Real Life Exponentials
Accessible for all HS grades
We, humans, are unable to grasp the concept of how massive or tiny an item on either end of the scale really is. Our brains are not programmed to visualize this perfectly, as evolution did not deem it necessary. This is why we can’t appreciate how impressive a skyscraper is or how incredible microprocessors are.
Assuming that all buildings are built on the same surface area of 300m², whereas a house would weigh in around 140 tons, a skyscraper is estimated to weigh over 350 THOUSAND tons. This is over 2500 times the weight exerted on the foundation of a house over the same area. This is something we would have never taken into consideration or put thought into unless prompted to do so. The same goes for every other example mentioned underneath.
Speaking of size, you’ve probably once pondered, “why don’t all animals grow eternally?” Well, the answer to that question is simple. Taking a Cube as an example, its surface area and volume formulas are, where A equals the length of an edge, SA = 6A², and V = A³, respectively. As the value of V has a greater exponent value than that of SA, as the value of A increases, the ratio between SA and V will also increase with respect to V. This is represented by the following graph: (Blue Line represents Volume wherein Red Line represents Surface Area)
As you can tell, at one point, the volume starts to increase drastically faster than the surface area. This leads to size; as you have a higher volume, you store more heat. However, animals will not release enough heat through their surface area due to this disproportionate growth, causing problems. These problems include overheating. As the increased volume, thus increased mass, increases the amount of energy, thus heat, the organism creates from respiration. This explains why large animals have some adaptation to counter this. (e.g., elephants and their ears or whales and icy waters). On the other hand, this also explains why the smallest of animals are cold-blooded.
On the flip side, we don’t just see such exponential growth in nature. Per Moore’s law, the number of transistors on a microchip doubles every two years. So every two years, the performance of computers double and this law has been in effect since the 1970s and thus the performance you can buy with the same $1000 (not accounting for inflation) has drastically increased since then. This growth can be represented by the graph below:
This graph represents the number of transistors in a thousand-dollar processor over time. This graph's formula is y = 2^x * 1000, wherein y is the number of transistors whilst x is the number of years.
This graph was made by first considering that Moore’s law states that the number of transistors doubles each year. This shows that, when starting with 2000 transistors, every year, the number of transistors would be exponential 2 times 1000, thus creating the formula.
Compared to the measly 2000-transistor-microchips we had in the 1970s, a modern high-end microchip such as the AMC Epyc Rome would contain nearly 50 billion transistors. To put this in comparison, a single transistor in a 1970s microchip would take up the space of 25 million transistors in an AMC Epyc Rome, showing how large the scale mathematics really goes.
There are many other incarnations of exponential formulas in our daily lives across many different fields besides the examples above. Among them are: Stock trading, where a firm's steady annual growth returns stockholders profit exponentially relative to the initial investment; Bank Interest, which roughly is equivalent in theory to stock trading; Population increase, consequently, market demand; and even COVID-19.
This shows how our unfathomable scale of advancements, production, and all else is linked with mathematics, from something relatively far from us like stocks to the chips in literally all our electronics, from the skyscrapers, hundreds of meters tall, to each transistor in a processor, just a couple nanometers wide, its scale can be fathomed with mathematics.
