Imaginary unit


The imaginary unit or unit imaginary number is a solution to the quadratic equation. Although there is no real number with this property, 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 in a complex number is.
Imaginary numbers are an important mathematical concept, which extend the real number system to the complex number system, which in turn provides at least one root for every nonconstant polynomial. The term "imaginary" is used because there is no real number having a negative square.
There are two complex square roots of, namely and, just as there are two complex square roots of every real number other than zero, which has one double square root.
In contexts where use of the letter is ambiguous or problematic, or the Greek iota| is sometimes used. In the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by instead of, because is commonly used to denote electric current.
For the history of the imaginary unit, see.

Definition

The imaginary number is defined solely by the property that its square is −1:
With defined this way, it follows directly from algebra that and 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 as an unknown quantity while manipulating an expression, and then using the definition to replace any occurrence of with −1. Higher integral powers of can also be replaced with, 1,, or −1:
Similarly, as with any non-zero real number:
As a complex number, is represented in rectangular form as, with a zero real component and a unit imaginary component. In polar form, is represented as , with an absolute value of 1 and an argument of /2. In the complex plane, which is a special interpretation of a Cartesian plane, is the point located one unit from the origin along the imaginary axis.

and

Being a quadratic polynomial with no multiple root, the defining equation has two distinct solutions, which are equally valid and which happen to be additive and multiplicative inverses of each other. More precisely, once a solution of the equation has been fixed, the value, which is distinct from, is also a solution. Since the equation is the only definition of, it appears that the definition is ambiguous. However, no ambiguity results as long as one or other of the solutions is chosen and labelled as "", with the other one then being labelled as. This is because, although and are not quantitatively equivalent, there is no algebraic difference between and. Both imaginary numbers have equal claim to being the number whose square is −1. If all mathematical textbooks and published literature referring to imaginary or complex numbers were rewritten with replacing every occurrence of , all facts and theorems would continue to be equivalently valid. The distinction between the two roots of with one of them labelled with a minus sign is purely a notational relic; neither root can be said to be more primary or fundamental than the other, and neither of them is "positive" or "negative".
The issue can be a subtle one. The most precise explanation is to say that although the complex field, defined as , is unique up to isomorphism, it is not unique up to a unique isomorphism — there are exactly two field automorphisms of which keep each real number fixed: the identity and the automorphism sending to. See also Complex conjugate and Galois group.

Matrices

A similar issue arises if the complex numbers are interpreted as 2 × 2 real matrices, because then both
are solutions to the matrix equation
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 ) has exactly two elementsthe identity and the automorphism which exchanges "CW" and "CCW" rotations. See orthogonal group.
All these ambiguities can be solved by adopting a more rigorous definition of complex number, and explicitly choosing one of the solutions to the equation to be the imaginary unit. For example, the ordered pair, in the usual construction of the complex numbers with two-dimensional vectors.
Consider the matrix equation
Then so the product xy is negative because thus the point lies in quadrant II or IV. Furthermore,

Proper use

The imaginary unit is sometimes written in advanced mathematics contexts. However, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the principal square root function, which is only defined for real, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal square root function to manipulate the principal branch of the complex square root function can produce false results:
Similarly:
The calculation rules
and
are only valid for real, positive values of and.
These problems are avoided by writing and manipulating expressions like, rather than. For a more thorough discussion, see Square root and Branch point.

Properties

Square roots

has two square roots, just like all complex numbers. These two roots can be expressed as the complex numbers:
Indeed, squaring both expressions:
Using the radical sign for the principal square root gives:

Cube roots

The three cube roots of are:
Similar to all of the roots of 1, all of the roots of are the vertices of regular polygons inscribed within the unit circle in the complex plane.

Multiplication and division

Multiplying a complex number by gives:
Dividing by is equivalent to multiplying by the reciprocal of :
Using this identity to generalize division by to all complex numbers gives:

Powers

The powers of repeat in a cycle expressible with the following pattern, where is any integer:
This leads to the conclusion that
where mod represents the modulo operation. Equivalently:

raised to the power of

Making use of Euler's formula, is
where, the set of integers.
The principal value is or approximately 0.207879576.

Factorial

The factorial of the imaginary unit is most often given in terms of the gamma function evaluated at :
Also,

Other operations

Many mathematical operations that can be carried out with real numbers can also be carried out with, such as exponentiation, roots, logarithms, and trigonometric functions. All of the following functions are complex multi-valued functions, and it should be clearly stated which branch of the Riemann surface the function is defined on in practice. Listed below are results for the most commonly chosen branch.
A number raised to the power is:
The root of a number is:
The imaginary-base logarithm of a number is:
As with any complex logarithm, the log base is not uniquely defined.
The cosine of is a real number:
And the sine of is purely imaginary:

Alternative notations