Complex numbers

Essay by pusiu January 2004

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The simplest way to define a complex or imaginary number is a number that is a multiple of i in which i is defined by the property of i squared equals -1.

This is puzzling to most people, because it is hard to imagine any number having a negative square. The result: it is tempting to believe that i doesn't really exist, but is just a convenient mathematical fiction.

This isn't the case. Imaginary numbers do exist. Despite their name, they are not really imaginary at all. (The name dates back to when they were first introduced, before their existence was really understood. At that point in time, people were imagining what it would be like to have a number system that contained square roots of negative numbers, hence the name "imaginary". Eventually it was realized that such a number system does in fact exist, but by then the name had stuck.)

When talking about numbers, there are many very different contexts that you could have in mind.

Here are the four most familiar ones:

*The Natural Numbers. These are the counting numbers 1, 2, 3, . . . that are possible answers to the question "how many?" They are abstract concepts that describe sizes of sets.

*The Integers. These are abstract concepts that describe, not sizes of sets, but the relative sizes of two sets. They are the possible answers to the question "how many more does A have than B has?" They include both positive numbers (meaning A has more than B) and negative numbers (meaning B has more than A).

*The Rational Numbers. These are abstract concepts that describe ratios of sizes of sets. They do not model sizes of sets the way that natural numbers do. If you say "I ate 3/4 of a pie",