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Imaginary numbers are a fundamental concept in mathematics, extending the real number system. They are based on the imaginary unit, denoted by the symbol 'i', which is defined as the square root of negative one (i² = -1). While real numbers can be visualized on a number line, imaginary numbers require a two-dimensional plane, known as the complex plane, for their representation.

A pure imaginary number is any real number multiplied by the imaginary unit 'i'. For example, 3i and -5i are imaginary numbers. The real part of such a number is zero. The concept of imaginary numbers arose from the need to solve equations that had no solutions within the real number system, such as x² + 1 = 0.

The introduction of imaginary numbers led to the development of complex numbers, which are numbers of the form a + bi, where 'a' is the real part and 'b' is the imaginary part. Complex numbers have wide-ranging applications in various fields, including electrical engineering, quantum mechanics, signal processing, and fluid dynamics. They provide a powerful tool for modeling and solving problems that involve oscillations, waves, and alternating currents.