The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition

Chapter 4

Counting Infinite Sets

4.1 Finite Sets

The natural numbers are the numbers introduced by the Hindu and Arabic cultures. Today they are written as

This is an implied list. The three dots,…, indicate that the pattern continues indefinitely, or without stopping. Although this set appears to be small, it contains numbers that we do not encounter in our daily lives. Here are some examples of natural numbers that you have probably not met before.

1. The number of people on the Earth is about 7,000,000,000 or 7 × 10⁹.
2. The bailouts funded by the federal government in 2010 amounted to over 700 billion, or 7 × 10¹¹, dollars.
3. The distance across the observable universe is about 15 × 10⁹ light years. This is 15 followed by 9 zeros. It takes light 15 × 10⁹ years to travel this distance. This number doesn't have a name, only a scientific significance.
4. The number of fish in the ocean on January 1, 2005 is quite large. Even though we can't write down this number, it is still a number and we can refer to it by the symbol, ϕ, pronounced fee.
5. The number of grains of sand on the Earth was first computed by Archimedes in about 100 BC. We will figure out how large this number is, but I think we can all agree that even though the number exists, it is impossible to write the exact number down.

Let us see how these numbers compare. To do this we will have to expand your imagination a bit. Once ...