The number i = √ –1 is the basis of any imaginary number, which, in general, is any real number times i. For example, 5i is an imaginary number and is equivalent to –1 ÷ 5. The real numbers are those numbers that can be expressed as terminating, repeating, or nonrepeating decimals; they include positive and negative numbers. The product of two negative real numbers is always positive. Thus, there is no real number that equals –1 when multiplied by itself—that is, no real number satisfies the equation x2 = –1 in the real number system. The imaginary number i was invented to provide a solution to this equation, and every imaginary number represents the solution to a similar equation (e.g., 5i is a solution to the equation x2 = –25).
In addition to providing solutions for algebraic equations, the imaginary numbers, when combined with the real numbers, form the complex numbers. Each complex number is the sum of a real number and an imaginary number, such as (6 + 9i). The complex numbers are very useful in mathematical analysis, the study of electricity and magnetism, the physics of quantum mechanics, and in the practical field of electrical engineering. In terms of the complex numbers, the imaginary numbers are equivalent to those complex numbers for which the real part is zero.
See also Square root.
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