Exponential Patterns and Large Numbers

Exponential patterns are common in natural systems, but the implication of exponential growth and decay can be hard to comprehend.

The story of the wheat and the chessboard is a good way to introduce the challenge.

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There are many variations of the story, but the basic idea is that a wise man once asked a king to reward him by giving him as much wheat as could fit on a chessboard based on the following pattern:

Place one grain of wheat on the first square,

two on the second,

four on the third,

and continue doubling the number of grains until the entire board is full.

Most people's intuition is that this won't add up to very much wheat.

That was the king's reaction, but after agreeing, he discovered that the 64th square of the board requires an astoundingly large number of wheat grains:

For perspective, this is equal to roughly five thousand years of U.S. wheat production - more than all the wheat that has ever been grown.

It is enough wheat to fill 168 million Olympic sized swimming pools.

The reason these questions trick us is that it is an exponential relationship and we don't have an intuitive sense of exponential patterns.

We see these small increases at the start of the set and fail to appreciate how quickly this continued doubling creates large numbers.

Exponential patterns are not limited to abstract situations like calculating the amount of wheat that can fit on a chessboard.

Many natural systems contain parameters that vary over ranges that make exponential notation useful in describing them.

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Electromagnetic energy is an example.

Types of electromagnetic energy range from long wavelength, low energy radio waves to extremely short wavelength, high energy x-ray and gamma radiation.

We measure, describe, and compare different types of electromagnetic energy using the energy's wavelength. The length of radio waves are measured in meters while gamma rays are measured in femtometers.

One femtometer is equal to 0.000000000000001 meters.

This range of values is almost as large as the difference between the number of grains of wheat on the first and 64th square of the chessboard.

Sound intensity is another scale that covers a huge range.

We measure sound using the decibel scale.

The scale ranges from one dB, which is the lower limit of human hearing to values well over 100 dB which are dangerously loud and can damage the human ear.

The Richter scale for earthquakes covers a range of motion from shakes that are below detection to ones that can destroy large buildings and are felt miles away.

pH, the scale we use to describe the acidity of solutions, also describes an exponential pattern.

The pH scale goes from 0 to 14 and describes the concentration of H+ ions in a solution.

The highest value, 14, has the lowest H+ concentration and the lowest value, 0, has the highest H+ concentration.

Each step single step down from 14 to 0 represents a concentration of hydrogen ions that is ten time greater than the previous one.

Just like the number of grains of wheat on the 64th square, all of these are examples of ranges and values that we have trouble conceptualizing even though we have well-developed methods for calculating and discussing them.

In order to describe values over such large ranges, scientists use scientific notation.

While many find it relatively easy to write numbers large and small numbers, it is important to also keep in mind the implications of needing this scientific notation.

These changes are huge. So huge that it is easy for us to be tricked by them. Our brains are not well suited to understanding values like 1,000,000,000,000 at an intuitive level.

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