Imaginary unit
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In mathematics, the imaginary unit i (sometimes also represented by j, but in this article i will be used exclusively) 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.)
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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 solution 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 (as well as in less advanced popular texts), but great 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 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1<math>
The calculation rule
- <math>\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}<math>
is only valid for real, positive values of a and b.
For a more thorough discussion of this phenomenon, see square root and branch. To avoid making mistakes when manipulating complex numbers, the best strategy is to never use a negative number under the square root sign.
Powers of i
The powers of i repeat in a cycle:
- <math>i^0 = 1<math>
- <math>i^1 = i<math>
- <math>i^2 = -1<math>
- <math>i^3 = -i<math>
- <math>i^4 = 1<math>
- <math>i^5 = i<math>
- <math>i^6 = -1<math>
This can be expressed with the following pattern where n is any integer:
- <math>i^{4n} = 1<math>
- <math>i^{4n+1} = i<math>
- <math>i^{4n+2} = -1<math>
- <math>i^{4n+3} = -i<math>
i and Euler's formula
Taking 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 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 N is any integer. From the number theorists point of view, <math>\imath<math> 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 <math>e^{-\pi/2}<math> in particular, are transcendental.
From the above identity
- <math>e^{i\pi/2} = i<math>
one arrives elegantly at Euler's identity
- <math>e^{i\pi} + 1 = 0<math>,
which relates five of the most significant mathematical entities in one simple expression.
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.
See also
de:Imaginäre Einheit et:Imaginaarühik he:מספר מרוכב it:Unità immaginaria nl:imaginaire eenheid pl:I (liczba urojona) pt:Unidade imaginária ja:虚数単位 sl:Imaginarna enota zh:虚数单位