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Mathematical Constants
Complex Numbers

How do you use imaginary numbers in real life?


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January 02, 2010 3:30AM

Although most of us do not use imaginary numbers in our daily life, in engineering and physics they are in fact used to represent physical quantities, just as we would use a real number to represent something physical like the length of a stick or the distance from your house to your school.

In general, an imaginary number is used in combination with a real number to form something called a complex number, a+bi where a is the real part (real number), and bi is the imaginary part (real number times the imaginary unit i). This number is useful for representing two dimensional variables where both dimensions are physically significant. Think of it as the difference between a variable for the length of a stick (one dimension only), and a variable for the size of a photograph (2 dimensions, one for length, one for width). For the photograph, we could use a complex number to describe it where the real part would quantify one dimension, and the imaginary part would quantify the other.

In electrical engineering, for example, alternating current is often represented by a complex number. This current requires two dimensions to represent it because both the intensity and the timing of the current is important. If instead it were a DC current, where the current was totally constant with no timing component, only one dimension is required and we don't need a complex number so a real number is sufficient. The two key points to remember are that the imaginary part of the complex number represents something physical, unlike it's name implies, and that the imaginary number is used in complex numbers to represent a second dimension.

Remember, a purely imaginary voltage in an AC circuit will shock you as badly as a real voltage - that's proof enough of it's physical existence. I'll put a link in the link area to a great interactive site (it's actually my site but for it's educational purposes only) that explains the imaginary number utility more visually with animations.