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Root Of Unity

Root of unity

In mathematics, the n^th roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. It can be shown that they are located on the unit circle of the complex plane and that in that plane they form the vertices of a n-sided regular polygon with one vertex on 1.

Definition

The complex numbers z which solve :

z^n = 1 \qquad (n = 1, 2, 3, \cdots  )

are called the n^th roots of unity. There are n different n^th roots of unity . :

e^ \qquad (k = 0, 1, 2, \cdots, n - 1).

(See Exponentiation#Powers_of_one)

Primitive roots

The n^th roots of unity form under multiplication a cyclic group of order n. A generator for this cyclic group is a primitive n^th root of unity. The primitive n^th roots of unity are

e^

where k and n are coprime. The number of different primitive n^th roots of unity is φ(n).

Examples

There is only one first root of unity, equal to 1. The second roots of unity are +1 and -1, of which only -1 is primitive. The third roots of unity are :

\left\ ,

where

i

is the imaginary unit; the latter two roots are primitive. The fourth roots of unity are :

\left\ ,

of which

+i

and

-i

are primitive.

Summation

As long as n is at least 2, the n^th roots of unity add up to 0. This fact arises in many areas of mathematics and can be proved in a number of ways. One elementary proof is to apply the formula for a geometric series: :

\sum_^ e^ = \frac = \frac = 0 .

Yet another reason for the zero summation is that the roots of unity, plotted in the complex plane, form the vertices of a regular polygon whose barycenter (by symmetry) lies at the origin.

Orthogonality

One can use the summation formula to prove an orthogonality relationship: :

\sum_^ e^ \cdot e^ = n \delta_

where

\delta

is the Kronecker delta. The

n

^th roots of unity can be used to form an

n \times n

matrix whose

(j,k)

^th entry is :

U_=n^ e^

From above, the columns of this matrix are orthonormal and thus the matrix is unitary. In fact, this matrix is precisely the discrete Fourier transform (although normalization and sign conventions vary). The n^th roots of unity form a irreducible representation of any cyclic group of order

n

. The orthogonality relationship then follows from group-theoretic principles as described in character group. The roots of unity appear as the eigenvectors of Hermitian matrices (for example, of a discretized one-dimensional Laplacian with periodic boundaries), from which the orthogonality property also follows (Strang, 1999).

Omega notation

The primitive root

e^

(or its conjugate

e^

) is often denoted

\omega_n

(or sometimes simply

\omega

), especially in the context of discrete Fourier transforms.

Cyclotomic polynomials

The n^th roots of unity are precisely the zeros of the polynomial p(X) = Xⁿ − 1; the primitive nth roots of unity are precisely the zeros of the nth cyclotomic polynomial :

\Phi_n(X) = \prod_^(X-z_k)\;

where z₁,...,z_φ(n) are the primitive n^th roots of unity. The polynomial Φ_n(X) has integer coefficients and is irreducible over the rationals (i.e., cannot be written as a product of two positive-degree polynomials with rational coefficients). The case of prime n, which is easier than the general assertion, follows from Eisenstein's criterion. Every n^th root of unity is a primitive d^th root of unity for exactly one positive divisor d of n. This implies that :

X^n - 1 = \prod_ \Phi_d(X).\;

This formula represents the factorization of the polynomial Xⁿ - 1 into irreducible factors and can also be used to compute the cyclotomic polynomials recursively. The first few are :Φ₁(X) = X − 1 :Φ₂(X) = X + 1 :Φ₃(X) = X² + X + 1 :Φ₄(X) = X² + 1 :Φ₅(X) = X⁴ + X³ + X² + X + 1 :Φ₆(X) = X² − X + 1 In general, if p is a prime number, then all p^th roots of unity except 1 are primitive p^th roots, and we have :

\Phi_p(X)=\frac=\sum_^ X^k

Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 1, −1, or 0; the first polynomial where this occurs is Φ₁₀₅, since 105=3×5×7 is the first product of three odd primes.

Cyclotomic fields

By adjoining a primitive n^th root of unity to Q, one obtains the n^th cyclotomic field F_n. This field contains all nth roots of unity and is the splitting field of the n^th cyclotomic polynomial over Q. The field extension F_n/Q has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring Z/nZ. As the Galois group of F_n/Q is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. In these cases Galois theory can be written out quite explicitly in terms of Gaussian periods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois. Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field — a theorem of Kronecker, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof.

