Imaginary unit

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; they extend the real number system to the complex number system , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.
There are two complex square roots of −1: and , just as there are two complex square roots of every real number other than zero (which has one double square root).
In contexts in which use of the letter is ambiguous or problematic, the letter is sometimes used instead. For example, in electrical engineering and control systems engineering, the imaginary unit is normally denoted by instead of , because is commonly used to denote electric current.
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 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 using the definition to replace any occurrence of with −1). Higher integral powers of can also be replaced with , 1 or −1:
or, equivalently,
Similarly, as with any non-zero real number:
As a complex number, can be represented in rectangular form as , with a zero real component and a unit imaginary component. In polar form, can be represented as (or just ), with an absolute value (or magnitude) of 1 and an argument (or angle) of radians. (Adding any multiple of to this angle works as well.) In the complex plane (also known as the Argand plane), which is a special interpretation of a Cartesian plane, is the point located one unit from the origin along the imaginary axis (which is orthogonal to the real axis).
i vs. −i
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. 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 (more precisely, not well-defined). However, no ambiguity will result as long as one or other of the solutions is chosen and labelled as "", with the other one then being labelled as . After all, although and are not quantitatively equivalent (they are negatives of each other), there is no algebraic difference between and , as both imaginary numbers have equal claim to being the number whose square is −1.
In fact, if all mathematical textbooks and published literature referring to imaginary or complex numbers were to be rewritten with replacing every occurrence of (and, therefore, every occurrence of replaced by ), all facts and theorems would remain 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. One way of articulating the situation is that although the complex field is unique (as an extension of the real numbers) up to isomorphism, it is not unique up to a unique isomorphism. Indeed, there are two field automorphisms of that keep each real number fixed, namely the identity and complex conjugation. For more on this general phenomenon, see Galois group.
Matrices
Using the concepts of matrices and matrix multiplication, imaginary units can be represented in linear algebra. The value of 1 is represented by an identity matrix and the value of is represented by any matrix satisfying . A typical choice is
More generally, a real-valued matrix satisfies if and only if has a matrix trace of zero and a matrix determinant of one, so can be chosen to be
whenever . The product is negative because ; thus, the points lie on hyperbolas determined by in quadrant II or IV.
Matrices larger than can be used. For example, could be chosen to be the identity matrix with chosen to be any of the three Dirac matrices for spatial dimensions, .
Regardless of the choice of representation, the usual rules of complex number mathematics work with these matrices because , and . For example,
Proper use
The imaginary unit is sometimes written in advanced mathematics contexts (as well as in less advanced popular texts). 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 (real) square root function to manipulate the principal branch of the complex square root function can produce false results:
Similarly:
Generally, the calculation rules
and
are guaranteed to be valid for real, positive values of and only.
When or is real but negative, these problems can be avoided by writing and manipulating expressions like , rather than . For a more thorough discussion, see square root and branch point.
Properties
Square roots
Just like all nonzero complex numbers, has two square roots: they are
Indeed, squaring both expressions yields:
Using the radical sign for the principal square root, we get:
Cube roots
The three cube roots of are:
and
Similar to all the roots of 1, all the roots of are the vertices of regular polygons, which are inscribed within the unit circle in the complex plane.
Multiplication and division
Multiplying a complex number by gives:
(This is equivalent to a 90° counter-clockwise rotation of a vector about the origin in the complex plane.)
Dividing by is equivalent to multiplying by the reciprocal of :
Using this identity to generalize division by to all complex numbers gives:
(This is equivalent to a 90° clockwise rotation of a vector about the origin in the complex plane.)
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:
Although we do not give the details here, if one chooses branch cuts and principal values to support it then this last equation would apply to all complex values of .
raised to the power of
Making use of Euler's formula, has infinitely many values
for any integer . A common principal value corresponds to and gives , which is .
Factorial
The factorial of the imaginary unit is most often given in terms of the gamma function evaluated at :
The magnitude of this number is
while its argument is
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. The following functions are well-defined, single-valued functions when is a positive real number.
A number raised to the power is:
The root of a number is:
The cosine of is:
which is a real number when is a real number.
The sine of is:
which is a purely imaginary number when is a real number.
In contrast, many functions involving , including those that depend upon or the logarithm of another complex number, are complex multi-valued functions, with different values on different branches of the Riemann surface the function is defined on. For example, if one chooses any branch where then one can write
when is a positive real number. When is not a positive real number in the above formulas then one must precisely specify the branch to get a single-valued function; see complex logarithm.
History
Designating square roots of negative numbers as "imaginary" is generally credited to René Descartes, and Isaac Newton used the term as early as 1670. The notation was introduced by Leonhard Euler.
See also
Euler's identity
Hyperbolic unit
Mathematical constant
Multiplicity (mathematics)
Root of unity
Unit complex number
Notes
References
Further reading
External links
at
Complex numbers
Algebraic numbers
Quadratic irrational numbers
Mathematical constants