- Reuters/Gleb Garanich
- Exponential growth is a basic concept in math that occurs when a quantity keeps doubling itself over and over.
- This can lead to astonishingly large numbers very quickly.
- Exponential growth is a big part of how technology has rapidly developed in the last several decades, and is an important thing to keep in mind when making decisions about saving and investment.
One of the more pervasive concepts in math that is often counterintuitive and leads to surprising results is exponential growth. Exponential growth is characterized by a quantity doubling itself in a particular time period over and over again, and this can lead to surprisingly large numbers very quickly.
There’s a classic legend about the invention of chess that beautifully illustrates the shocking consequences of exponentials. The idea is that a vizier or minister invents the game in ancient India and presents it to his king. The king asks the vizier what he wants as a reward, and the vizier requests that the king place one grain of wheat on the first square of a chessboard, then two grains on the second square, four grains on the third, eight grains on the fourth, and so on for each of the 64 squares on the 8×8 square board.
The king happily obliges this request, but the situation quickly becomes untenable due to the constant doubling. On the 11th tile of the board, the king needs to place 1,024 grains of wheat. On the 21st, we pass the 1 million grains mark. On the final tile of the board, the king needs to place a virtually unfathomable 9,223,372,036,854,775,808 grains of wheat. In total, the chess board has something on the order of 18 quintillion grains of wheat.
Assuming a grain of wheat weighs about 0.065 grams, as suggested by the Encyclopedia Britannica, that is about 1.2 trillion tons metric tons of wheat, or about 1,600 times the total production of wheat worldwide in 2017/2018, according to Statista.
The unfathomably large numbers that can emerge from exponential processes can make one’s mind reel. In our day to day life, we are more used to seeing things that develop in a more linear or similarly sub-exponential way, and so situations where there is constant doubling can throw us off.
Exponential growth is in front of us every day, in our retirement accounts and investments
Exponential growth is key to saving and investing. Compound interest is a prime example of an exponential growth process. Because you earn further on the interest you have already made, starting to save early can be very fruitful.
As an example, the following chart compares two savers. Both invest $100 per month at a 5% annual compound rate of return. The first saver begins investing at age 25 until retiring at 65, while the second begins saving at age 35. But because of the exponential nature of compound interest, that extra ten years of saving means that the early saver has nearly twice as much in her bank account as the later saver by the time she is 65:
- Business Insider/Andy Kiersz
We also see exponential growth in tech
Exponential growth shows up in many actual, real-world situations. One of the most famous is Moore’s Law from computer technology. Intel cofounder Gordon Moore observed in the 1960s that the number of transistors that could be placed on a computer chip tends to double about every two years. That doubling has led to the exponential growth of computing power in the last several decades that has revolutionized the global economy and our lives.
Other technologists have generalized the idea of exponentially growing technology. Futurist Ray Kurzweil described a broad “Law of Accelerating Returns,” pointing out exponential increases in other technological areas like memory, hard drive storage, internet speeds, and DNA sequencing.
Kurzweil goes further and suggests that, should the exponential growth of technology continue, the rate of change could become so fast that it escapes our present understanding in a “technological singularity.” Kurzweil predicted, “it is not the case that we will experience a hundred years of progress in the twenty-first century; rather we will witness on the order of twenty thousand years of progress (at today’s rate of progress, that is).”
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