Imaginary unit - Definition


Imaginary unit - Definition

In mathematics, the imaginary unit i allows the real number system <math>\mathbb{R}<math> to be extended to the complex number system <math>\mathbb{C}<math>. 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.)

Contents

5 i and Euler's Formula

6 Alternate notation

7 See also

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.

i and −i

The above equation actually has two distinct solutions which are additive inverses. More precisely, once a solution i of the equation has been fixed, −i ≠ i is also a solution. Since the equation is the only definition of i, it appears that the definition is ambiguous (more precisely, not well-defined). However, no ambiguity results, as long as we choose a soluton and fix it forever as "positive i".

The issue is a subtle one. The most precise explanation is to say that although the complex field, defined as R[X]/(X² + 1), (see complex number) is unique up to isomorphism, it is not unique up to a unique isomorphism — there are exactly 2 field automorphisms of R[X]/(X² + 1), the identity and the automorphism sending X to −X. (It should be noted that these are not the only field automorphisms of C; they are the only field automorphisms of C which keep each real number fixed.) See complex number, complex conjugation, field automorphism, and Galois group.

A similar problem appears to occur if the complex numbers are interpreted as 2 × 2 real matrices (see complex number), because then both

<math>

\begin{pmatrix}

 0 &     -1  \\
 1 & \;\; 0

\end{pmatrix} \mbox{ and } \begin{pmatrix}

  0 &      1  \\
 -1 & \;\; 0

\end{pmatrix} <math> are solutions to the equation x² = −1. In this case, the ambiguity results from the geometric choice of which "direction" around the unit circle is "positive". A more precise explanation is to say that the automorphism group of the special orthogonal group SO(2, R) has exactly 2 elements — the identity and the automorphism which exchanges "CW" (clockwise) and "CCW" (counter-clockwise) rotations. See orthogonal group.

Warning

The imaginary unit is sometimes written <math>\sqrt{-1}<math> in advanced mathematics contexts, but care needs to be taken when manipulating formulas involving radicals. The 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 = \mathrm{i} \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1<math>

The calculation rule

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

For a more thorough discussion of this phenomenon, see square root and branch_(complex analysis).

Powers of i

The powers of i repeat in a cycle:

<math>\mathrm{i}^1 = \mathrm{i}<math>

<math>\mathrm{i}^2 = -1<math>

<math>\mathrm{i}^3 = -\mathrm{i}<math>

<math>\mathrm{i}^4 = 1<math>

<math>\mathrm{i}^5 = \mathrm{i}<math>

<math>\mathrm{i}^6 = -1<math>

This can be expressed with the following pattern where n is any integer:

<math>\mathrm{i}^{4n} = 1<math>

<math>\mathrm{i}^{4n+1} = \mathrm{i}<math>

<math>\mathrm{i}^{4n+2} = -1<math>

<math>\mathrm{i}^{4n+3} = -\mathrm{i}<math>

i and Euler's Formula

Taking Euler's formula <math>e^{\mathrm{i}x} = \cos\mbox{ }x + \mathrm{i}\mbox{ }\sin\mbox{ }x<math>, and substituting <math>\pi/2<math> for <math>x<math>, one arrives at

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

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

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

where N is any integer. From the number theorists point of view, i is a quadratic irrational number, like √2, and by applying the Gelfond-Schneider theorem, we can conclude that all of the values we obtained above, and e^-π/2 in particular, are transcendental.

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.