References

- Gilbert Strang, "[http://epubs.siam.org/sam-bin/dbq/article/33674 The discrete cosine transform]," SIAM Review 41 (1), 135-147 (1999). Category:Field theory Category:Algebraic number theory ja:1の冪根

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Complex numbers

In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. For example, :3 + 2i is a complex number, where 3 is called the real part and 2 the imaginary part. Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane. The set of all complex numbers is usually denoted by C, or in blackboard bold by

\mathbb

. It includes the real numbers because every real number can be regarded as complex: a = a + 0i. Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i ² = −1: :(a + bi) + (c + di) = (a+c) + (b+d)i :(a + bi) − (c + di) = (a−c) + (b−d)i :(a + bi)(c + di) = ac + bci + adi + bd i ² = (ac−bd) + (bc+ad)i Division of complex numbers can also be defined (see below). Thus the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed. In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

Definition

The complex number field

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:
-

( a , b ) + ( c , d ) = ( a + c , b + d ) \,

( a , b ) \cdot ( c , d ) = ( ac - bd , bc + ad ). \,

So defined, the complex numbers form a field, the complex number field, denoted by C. We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0, 1). In C, we have:
- additive identity ("zero"): (0, 0)
- multiplicative identity ("one"): (1, 0)
- additive inverse of (a,b): (−a, −b)
- multiplicative inverse (reciprocal) of non-zero (a, b):

\left(,\right).

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.

The complex plane

image:complex.png

A complex number can be viewed as a point or a position vector on a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand). The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the circular coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have :

z = x + iy = r (\cos \phi + i\sin \phi ) = r e^. \,

Additionally the notation r cis φ is sometimes used. Note that the complex argument is unique modulo 2π, that is, if any two values of the complex argument exactly differ by an integer multiple of 2π, they are considered equivalent. By simple trigonometric identities, we see that :

r_1 e^ \cdot r_2 e^ 
= r_1 r_2 e^ \,

and that :

\frac

= \frac e^. \,

Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching. Multiplication with i corresponds to a counter clockwise rotation by 90 degrees (

\pi/2

radians). The geometric content of the equation i² = −1 is that a sequence of two 90 degree rotations results in a 180 degree (

\pi

radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = r e^iφ is defined as |z| = r. Algebraically, if z = a + ib, then

| z | = \sqrt.

One can check readily that the absolute value has three important properties: :

| z | = 0 \,

iff

z = 0 \,

| z + w | \leq | z | + | w | \,

| z w | = | z | \; | w | \,

for all complex numbers z and w. It then follows, for example, that

| 1 | = 1

and

|z/w|=|z|/|w|

. By defining the distance function d(z, w) = |z − w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers. The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as

\bar

z^ - \,

. As seen in the figure,

\bar

is the "reflection" of z about the real axis. The following can be checked: :

\overline = \bar + \bar

\overline = \bar\bar

\overline = \bar/\bar

\bar=z

\bar=z

iff z is real :

|z|=|\bar|

|z|^2 = z\bar

z^ = \bar|z|^

if z is non-zero. The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates. That conjugation commutes with all the algebraic operations (and many functions; e.g.

\sin\bar z=\overline

) is rooted in the ambiguity in choice of i (−1 has two square roots); note, however, that conjugation is not differentiable (see holomorphic).

Complex number division

Given a complex number (a + ib) which is to be divided by another complex number (c + id) whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easy to derive. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. This causes the denominator to simplify into a real number: :

=  =

:::

= \left(\right) + i\left(  \right).

Matrix representation of complex numbers

While usually not useful, alternative representations of complex fields can give some insight into their nature. One particularly elegant representation interprets every complex number as 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form :

\begin
  a &   -b  \\
  b & \;\; a  
\end

with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as :

\begin
  a &     -b  \\
  b & \;\; a  
\end
=
a \begin
  1 & \;\; 0  \\
  0 & \;\; 1 
\end
+
b \begin
  0 &     -1  \\
  1 & \;\; 0 
\end

which suggests that we should identify the real number 1 with the matrix :

\begin
  1 & \;\; 0  \\
  0 & \;\; 1 
\end

and the imaginary unit i with :

\begin
  0 &     -1  \\
  1 & \;\; 0  
\end

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to −1. The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z. If the matrix elements are themselves complex numbers, then the resulting algebra is that of the quaternions. In this way, the matrix representation can be seen as a way of expressing the Cayley-Dickson construction of algebras.

