Imaginary unit

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. 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 2 + 3i. The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. 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 2 + 3i. Imaginary numbers are an important mathematical concept, which extend the real number system ℝ to the complex number system ℂ, which in turn provides at least one root for every nonconstant polynomial P(x). (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 −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used (see § Alternative notations). In the disciplines of 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. For the history of the imaginary unit, see Complex number § History. 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 −i 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 i as an unknown quantity while manipulating an expression, and then using the definition to replace any occurrence of i2 with −1. Higher integral powers of i can also be replaced with −i, 1, i, or −1: