# Imaginary Number

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 x^{2} = –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 x^{2} = –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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