- Exponential growth is a basic concept in mathematics that occurs when a quantity keeps doubling.
- This can lead to surprisingly high numbers very quickly.
- Exponential growth is a big part of how technology has developed rapidly over the past few decades, and is an important thing to keep in mind when making decisions about saving and investing.

One of the most prevalent concepts in mathematics that is often counterintuitive and leads to surprising results is exponential growth. Exponential growth is characterized by an amount that doubles over and over again over a period of time, which can lead to surprisingly high numbers very quickly.

There is a classic legend about the invention of chess that beautifully illustrates the shocking consequences of exponentials. The idea is that a vizier or a 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 asks that the king place a 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 must place 1,024 grains of wheat. On the 21st, we pass the milestone of one million grains. On the last tile of the board, the king must place a 9 223 372 036 854 775 808 grains of wheat. In total, the board has something on the order of 18 quintillion grains of wheat.

Assuming a grain of wheat weighs around 0.065 grams, as the Encyclopedia Britannica suggests, this represents around 1.2 trillion metric tonnes of wheat, or roughly 1,600 times the total wheat production in the world in 2017 / 2018, according to Statista.

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The unfathomable numbers that can emerge from exponential processes can make the mind waver. In our day to day life, we are more used to seeing things that develop in a more linear or similar sub-exponential way, and therefore situations where there is a constant doubling can throw us off balance.

## Exponential growth is before us every day, in our retirement accounts and our investments

Exponential growth is the key to saving and investing. Compound interest is a great example of an exponential growth process. Because you earn more on the interest you’ve already earned, starting to save early can be very fruitful.

As an example, the following table compares two savers. Both invest $ 100 per month at an annual compound rate of return of 5%. The first saver starts investing at 25 until he retires at 65, while the second starts saving at 35. But due to the exponential nature of

compound interest

, these ten additional years of savings means that the early saver has almost twice as much in their bank account as the subsequent saver at age 65:

## We are also seeing exponential growth in technology

Exponential growth appears in many real and real situations. One of the most famous is Moore’s Law, which originated in computer technology. Intel co-founder Gordon Moore observed in the 1960s that the number of transistors that could be placed on a computer chip tended to double every two years or so. This doubling has led to an exponential growth in computing power over the past decades, which has revolutionized the global economy and our lives.

Other technologists have generalized the idea of an exponentially growing technology. Futurist Ray Kurzweil described a broad “law of accelerated returns”, pointing to exponential increases in other areas of technology like memory, hard disk storage, Internet speeds and DNA sequencing.

Kurzweil goes further and suggests that, if the exponential growth of technology continues, the rate of change could become so rapid that it escapes our current understanding in a “technological singularity.” Kurzweil predicted, “We will not live a hundred years of progress in the 21st century; rather we will witness the order of twenty thousand years of progress (at *today* rate of progress, i.e.).