How to Understand Imaginary Numbers

Any mathematician will tell you that imaginary numbers-despite their name-are every bit as "real" as the counting numbers such as one, two or three. However, there is something decidedly odd about imaginary numbers. They are the square roots of negative quantities, which according to the basic arithmetic kids are taught in grade school shouldn't be allowed to exist. Here is how to wrap your mind around this elusive concept.


    • 1

      In basic arithmetic, every number "squares" to a positive number. The number 4 squared (that is, multiplied by itself four times) is 16, and the number negative 4 squared is also 16. This is fine as far as it goes, but over the last few centuries mathematicians realized that it was useful to work with another quantity, which they defined as the square root of negative one (symbolized by "i")-what they called an imaginary number.

    • 2

      Any imaginary number can be formed from multiples of i. Although it is technically possible to assign symbols to "the square root of minus 2" or "the square root of minus one-half," it is formally easier just to write these imaginary numbers as multiples of the square root of negative one. So, the square root of minus 2 is denoted as 2i, and the square root of minus one-half as 1/2i.

    • 3

      An imaginary number is one component of a "Complex Number." Mathematicians and physicists rarely work with purely imaginary numbers; rather, they juggle what are known as complex numbers, which are made of a real number combined with an imaginary number. Examples of complex numbers include 5 + 2i or minus 17 + 7.5i. Conveniently, complex numbers can be added, subtracted, multiplied and divided, just like counting numbers.

    • 4

      Imaginary and complex numbers aren't part of the classic number line. On a straight number line, it's a simple matter to determine that (for example) the number 17 is larger than the number 3. However, imaginary and complex numbers are charted on what's known as the "Complex Plane," an extension of the number line to two dimensions. For this reason, there's no sense in which (say) the number 5i is greater than or lesser than the number 2.

    • 5

      Imaginary numbers aren't just a human invention. Part of the reason imaginary numbers are so useful is that nature makes use of them in all sorts of ways, ranging from relativity theory to quantum physics. That's why physicists will tell you that imaginary or complex numbers are every bit as real as snowflakes, rather than quantities that have been made up by mathematicians to amuse themselves!

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