Exponential growth explains how technology has developed in recent decades. At the same time, it is important to keep this in mind when making decisions about saving and investing.

One of the most ubiquitous concepts in mathematics, often counterintuitive but leading to surprising results, is exponential growth. Exponential growth is characterized by an amount that doubles over a period of time over and over again. And that leads to surprisingly large numbers very quickly.

There is a classic legend about the invention of chess that illustrates the consequences of exponentials. A minister invents the game in ancient India and presents it to his king. The king asks what he wants as a reward and the minister asks him to place a grain of wheat in the first square of a chessboard, two grains in the second square, four grains in the third, eight grains in the fourth, and so on. Successively for each of the 64 squares on the 8 × 8 square board.

## Is it possible to travel in a time machine? A mathematical model can make it viable

The king accepts, but the situation becomes untenable due to the constant duplication. On the eleventh tile of the board, the king needs to place 1,024 grains of wheat. On the 21st, we surpassed the million grain mark. On the final tile of the board, the king needs to place 9.223.372.036.854.775.808 wheat grains.

The large numbers that arise from the exponential processes cause our mind to wobble. In everyday life, we are more accustomed to seeing things that develop more linearly or sub exponentially, so that situations of constant duplication mislead us.

## Exponential growth is in front of us every day, in our retirement and investment accounts

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

## The 17 mathematical equations that changed the world

As an example, the following table compares two savers. Both invest \$ 100 per month at a compound annual rate of return of 5%. The first saver begins to invest at age 25 until retirement at 65, while the second begins saving at age 35. But due to the exponential nature of compound interest, those ten additional years of saving mean that, at age 65 , the first saver has almost twice as much in his account as the second.

## We also see an exponential growth of technology

Exponential growth appears in many situations in the real world. One of the most famous is Moore’s Law of technology. Intel co-founder Gordon Moore observed in the 1960s that the number of transistors that can be placed on a computer chip tends to double every two years. This duplication has led to the exponential growth of computing power in recent decades that has revolutionized the global economy and our lives.

Other technologists have generalized the idea of ​​an exponentially growing technology. The futurist Ray Kurzweil described a broad “Law of Return Acceleration”, pointing to exponential increases in other technological areas such as 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 understanding. This is known as “technological uniqueness”.

Kurzweil was predicting: “it is not true that we will experience one hundred years of progress in the 21st century, but that we will witness the order of twenty thousand years of progress”