’s another side to this equation. The same mathematics that compounds financial burdens and resource demands can help produce extraordinary advances in science and technology. We’re now witnessing developments in artificial intelligence that would’ve seemed like science fiction a generation ago. That brings Bartlett’s half-century-old warning directly into the world we’re living in today.

Artificial Intelligence: Exponential Growth Meets the Twenty-First Century

If Albert Bartlett were alive today, he’d have a fascinating new subject for his lecture. When he first warned about exponential growth in 1969, computers filled rooms, the internet was in its infancy, and artificial intelligence was largely confined to research laboratories. Today, AI systems can write computer programs, analyze complex information, solve advanced mathematical problems, and assist scientists with discoveries that would’ve taken years using conventional methods. And the technology’s improving at a pace that’s difficult to comprehend.

Consider what Stanford University’s annual AI Index has documented. Its 2025 report found that the computing power used to train notable AI models had been doubling approximately every five months. Meanwhile, the cost of running a model capable of performing at a benchmark level comparable to GPT-3.5 fell more than 280-fold between November 2022 and October 2024. In other words, we’re developing increasingly powerful technology while making many of its capabilities dramatically cheaper and more accessible.

Stanford’s 2026 report paints an equally remarkable picture. Artificial intelligence reached 53 percent adoption within three years, according to the report’s global adoption measure, considerably faster than personal computers or the internet achieved comparable penetration. AI systems designed to operate computers independently improved their success rate on one established benchmark from approximately 12 percent to 66 percent in a single year. That’s substantial progress, although it also means those systems were still failing roughly one-third of the tested tasks.

But here’s where we need to apply some of Bartlett’s mathematical discipline. Not every improvement in artificial intelligence follows a consistent exponential curve, and nobody can confidently predict how long present rates of advancement will continue. Some capabilities are improving extraordinarily quickly, while others remain frustratingly unreliable. An AI system might solve an advanced mathematical problem yet struggle with something a reasonably competent human would find straightforward. Technological progress doesn’t always follow a smooth or predictable path.

What makes this development particularly significant is how improvements can reinforce one another. Better computing equipment enables more ambitious research, improved algorithms make existing hardware more productive, and cheaper AI services encourage wider adoption and further investment. The potential benefits for medicine, education, scientific research, and economic productivity are enormous, but so are the challenges of adapting to changes occurring faster than many institutions can respond.

Bartlett would’ve recognized both the mathematics and the warning signs. What we’re witnessing isn’t simply another technological invention. It’s a rapidly evolving collection of capabilities that could fundamentally change how humanity learns, works, discovers, and solves problems. But even the most sophisticated artificial intelligence depends on something decidedly old-fashioned: electricity, physical resources, and the infrastructure needed to keep it running.

Is artificial intelligence advancing exponentially, and what are the consequences? Some aspects of artificial intelligence have advanced at approximately exponential rates over particular periods, including the computing power used to train leading AI models. However, AI capability, affordability, and adoption don’t follow one universal exponential curve. Physical resources, electricity, computing infrastructure, and technological limitations constrain progress. Understanding these trends helps societies anticipate potential disruptions while recognizing opportunities for scientific discovery, improved healthcare, education, and greater human capability.

When Exponential Progress Meets Physical Reality

Artificial intelligence might seem almost magical when we’re sitting at a computer watching it perform tasks that would’ve been impossible a few years ago. But behind every AI interaction is an enormous physical infrastructure of computer chips, servers, data centres, electrical grids, and cooling systems. These machines consume electricity, generate heat, and require substantial resources to manufacture and operate. The digital world might appear limitless, but it still has to obey the laws of physics.

According to the International Energy Agency, the world’s data centres consumed approximately 485 terawatt-hours of electricity in 2025. That’s expected to nearly double to 950 terawatt-hours by 2030, representing roughly three percent of global electricity demand. These figures include conventional data centres as well as those dedicated to AI, but artificial intelligence is a major driver of the increase. And those numbers are projections, not guarantees.

Here’s where things get complicated. Software can improve almost overnight, but building power plants, manufacturing electrical transformers, and expanding transmission networks can take years. Data centres also need water or other cooling arrangements, suitable locations, and access to reliable power. In some regions, rapidly increasing demand could strain electrical grids, raise costs, and compete with other essential infrastructure requirements. You can’t solve every physical problem with a software update.

There’s another interesting twist. As computer chips and AI algorithms become more efficient, individual tasks often require less electricity. That’s encouraging, but cheaper technology can also encourage greater use, and increasingly sophisticated applications may consume substantially more energy. Improvements in efficiency don’t necessarily translate into lower overall consumption. It’s another example of why understanding the interaction between growth and resources matters.

Bartlett would’ve recognized this conflict immediately. Exponential progress in one area doesn’t mean everything supporting it can expand at the same rate. Physical limits, economic realities, and human decisions all influence how far and how quickly technology can advance. But those constraints don’t mean progress has to stop. In fact, artificial intelligence might help us overcome some of the very problems its rapid development is creating.

The Other Side of the Exponential Curve

After looking at the growing demands on energy and natural resources, it’d be easy to conclude that exponential growth is mostly bad news. It isn’t. The same mathematics that can turn a small problem into a major headache can also magnify the benefits of scientific discovery, education, and technological progress. One improvement creates opportunities for another, and before long we’re doing things that would’ve seemed impossible a generation ago.

Take medical research. Google’s DeepMind developed an artificial intelligence system called AlphaFold that’s helped scientists predict the structures of hundreds of millions of proteins. That’s information researchers can use to better understand diseases and investigate potential treatments. It doesn’t mean AI has suddenly cured everything, but it’s opened possibilities that would’ve required enormous amounts of conventional research.

Or consider weather forecasting. European researchers have developed AI systems that can outperform traditional forecasting methods on several important measures, including predicting storm tracks. Better forecasts can’t stop a hurricane, but they can give people more time to prepare. In education, AI can provide personalized explanations and learning assistance that would’ve been difficult or expensive to obtain just a few years ago.

And here’s where things become particularly interesting. Artificial intelligence isn’t simply another invention that makes existing tasks faster. It can help humans investigate problems, test ideas, develop knowledge, and improve the very methods we use to make further discoveries. That creates the possibility of progress building on progress, although it doesn’t guarantee that every advance will produce another.

Of course, none of this happens automatically. Technology can make mistakes, create dependencies, and produce unintended consequences. Human judgment, critical thinking, and responsible decisions remain essential. But understanding exponential growth gives us something more valuable than the ability to predict trouble. It gives us an opportunity to recognize possibilities, prepare for change, and deliberately influence the future we’re creating.

Understanding the Curve Before It Overtakes Us

Albert Bartlett spent decades trying to get one simple message across. Exponential growth isn’t difficult to calculate, but its consequences can be extraordinarily difficult to imagine. He wasn’t asking everyone to become a mathematician. He wanted us to recognize the pattern, question our assumptions, and understand what might happen before it’s too late to do something about it.

That’s particularly important today as artificial intelligence and other technologies change how we live, work, and understand the world. We can’t predict the future simply by extending a mathematical curve, but we can become better at recognizing emerging problems and extraordinary opportunities.

There’s every reason to be optimistic about what lies ahead, provided we approach change with knowledge, curiosity, and sound judgment. Bartlett’s warning wasn’t a declaration that humanity was doomed. It was a reminder that understanding reality gives us a better chance of shaping our future rather than being overtaken by it.