Geometric interpretation of the operations on complex numbers

Cayley-Dickson construction Choose a point in the plane which will be the origin,

0

. Given two points A and B in the plane, their sum is the point X in the plane such that the triangles with vertices 0, A, B and X, B, A are similar. similar Choose in addition a point in the plane different from zero, which will be the unity, 1. Given two points A and B in the plane, their product is the point X in the plane such that the triangles with vertices 0, 1, A, and 0, B, X are similar. similar Given a point A in the plane, its complex conjugate is a point X in the plane such that the triangles with vertices 0, 1, A and 0, 1, X are mirror image of each other.

Some properties

Real vector space

C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. R-linear maps C → C have the general form :

f(z)=az+b\overline

with complex coefficients a and b. Only the first term is C-linear; also only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations. The function :

f(z)=az\,

corresponds to rotations combined with scaling, while the function :

f(z)=b\overline

corresponds to reflections combined with scaling.

Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A most striking result is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and shows that the complex numbers are an algebraically closed field. Indeed, the complex number field is the algebraic closure of the real number field, and Cauchy constructed complex numbers in this way. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X² + 1: :

\mathbb = \mathbb[ X ] / ( X^2 + 1). \,

This is indeed a field because X² + 1 is irreducible, hence generating a maximal ideal, in R[X]. The image of X in this quotient ring becomes the imaginary unit i.

Algebraic characterization

The field C is (up to field isomorphism) characterized by the following three facts:
- its characteristic is 0
- its transcendence degree over the prime field is the cardinality of the continuum
- it is algebraically closed Consequently, C contains many proper subfields which are isomorphic to C. Another consequence of this characterization is that the Galois group of C over the rational numbers is enormous, with cardinality equal to that of the power set of the continuum.

Characterization as a topological field

As noted above, the algebraic characterization of C fails to capture some of its most important properties. These properties, which underpin the foundations of complex analysis, arise from the topology of C. The following properties characterize C as a topological field:
- C is a field.
- C contains a subset P of nonzero elements satisfying:
- P is closed under addition, multiplication and taking inverses.
- If x and y are distinct elements of P, then either x-y or y-x is in P
- If S is any nonempty subset of P, then S+P=x+P for some x in C.
- C has a nontrivial involutive automorphism x->x
- , fixing P and such that xx
- is in P for any nonzero x in C. Given these properties, one can then define a topology on C by taking the sets
-

B(x,p) = \

as a base, where x ranges over C, and p ranges over P. To see that these properties characterize C as a topological field, one notes that P ∪ ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization. Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting the nonzero complex numbers are connected whereas the nonzero real numbers are not.

Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Applications

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are
- in the right half plane, it will be unstable,
- all in the left half plane, it will be stable,
- on the imaginary axis, it will be marginally stable. If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis

Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency. If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as the real part of complex valued functions of the form :

f ( t ) = z e^ \,

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above. In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and Wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Improper integrals

In applied fields, the use of complex analysis is often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this, see methods of contour integration.

Quantum mechanics

The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.

Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.

Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = e^rt.

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in 2d.

Fractals

Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set.

History

The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation

x^3-x=0

: :

\frac\left(\sqrt^+\frac\right).

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation

z^3=i

has solutions −i,

\frac+\fraci

and

\frac+\fraci

. Substituting these in turn for

\sqrt^

into the cubic formula and simplifying, one gets 0, 1 and -1 as the solutions of

x^3-x=0.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in the 17th century and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation

\sqrt^2=\sqrt\sqrt=-1

seemed to be capriciously inconsistent with the algebraic identity

\sqrt\sqrt=\sqrt

, which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity

\frac=\sqrt

) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of

\sqrt

to guard against this mistake. The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To De Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula: :

(\cos \theta + i\sin \theta)^ = \cos n \theta + i\sin n \theta \,

and to Euler (1748) Euler's formula of complex analysis: :

\cos \theta + i\sin \theta = e ^. \,

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus. Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that

\pm\sqrt

should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known. The common terms used in the theory are chiefly due to the founders. Argand called

