﻿ ﻿Stumped by a Sheet of Paper

# Stumped by a Sheet of Paper

Exponential Growth

A piece of paper is folded in two, then in half again, and again and again. How thick will it be after fifty folds? Write down your guess before you continue reading.

Second task. Choose between these options: (a) Over the next thirty days, I will give you \$1,000 a day. (b) Over the next thirty days, I will give you a cent on the first day, two cents on the second day, four cents on the third day, eight cents on the fourth day, and so on. Don’t think too long about it: A or B?

Are you ready? Well, if we assume that a sheet of copy paper is approximately 0.004 inches thick, then its thickness after fifty folds is a little over sixty million miles. This equals the distance between the earth and the sun, as you can check easily with a calculator. With the second question, it is worthwhile choosing option B, even though A sounds more tempting. Selecting A earns you \$30,000 in thirty days; choosing B gives you more than \$5 million.

Linear growth we understand intuitively. However, we have no sense of exponential (or percentage) growth. Why is this? Because we didn’t need it before. Our ancestors’ experiences were mostly of the linear variety. Whoever spent twice the time collecting berries earned double the amount. Whoever hunted two mammoths instead of one could eat for twice as long. In the Stone Age, people rarely came across exponential growth. Today, things are different.

“Each year, the number of traffic accidents rises by 7 percent,” warns a politician. Let’s be honest: We don’t intuitively understand what this means. So, let’s use a trick and calculate the “doubling time.” Start with the magic number of 70 and divide it by the growth rate in percent. In this instance: 70 divided by 7 = 10 years. So what the politician is saying is: “The number of traffic accidents doubles every ten years.” Pretty alarming. (You may ask: “Why the number 70?” This has to do with a mathematical concept called logarithm. You can look it up in the notes section.)

Another example: “Inflation is at 5 percent.” Whoever hears this thinks: “That’s not so bad, what’s 5 percent anyway?” Let’s quickly calculate the doubling time: 70 divided by 5 = 14 years. In fourteen years, a dollar will be worth only half what it is today—a catastrophe for anyone who has a savings account.

Suppose you are a journalist and learn that the number of registered dogs in your city is rising by 10 percent a year. Which headline do you put on your article? Certainly not: “Dog Registrations Increasing by 10 Percent.” No one will care. Instead, announce: “Deluge of Dogs: Twice as Many Mutts in Seven Years’ Time!”

Nothing that grows exponentially grows forever. Most politicians, economists, and journalists forget that. Such growth will eventually reach a limit. Guaranteed. For example, the intestinal bacterium Escherichia coli divides every twenty minutes. In just a few days, it could cover the whole planet, but since it consumes more oxygen and sugar than is available, its growth has a cutoff point.

The ancient Persians were well aware that people struggled with percentage growth. Here is a local tale: There was once a wise courtier who presented the king with a chessboard. Moved by the gift, the king said to him: “Tell me how I can thank you.” “Your highness, I want nothing more than for you to cover the chess board with rice, putting one grain of rice on the first square, and then on every subsequent square, twice the previous number of grains.” The king was astonished: “It is an honor to you, dear courtier, that you present such a modest request.” But how much rice is that? The king guessed about a sack. Only when his servants began the task—placing a grain on the first square, two grains of rice on the second square, four grains of rice on the third, and so on—did he realize that he would need more rice than was growing on earth.

When it comes to growth rates, do not trust your intuition. You don’t have any. Accept it. What really helps is a calculator or, with low growth rates, the magic number of 70.

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