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  Imaginary Numbers  

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What is an imaginary number?  What is i?

Contrary to what some people might tell you, imaginary numbers are not numbers that only exist in the brains of weird people. Or maybe they are; all numbers in math are "imaginary" in the sense that you can't touch them or experience them directly.

But this is not what people mean when they talk about imaginary numbers. Imaginary numbers are numbers that can be written as a real number times i.

So what is a real number, and what is i?

Well, the real numbers are all the positive numbers, negative numbers, and zero. These are numbers like

      5
     -2.9
     -4/3
      0
      1.11211211111221312211131122211113213211...
       (bonus points if you can tell us
       what the next few digits are and why!)

So the real numbers are the numbers that you probably already know: they're the ones on the number line.

What is i? It's the square root of -1 (see footnote below). And it's NOT a real number.  i was invented because people wanted to be able to take square roots of negative numbers, and you can't do that if you limit yourself to real numbers.

So we can make an imaginary number by taking a real number like 5 and multiplying it by i. That gives us 5i. Some other imaginary numbers are

     37.3i
      1.11211211111221312211131122211113213211i
     Pi*i.

Note that the square of any imaginary number (except 0) is a negative number.

Complex numbers are numbers like 7 + .4i; they're a real number plus an imaginary number.

Footnote: actually, there are TWO numbers that are the square root of -1, and those numbers are i and -i , just as there are two numbers that are the square root of 4, 2 and -2.

Browse the Dr. Math archives to find answers written for a variety of levels. Here are five:

And don't miss this commentary on the word "imaginary number":

Search the Dr. Math archives using the words "imaginary number" (that exact phrase; just the words, not the quotes) to find more questions and answers.

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