Imaginary unit

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation . Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is . Imaginary numbers are an important mathematical concept; they extend the real number system to the complex number system , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square. There are two complex square roots of −1: and , just as there are two complex square roots of every real number other than zero (which has one double square root). In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. The imaginary number i is defined solely by the property that its square is −1: With i defined this way, it follows directly from algebra that i 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 valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of with −1). Higher integral powers of i can also be replaced with , 1, i, or −1: or, equivalently, Similarly, as with any non-zero real number: As a complex number, i can be represented in rectangular form as , with a zero real component and a unit imaginary component.