The imaginary unit or unit imaginary number () is a solution to the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...
. Although there is no
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
with this property, can be used to extend the real numbers to what are called
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, 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
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
for every nonconstant
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
exists (see
Algebraic closure and
Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
having a negative
square.
There are two complex square roots of −1: and
, just as there are two complex
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
s 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 or the Greek
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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
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 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, is represented in
rectangular form as , with a zero real component and a unit imaginary component. In
polar form, is represented as
(or just
), with an
absolute value (or magnitude) of 1 and an
argument (or angle) of
. 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 inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
s 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
A similar issue arises if the complex numbers are interpreted as
real matrices (see
matrix representation of complex numbers), because then both
and
would be 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 elements: The identity and the automorphism which exchanges "CW" (clockwise) and "CCW" (counter-clockwise) rotations. For more, see
orthogonal group.
All these ambiguities can be solved by adopting a more rigorous
definition of complex number, and by explicitly ''choosing'' one of the solutions to the equation to be the imaginary unit. For example, the ordered pair (0, 1), in the usual construction of the complex numbers with two-dimensional vectors.
Consider the matrix equation
Here,
, so the product is negative because
; thus, the point
lies in quadrant II or IV. Furthermore,
so
is bounded by the hyperbola
.
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
Radical may refer to:
Politics and ideology Politics
*Radical politics, the political intent of fundamental societal change
*Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
. 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:
:
The calculation rules
:
and
:
are only valid for real, positive values of and .
These problems can be avoided by writing and manipulating expressions like
, rather than
. For a more thorough discussion, see
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
and
branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that 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
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
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
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
. Equivalently:
:
raised to the power of
Making use of
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
,
is
:
where , the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s.
The
principal value (for
) is
, or approximately 0.207879576.
Factorial
The
factorial of the imaginary unit is most often given in terms of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
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:
:
History
See also
*
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
*
Mathematical constant
*
Multiplicity (mathematics)
*
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
*
Unit complex number
Notes
References
Further reading
*
External links
* at {{cite web , title=Convergence , website=mathdl.maa.org , publisher=Mathematical Association of America , url=http://mathdl.maa.org/convergence/1/ , url-status=dead , archive-url=https://web.archive.org/web/20070713083148/http://mathdl.maa.org/convergence/1/ , archive-date=2007-07-13
Complex numbers
Algebraic numbers
Quadratic irrational numbers
Mathematical constants