\cos \phi + i  -  \sin \phi

the direction factor, and

r = \sqrt

the modulus; Cauchy (1828) called

\cos \phi + i  - \sin \phi

the reduced form (l'expression réduite); Gauss used i for

\sqrt

, introduced the term complex number for

a+bi

, and called

a^2+b^2

the norm. The expression direction coefficient, often used for

\cos \phi + i - 
\sin \phi

, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass. Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers. A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. Gauss studied complex numbers of the form

a + bi

, where a and b are integral, or rational (and i is the root of

x^2 + 1 = 0

). His student, Ferdinand Eisenstein, studied the type

a + b\omega

, where

\omega

is a complex root of

x^3 - 1 = 0

. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity

x^k - 1 = 0

for higher values of

k

. This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation :

\ F(x) = 0

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Berloty, Henri Poincaré, Eduard Study, and Alexander MacFarlane. The formally correct definition using pairs of real numbers was given in the 19th century.

External links

- Complex numbers at Wikibooks
- [http://mathforum.org/johnandbetty/ John and Betty's Journey Through Complex Numbers]
- [http://mathworld.wolfram.com/ComplexNumber.html Complex Number from MathWorld]
- [http://www.sosmath.com/complex/complex.html SOS Math - Complex Variables]
- [http://www.binarythings.com/hidigit/ Windows calculator that supports complex numbers] Category:Complex numbers Category:Elementary mathematics ko:복소수 ja:複素数 nb:Komplekst tall th:จำนวนเชิงซ้อน

Exponentiation

In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. (The next operation after exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function.)

Positive integer exponents

The simplest case involves a positive integer exponent: For example, 3⁵ = 3 × 3 × 3 × 3 × 3 = 243. Here, 3 is the base, 5 is the exponent (written as a superscript), and 243 is 3 raised to the 5th power or 3 raised to the power 5. (The word "raised" is usually omitted, and most often "power" as well, so 3⁵ is typically pronounced "three to the fifth" or "three to the five".) Notice that the base 3 appears 5 times in the repeated multiplication, because the exponent is 5.

Notation

In contexts where superscripts are not available, such as computer languages and e-mail, 3⁵ is commonly written "3^5" (with a caret see Bc_%28Unix%29), and sometimes as "3
- 5" (with two asterisks, see Fortran). Another way of writing it, requiring Unicode encoding, is "3↑5" (with an up-arrow; HTML ↑). The exponent 1 is not normally written, since any number to the power 1 is itself. The exponents 2 and 3 occur so commonly that there are short words for them: the powers are called the square and cube of the base, respectively. 3² is pronounced "three squared," and 3³ is "three cubed." The exponents 1/2 and 1/3 occur so commonly that there are short words for them too: the powers are called the Square root and Cube root of the base, respectively.

Computation

aⁿ can be computed by means of n-1 multiplications, but, if n is a large number, the work required can be much reduced by the following trick. :aⁿ= a^b(a²)^c where :n=b+2c ; b=n mod 2 ; c=(n-b)/2. For example :2¹³=(2)(4)⁶=(2)(16)³=(2)(16)(256)¹=(32)(256)=8192 The number of multiplications performed was 5, rather than 12. The article Exponentiation by squaring provides more detail.

Zero exponents

The meaning of 3⁵ may also be viewed as 1 × 3 × 3 × 3 × 3 × 3: the starting value 1 (the identity element of multiplication) is multiplied by the base, as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to zero exponents: any number to the 0 power is 1. :

\ x^0=1

0⁰ is sometimes taken as undefined, but is sensibly defined as 1. See Empty_product#0_raised_to_the_0th_power.

Negative integer exponents

A negative exponent indicates repeated division by the base. Thus 3^-5 = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3 = 1/243. Raising any nonzero number to the -1 power produces its reciprocal. :

\ x^=\frac

\ x^=(x^n)^=\frac

Raising 0 to a negative power would imply division by 0, and so is undefined.

Identities and properties

Important identities satisfied by exponentiation include:
- x^m+n = x^mxⁿ
- x^m-n = x^m/xⁿ
- (x^m)ⁿ = x^mn Whereas addition or multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2×3 = 6 = 3×2), this is not true of exponentiation: 2³ = 8 while 3² = 9. Similarly, whereas addition or multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2×3)×4 = 24 = 2×(3×4)), this is not true of exponentiation either: 2³ to the 4th power is 8⁴ or 4,096, while 2 to the 3⁴ power is 2⁸¹ or 2,417,851,639,229,258,349,412,352.

