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Imaginary unit

In mathematics, the imaginary unit i allows the real number system R to be extended to the complex number system C. Its precise definition is dependent upon the particular method of extension.

The primary motivation for this extension is the fact that not every polynomial equation f(x) = 0 has a solution in the real numbers. In particular, the equation x² + 1 = 0 has no real solution. However, if we allow complex numbers as solutions, then this equation, and indeed every polynomial equation f(x) = 0 does have a solution. (See algebraic closure and fundamental theorem of algebra.)

Definition

By definition, the imaginary unit i is a solution of the equation

x² = −1

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 occurrences of i² with -1.

The above equation actually has two distinct solutions which are additive inverses. Since the equation is the only definition of i, no mathematical test can distinguish one solution from the other, and they can be swapped in any equation. However, when imaginary numbers are used to describe physical quantities, such as phase angles, the quantities corresponding to i and -i must be specified so that all conversions between quantities and mathematical values are consistent. (See complex conjugation and field automorphism.)

We can also construct an isomorphism between the complex numbers and the set of matrices in the following form below

<math>\begin{pmatrix} x & -y \\ y & x \end{pmatrix},\ x,y \in \Bbb{R}<math>

and then observing that any matrix of this form can be written as

<math>x \begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix} + y \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}<math>

and using the symbol i for the matrix <math>\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}<math>, and 1 for the identity matrix.

Warning

The imaginary unit should not be written down or treated as √(−1). This notation is reserved either for the principal square root function, which is only defined for real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function will produce false results:

<math>-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1<math>

The calculation rule

<math>\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}<math>

is only valid for real, non-negative numbers a and b.

For a more thorough discussion of this phenomenon, see square root and Euler's formula <math>e^{ix} = \cos\mbox{ }x + i\mbox{ }\sin\mbox{ }x<math>, and substituting <math>\pi/2<math> for <math>x<math>, one arrives at

<math>e^{i\pi /2} = i<math>

If both sides are raised to the power <math>i<math>, remembering that <math>i^2 = -1<math>, one obtains this remarkable identity:

<math>i^i = e^{-\pi /2} = 0.2078795763\dots<math>

In fact, it is easy to determine that <math>i^i<math> has an infinite number of solutions in the form of

<math>i^{i} = e^{-\pi / 2 + 2 \pi N}<math>

where <math>N<math> is any integer.

Alternate notation

In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with a changing current, traditionally denoted by i.

Also see:

• imaginary number
• complex number