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Among other things, complex numbers play an important role:* In electrical circuits - quantities in AC circuits are described by complex numbers.
* In quantum mechanics - the "probability amplitude" is an important concept in quantum mechanics, and it is described by a complex number.
* In art - for example, the Mandelbrot set is based on calculations with complex numbers.
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Rafa
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ReneImaginary and complex numbers are simply extensions of the real number system in much the same way as negative integers were an extension to the set of natural numbers, or rationals were an extension to integers. Negatives provided closure for subtraction, while rationals gave closure for division of integers by (non-zero) integers. Imaginary and complex numbers give closure to taking roots of any real number.
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What is the difference between imaginary numbers and complex numbers?
No difference. The set of complex numbers includes the set of imaginary numbers.
Can a complex number be imaginary?
Yes, imaginary numbers are a subset of complex numbers.
Can a number be both complex and imaginary?
A complex number has a real part and a (purely) imaginary part, So imaginary numbers are a subset of complex numbers. But the converse is not true. A real number is also a member of the complex domain but it is not an imaginary number.
What do imaginary numbers represent?
The real numbers together with the imaginary numbers form a sort of vector. What these complex numbers (complex means the combination of real and imaginary numbers) represent depends on the specific situation. Just as there are situations where it doesn't make sense to use negative numbers, or fractional numbers, in many common situations it doesn't make sense to use complex numbers. In an electrical circuit (specifically, AC), the real numbers might represent resistance, while the imaginary number represent reactance - and voltages and currents are also represented by complex numbers, with the angle of the complex number representing how much one quantity is ahead or behind another quantity (the "phase angle"). In quantum mechanics, the complex numbers might not represent anything (perhaps they don't, I am not sure...), but they are useful for calculations.
Why are complex numbers imaginary?
An imaginary number has the form b*i, where b is any real number, and i is the imaginary unit number sqrt(-1). A complex number is in the form a + b*i, where a and b are any real number. Complex numbers are called complex, because they combine numbers from the real domain and the imaginary domain.I thought the Wikipedia article was very good. Also, Answers.com has a video about complex and imaginary numbers.