Powers of ten

Powers of 10 are easy to compute because we use a base ten number system: for example 10⁶ = 1 million, which is 1 followed by 6 zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 (the speed of light in a vacuum, in meters per second) can be written as 2.99792458 × 10⁸ and then approximated as 2.998 × 10⁸ if this is useful. SI prefixes are also used to describe small or large quantities, and these are also based on powers of 10; for example, the prefix kilo means 10³ = 1000, so a kilometre is 1000 metres.

Powers of two

Powers of 2 are important in computer science; for example, there are 2ⁿ possible values for a variable that takes n bits to store in memory. They occur so commonly that SI prefixes are commonly reinterpreted to refer to them: 1 kilobyte = 2¹⁰ = 1024 bytes. As the standard meanings of the prefixes also occur, confusion may result, and in 1998 the International Electrotechnical Commission approved a set of binary prefixes. For instance, the prefix for multiples of 1024 is kibi-, so 1024 bytes is 1 kibibyte. Other prefixes are mebi-, gibi-, and tebi-.

Powers of e

A big power of a number close to one can be written in the form :

\ \left(1+\frac\right)^n

Here

\ n

is an integer, so the power is well defined. Substituting :

\ n=mx

gives :

\ \left(1+\frac\right)^n= \left(1+\frac\right)^= \left( \left(1+\frac \right) ^m\right) ^x

. The limiting value when

\ m

goes to infinity has got the name e :

\ e=\lim_ \left(1+\frac \right) ^m

. Any power of e is defined by :

\ e^x=\lim_ \left(1+\frac \right) ^n

. (For further reading, see Exponential function).

Powers of one

The above definition of

e^x

also apply when the exponent

x

is a complex number. If the exponent is an imaginary number,

i x

, then

e^

is a complex number on the unit circle, a direction. (See Euler's formula). The real number

x

is an angle measured in radian. The angle

2\pi

radian is a turn: :

\ e^=1

2\pi i

is a solution to the equation

e^x=1

. It is not the only one. The solutions are simply

2\pi i n

where

n

is an integer (see Pi). The expression :

\ e^

is an x 'th power of one. If x is an integer the result is 1. If x is a rational number the result is a root of unity. If x is a real number the result is a direction.

Powers of zero

If the exponent

\ x

is positive, the power of zero is zero:

\ 0^x=0

. If the exponent is negative, the power of zero

\ 0^

is undefined. If the exponent is zero, the power is always one.

\ 0^0=1

. (See Empty_product#0_raised_to_the_0th_power)

Fractional exponents

Exponentiation with a fractional exponent

\ \frac

, where

n\ge 2

, is a solution to the equation :

\ x^n = a.

For

a \ne 0

this equation has

n

solutions. If

\ a

is a positive real number, then one of the solutions is also a positive real number. The positive solution to

\ x^n = a

is called a radical,

\sqrt[n]

for

a\ne 1.

The positive solution to

\ x^n = 1

is (of course)

\ 1

. All the solutions are given by: :

e^\sqrt[n]

for

k=0,\ldots,n-1

. If

a

is a complex number which is not a positive real number, then

\sqrt[n]

is a multivalued function of

a

. Exponentiation with a fractional exponent

m/n

a^ = \left( \sqrt[n]\right)^m.

For example: 8^2/3 = 4. It was once believed that exponentiation with fractional exponents leads to finding roots of polynomials. That this is not true in general is the assertion of the Abel-Ruffini theorem. For example, the solutions of the equation :

\ x^5=x+1

cannot be expressed in terms of fractional exponents. For solving any equation of the n^th degree, see Root-finding algorithm#How to solve an algebraic equation.

