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The term imaginary unit or unit imaginary number refers to a solution to the equation . By convention, the solution is usually denoted . Since there is no real number with this property, it extends the real numbers, and under the assumption that the familiar properties of addition and multiplication (namely closure, associativity, commutativity and distributivity) continue to hold for this extension, the complex numbers are generated by including it. Imaginary numbers are an important mathematical concept, which extends the real number system to the complex number system , which in turn provides at least one root for every nonconstant polynomial . (See Algebraic closure and Fundamental theorem of algebra.) The term "imaginary" is used because there is no real number having a negative square. There are two complex square roots of , namely and , just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where is ambiguous or problematic, or the Greek is sometimes used (see ). In the disciplines of electrical engineering and control systems engineering, the imaginary unit is often denoted by instead of , because is commonly used to denote electric current. For the history of the imaginary unit, see . ==Definition== The imaginary number is defined solely by the property that its square is −1: : With defined this way, it follows directly from algebra that and are both square roots of −1. Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers by treating as an unknown quantity while manipulating an expression, and then using the definition to replace any occurrence of with −1. Higher integral powers of can also be replaced with , 1, , or −1: : : : Similarly, as with any non-zero real number: : As a complex number, is represented in rectangular form as , having a unit imaginary component and no real component (i.e., the real component is zero). In polar form, is represented as , having an absolute value (or magnitude) of 1 and an argument (or angle) of π/2. In the complex plane (also known as the Cartesian plane), is the point located one unit from the origin along the imaginary axis (which is at a right angle to the real axis). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「imaginary unit」の詳細全文を読む スポンサード リンク
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