Imaginary unit

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In mathematics, the imaginary unit i (or sometimes the Latin j or the Greek iota, see below) allows the real number system \mathbb{R} to be extended to the complex number system \mathbb{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 (see "Definition", below). 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.)

For a history of the imaginary unit, see the history of complex numbers.

The imaginary unit is often loosely referred to as the "square root of negative one" or the "square root of minus one", but see below for difficulties that may arise from a naive use of this idea.

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[edit] Definition

By definition, the imaginary unit i is one solution of the quadratic equation

x^2 + 1 = 0 \

or equivalently

x^2 = -1 \.

Since there is no real number that squares to any negative real number, we imagine such a number and assign to it the symbol i.

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 i 2 with −1. Higher integral powers of i can also be replaced with −i , 1, i, or −1.

[edit] i and − i

Being a 2nd order polynomial with no multiple real root, the above equation has two distinct solutions that are equally valid and that happen to be additive inverses of each other. More precisely, once a solution i of the equation has been fixed, the value −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 one of the solutions is chosen and fixed as the "positive i". This is because, although −i and i are not quantitatively equivalent (they are negatives of each other), there is no qualitative difference between i and −i (that cannot be said for −1 and +1). Both imaginary numbers have equal claim to square to −1. If all mathematical textbooks and published literature referring to imaginary or complex numbers were rewritten with −i replacing every occurrence of +i (and therefore every occurrence of −i replaced by −(−i) = +i), all facts and theorems would continue to be equivalently valid. The distinction between the two roots x of x2 + 1 = 0 with one of them as "positive" is purely a notational relic; neither root can be said to be more important than the other.

The issue can be 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. (These are not the only field automorphisms of C, but 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 issue arises if the complex numbers are interpreted as 2 × 2 real matrices (see complex number), because then both

X = \begin{pmatrix} 0 & -1 \\ 1 & \;\; 0 \end{pmatrix}

and

X = \begin{pmatrix} 0 & 1 \\ -1 & \;\; 0 \end{pmatrix}

are solutions to the matrix equation

X^2 = -I \ .

In this case, the ambiguity results from the geometric choice of which "direction" around the unit circle is "positive" rotation. 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.

[edit] Warning

The imaginary unit is sometimes written \sqrt{-1} 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:

-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1

The calculation rule

\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}

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

For a more thorough discussion of this phenomenon, see square root and branch.

To avoid making such mistakes when manipulating complex numbers, a strategy is never to use a negative number under a square root sign. This means to avoid writing expressions like \sqrt{-7}, but to write i\sqrt{7} instead. That is what the imaginary unit is intended for.

[edit] Square root of the imaginary unit

One might assume that a further set of imaginary numbers need to be invented to account for the square root of i. However this is not necessary as it can be expressed as either of two complex numbers: \sqrt{i} = \pm \frac{\sqrt{2}}{2} (1 + i). This can be shown to be valid from:

\left( \pm \frac{\sqrt{2}}{2} (1 + i) \right)^2 \ = \left( \pm \frac{\sqrt{2}}{2} \right)^2 (1 + i)^2 \
= (\pm 1)^2 \frac{2}{4} (1 + i)(1 + i) \
= 1 \times \frac{1}{2} (1 + 2i + i^2) \quad \quad (i^2 = -1) \
= \frac{1}{2} (2i) \
= i \

[edit] Powers of i

The powers of i repeat in a cycle:

i^{-3} = i\,
i^{-2} = -1\,
i^{-1} = -i\,
i^0 = 1\,
i^1 = i\,
i^2 = -1\,
i^3 = -i\,
i^4 = 1\,
i^5 = i\,
i^6 = -1\,

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

i^{4n} = 1\,
i^{4n+1} = i\,
i^{4n+2} = -1\,
i^{4n+3} = -i\,

[edit] i and \sqrt{2}

Using the semicircle equation of

r\sin(\cos^{-1}(x/r)) = \sqrt{r^{2}-x^{2}}

We can determine that

\sin(\cos^{-1}(i)) = \sqrt{2}

This can easily be generalized to show that

\sin(\cos^{-1}(xi)) = \sqrt{x^2 + 1} = \cos(\sin^{-1}(xi))

[edit] i and Euler's formula

Taking Euler's formula

e^{ix} = \cos\; x + i\sin\; x\, ,

and substituting \pi/2\, for x\,, one arrives at

e^{i\pi/2} = i\,

and

ln(i) = i \pi/2\,

If both sides of the first equation are raised to the power i, remembering that i^2 = -1\,, one obtains this identity:

i^i = e^{i ln(i)} = e^{-\pi/2} = 0.2078795763\dots\,

In fact, ii has an infinite number of solutions in the form of

i^i = e^{-\pi/2 + 2\pi N}\,

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

From the above identity

e^{i\pi/2} = i\,

one arrives elegantly at Euler's identity

e^{i\pi} + 1 = 0\,,

which remarkably relates the five most significant mathematical entities, along with the principle of equality and the operations of addition, multiplication, and exponentiation, in one simple expression.

[edit] Operations with i

Many mathematical operations that can be carried out with real numbers can also be carried out with i, such as exponentation, roots and logarithms.

A number raised to the ni power is:

\!\ x^{ni} = \cos(\ln(x^n)) + i \sin(\ln(x^n))

The nith root of a number is:

\!\ \sqrt[ni]{x} = \cos(\ln(\sqrt[n]{x})) - i \sin(\ln(\sqrt[n]{x}))

The log base i of a number is:

\log_i(x) = {{-2 \ln(x) i} \over \pi}

[edit] Alternative notation

In electrical engineering and related fields, the imaginary unit is often written as j to avoid confusion with electrical current as a function of time, traditionally denoted by i(t) or just i. The Python programming language also uses j to denote the imaginary unit.

Some extra care needs to be taken in certain textbooks which define j=−i, in particular to travelling waves (e.g. a right travelling plane wave in the x direction ei(kx − ωt) = ej(ωt − kx)).

Some texts use the Greek letter iota to write the imaginary unit to avoid confusion. For example: Biquaternion.

[edit] See also

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