Arbitrary real and complex exponents

Exponentiation to an arbitrary real exponent can then be defined by continuity. With real numbers, the exponential function exp is the same as raising the transcendental number e to the indicated power: exp x = e^x. Exponentiation of real numbers, and even complex numbers, can be understood with the aid of the exponential function and its inverse, the natural logarithm; in general, we can define : x^y = exp(y ln x). However, in the complex number field, it should be noted that logarithms are always multi-valued functions, usually with an imaginary periodicity. Therefore, expressions such as x^y are always similarly multi-valued. A "branch cut" may be created in the complex plane, if a single-valued logarithm or power is desired. Most often, this branch cut is made along the negative real axis. The use of e^x in this context is usually assumed to use this "principal branch" of the logarithm, so the results correspond with that of the exponential function which satisfies analyticity constraints. For more on exponents in real and complex numbers, and other situations relevant to mathematical analysis, see Exponential function. That article also lists certain exponential laws (more general than the algebraic laws listed below) that apply in these situations.

Exponents on function names

When the name or symbol of a function is given an integer superscript, as if being raised to a power, this commonly refers to repeated function composition rather than repeated multiplication. Thus f³(x) may mean f(f(f(x))); in particular, f^-1(x) usually denotes f's inverse function. A special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of -1 indicates the inverse function. That is, sin²x is just a shorthand way to write (sin x)² without using parentheses, whereas sin^-1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand of this kind for reciprocal trigonometric functions since they each have their own name and abbreviation already: (sin x)^-1 is normally just written as csc x.

Exponentiation in abstract algebra

Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers. Specifically, suppose that X is a set with a power-associative binary operation, which we will write multiplicatively. In this very general situation, we can define xⁿ for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn't matter in which order we perform the multiplications. Now additionally suppose that the operation has an identity element 1. Then we can define x⁰ to be equal to 1 for any x. Now xⁿ is defined for any natural number n, including 0. Finally, suppose that the operation has inverses, and that the multiplication is associative (so that the magma is a group). Then we can define x^-n to be the inverse of xⁿ when n is a natural number. Now xⁿ is defined for any integer n. Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):
-

\ x^=x^mx^n

\ x^=x^m/x^n

\ x^=1/x^n

\ x^0=1

\ x^1=x

\ x^=1/x

\ (x^m)^n=x^

Here, we use a division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol x^-1 for raising x to the power -1, rather than the inverse of x. However, as one of the laws above states, x^-1 is always equal to the inverse of x, so the notation doesn't matter in the end. If in addition the multiplication operation is commutative (so that the magma is an abelian group), then we have some additional laws:
- (xy)ⁿ = xⁿyⁿ
- (x/y)ⁿ = xⁿ/yⁿ Notice that in this algebraic context, 0⁰ is always equal to 1. When 0⁰ is attained as a limit, however, it may be more useful to leave 0⁰ undefined. However, when exponentiation is purely algebraic, that is when the exponents are taken only to be integers, then it is generally most useful to let 0⁰ be 1, just like every other case of x⁰. For example, if you expand (0 + x)ⁿ using the binomial theorem, you'll want to use 0⁰ = 1. If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication. When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x^{- n} is x
- ···
- x, while x^#n is x # ··· # x, whatever the operations
- and # might be. Exponential notation is also used, especially in group theory, to indicate conjugation. That is, g^h = h^-1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

Exponentiation over sets

The above algebraic treatment of exponentiation builds a finitary operation out of a binary operation. In more general contexts, one may be able to define an infinitary operation directly on an indexed set. For example, in the arithmetic of cardinal numbers, it makes sense to say :

\prod_ k_

for any index set I and cardinal numbers k_i. By taking k_i = k for every i, this can be interpreted as a repeated product, and the result is k^I. In fact, this result depends only on the cardinality of I, so we can define exponentiation of cardinal numbers so that k^l is k^I for any set I whose cardinality is l. This can be done even for operations on sets or sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of :

\bigoplus_ V_,

where each V_i is a vector space. Then if V_i = V for each i, the resulting direct sum can be written in exponential notation as V^(+)I, or simply V^I with the understanding that the direct sum is the default. We can again replace the set I with a cardinal number k to get V^k, although without choosing a specific standard set with cardinality k, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and k to be some natural number n, we get the vector space that is most commonly studied in linear algebra, the Euclidean space Rⁿ. If the base of the exponentiation operation is itself a set, then by default we assume the operation to be the Cartesian product. In that case, S^I becomes simply the set of all functions from I to S. This fits in with the exponentiation of cardinal numbers once gain, in the sense that |S^I| = |S|^|I|, where |X| is the cardinality of X. When I=2=, we have |2^X| = 2^|X|, where 2^X, usually denoted by PX, is the power set of X. (This is where the term "power set" comes from.) Note that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, a^b is the smallest ordinal number greater than a^c for c < b when b is a limit ordinal, and of course a^b+1 := a^ba. In category theory, we learn to raise any object in a wide variety of categories to the power of a set, or even to raise an object to the power of an object, using the exponential.

Syntax in some common languages / applications

- x ^ y: Basic, Matlab and many others
- x
- y: Fortran, Perl, Python, Ruby
- Power(x, y): Pascal
- pow(x, y): C, C++
- Math.pow(x, y): Java, JavaScript Note that C, C++, Java and JavaScript represent bitwise XOR with ^ .

Table

Table of kⁿ, with k on the left and n at the top.

External links

- [http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ sci.math FAQ: What is 0⁰?]
- Category:Exponentials ja:冪乗

Complex plane

In mathematics, the complex plane is a way of visualising the space of the complex numbers. It can be thought of as a modified cartesian plane, with the real part represented in the x-axis and the imaginary part represented in the y-axis. The x-axis is also called the real axis and the y-axis is called the imaginary axis. The complex plane is sometimes called the Argand plane for its use in Argand diagrams. Its creation is generally credited to Jean-Robert Argand, although it was first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel. The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, and the multiplication of complex numbers can be expressed simply using polar coordinates, where the magnitude of the product is the product of those of the terms, and the angle from the real axis of the product is the sum of those of the terms. Argand diagrams are frequently used to plot the positions of poles and zeros of a function in the complex plane.

Use of the complex plane in control theory

In control theory, one use of a complex plane is that known as the 's-plane'. It is used to visualise the roots of the equation describing a system's behaviour (the characteristic equation) graphically. The equation is normally expressed as a polynomial in the parameter 's' of the Laplace transform, hence the name 's' plane. In addition, a related use of the complex plane is with the Nyquist stability criterion. This is a geometric principle which allows the stability of a control system to be determined from inspection of a Nyquist plot of its frequency-phase response (Transfer function) in the complex plane. The 'z-plane' is a discrete-time version of the s-plane, where z-transforms are used instead of the Laplace transformation.

External links

- Category:Complex numbers Category:Control theory

Vertex

In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). In graph theory, a graph describes a set of connections between objects. Each object is called a node or vertex. The connections themselves are called edges or arcs. In 3D computer graphics, a vertex is a point in 3D space with a particular location, usually given in terms of its x, y, and z coordinates. It is one of the fundamental structures in polygonal modelling: two vertices, taken together, can be used to define the endpoints of a line; three vertices can be used to define a planar triangle. Vertices are commonly confused with vectors because a vertex can be described as a vector from a coordinate system's origin. They are, however, two completely different things. In anatomy, the vertex is the highest point of the skull in the anatomical position (i.e. standing upright). It lies between the parietal bones in the median sagittal plane. In Astrology, the Vertex is a point in 3D space, seen near the Eastern horizon and the Ascendant in the (two-dimensional) starchart. Planets close to it lend their personality to the event(s) the chart captures.

Regular polygon

:For other use please see Polygon (disambiguation) A polygon (literally "many angle", see Wiktionary for the etymology) is a closed

Root of unity

Definition

Primitive roots

Examples

Summation

Orthogonality

Omega notation

Cyclotomic polynomials

Cyclotomic fields

References

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

Common misconceptions

See also

Bibliography

External links

Complex numbers

Definition

The complex number field

The complex plane

Absolute value, conjugation and distance

Complex number division

Matrix representation of complex numbers

Geometric interpretation of the operations on complex numbers

Some properties

Real vector space

Solutions of polynomial equations

Algebraic characterization

Characterization as a topological field

Complex analysis

Applications

Control theory

Signal analysis

Improper integrals

Quantum mechanics

Relativity

Applied mathematics

Fluid dynamics

Fractals

History

See also

Further reading

External links

Exponentiation

Positive integer exponents

Notation

Computation

Zero exponents

Negative integer exponents

Identities and properties

Powers of ten

Powers of two

Powers of e

Powers of one

Powers of zero

Fractional exponents

Arbitrary real and complex exponents

Exponents on function names

Exponentiation in abstract algebra

Exponentiation over sets

Syntax in some common languages / applications

Table

See also

External links