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Mathematical Constant

Mathematical constant

A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. Unlike physical constants, mathematical constants are defined independently of any physical measurement. Apart from the obvious cases of zero and unity, many particular numbers have special significance in mathematics, and arise in many different contexts. For example, up to multiplication with nonzero complex numbers, there is a unique holomorphic function f with f = f. Therefore, f(1)/f(0) is a mathematical constant, the constant e. f is also a periodic function, and the absolute value of its period is another mathematical constant, 2π. Mathematical constants are typically elements of the field of real numbers or complex numbers. Mathematical constants that one can talk about are definable numbers (and almost always also computable). However, there are still some mathematical constants for which only very rough estimates are known. An alternate sorting may be found at Mathematical constant (sorted by continued fraction representation)
Table of selected mathematical constants
Abbreviations used: : I - irrational number, A - algebraic number, T - transcendental number, ? - unknown : Gen - General, NuT - Number theory, ChT - Chaos theory, Com - Combinatorics, Inf - Information theory, Ana - Mathematical analysis
See also

- An alternative sorting based on the continued fraction representations
- Physical constants
External links

- Steven Finch's page of mathematical constants: http://pauillac.inria.fr/algo/bsolve/constant/constant.html
- Steven Finch's alternative indexing: http://pauillac.inria.fr/algo/bsolve/constant/table.html
- Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms: http://numbers.computation.free.fr/Constants/constants.html
- Simon Plouffe's inverter: http://pi.lacim.uqam.ca/eng/
- CECM's Inverse symbolic calculator (ISC) (tells you how a given number can be constructed from mathematical constants): http://www.cecm.sfu.ca/projects/ISC/
- ko:수학 상수 ja:数学定数

Real number
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247… The three dots indicate that there would still be more digits to come, no matter how many more might be added at the end. Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. Mathematicians use the symbol R (or alternatively, $\Bbb$ , the letter "R" in blackboard bold) to represent the set of all real numbers. The notation Rⁿ refers to an n-dimensional space of real numbers; for example, a value from R³ consists of three real numbers and specifies a location in 3-dimensional space. In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.
History
Vulgar fractions had been used by the Egyptians around 1000 BC; around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers were invented by Indian mathematicians around 600 AD, and then possibly reinvented in China shortly after. They were not used in Europe until the 1600s, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic. The development of calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.
Definition

Construction from the rational numbers
The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.
Axiomatic approach
Let R denote the set of all real numbers. Then:
- The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
- The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
- if x ≥ y then x + z ≥ y + z;
- if x ≥ 0 and y ≥ 0 then xy ≥ 0.
- The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R₁ and R₂, there exists a unique field isomorphism from R₁ to R₂, allowing us to think of them as essentially the same mathematical object.
Properties

Completeness
The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x_m| is less than ε provided that n and m are both greater than N. In other words, a sequence is a Cauchy sequence if its elements x_n eventually come and remain arbitrarily close to each other. A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x. It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true: :Every Cauchy sequence of real numbers is convergent. That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example, the standard series of the exponential function : $\mathrm^x = \sum_^ \frac$ converges to a real number because for every x the sums : $\sum_^ \frac$ can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we do not know ahead of time what the limit is.
"The complete ordered field"
The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore the field structure. However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way. But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.
Advanced properties
The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2^ω, i.e., the cardinality of the set of subsets of the natural numbers. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. The real numbers form a metric space: the distance between x and y is defined to be the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected and simply connected), separable metric space of dimension 1, and are everywhere dense. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, continuous, and separable order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra. The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1. The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim-Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of hyperreal numbers is much bigger than R but also satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.
Generalizations and extensions
The real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean. Occasionally, the two formal elements +∞ and −∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers. Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers. Category:Elementary mathematics Category:Real numbers Category:Set theory ko:실수 ja:実数 th:จำนวนจริง

Complex number

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

Physical constants

In science, a physical constant is a physical quantity whose numerical value does not change. It can be contrasted with a mathematical constant, which is a fixed value that does not directly involve a physical measurement. There are many physical constants in science, some of the most famous being Planck's constant

\hbar \

, the gravitational constant

G \

, the speed of light

c \

, the electric constant

\epsilon_0 \

, and the elementary charge

e \

. Constants can take many forms: the speed of light in a vacuum signifies a maximum speed limit of the universe; while the fine-structure constant

\alpha \

characterizes the interaction between electrons and photons, is dimensionless. Beginning with Paul Dirac in 1937, some scientists have speculated that physical constants may actually decrease in proportion to the age of the universe. Scientific experiments have not yet pinpointed any definite evidence that this is the case, although they have placed upper bounds on the maximum possible relative change per year at very small amounts (roughly 10^-5 per year for the fine structure constant

\alpha \,\!

and 10^-11 for the gravitational constant

G \

). However it is currently disputed [http://www.arxiv.org/abs/hep-th/0208093] [http://xxx.lanl.gov/pdf/physics/0110060] that any changes in dimensionful physical constants such as

G \

c \

\hbar \

, or

\epsilon_0 \

are operationally meaningless, however a change in a dimensionless constant such as

\alpha \

is something that would definitely be noticed. If a measurement indicated that a dimensionful physical constant had changed, that is the result or interpretation of a more fundamental dimensionless constant changing, which is the salient metric. Unless the system of natural units is used, the numerical values of dimensionful physical constants are artifacts of the unit system used, such as SI or cgs; that is, they are are essentially conversion factors of human construct. While some properties of materials and particles are constant, they do not show up on this page because they are specific to their respective materials or properties alone. Constants that are independent of systems of units are typically dimensionless numbers, are known as fundamental physical constants, and are truly meaningful parameters of nature, not merely human constructs. The fine-structure constant

\alpha \

is probably the most well known dimensionless fundamental physical constant. The dimensionless ratios of masses (or other like dimensioned properties) of fundamental particles are also fundamental physical constants as are the measure of these properties in terms of natural units. Some people claim that if the physical constants had slightly different values, our universe would be so different that intelligent life would probably not have emerged, and that our universe seems to be fine-tuned for intelligent life. The weak anthropic principle simply says that only because these fundamental constants came out to be the values they are, that there was sufficient order and richness in elemental diversity, that life could have formed and eventually evolved with sufficient intelligence to observe that these constants have taken on the values they have.

Table of universal constants

Table of electromagnetic constants

Table of atomic and nuclear constants

Table of physico-chemical constants

Table of adopted values

Notes

¹The values are given in the so-called concise form; the number in brackets is the standard uncertainty, which is the value multiplied by the relative standard uncertainty.
²This is the value adopted internationally for realizing representations of the volt using the Josephson effect.
³This is the value adopted internationally for realizing representations of the ohm using the quantum Hall effect.

References

- [http://physics.nist.gov/cuu/Constants CODATA Recommendations] - 2002 CODATA Internationally recommended values of the Fundamental Physical Constants Category:Measurement ko:물리 상수 ja:物理定数

Unity (number)

: This article discusses the number one. For the year AD 1, see 1. For other uses of 1, see 1 (disambiguation) 1 (one) is a number, numeral, and the name of the glyph representing that number. It is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity, and unit is sometimes used as an adjective in this sense. (For example, a line segment of "unit length" is a line segment of length 1.)

History

Some Ancient Greeks did not consider one as a number: they considered it to be the unit, two being the first proper number as it represented a multiplicity.

In mathematics

For any number x: :x·1 = 1·x = x (This expresses the fact that 1 is the multiplicative identity.) As a consequence of this, 1 is a 1-automorphic number in any place-based numbering system. :x/1 = x (see division) :x¹ = x, 1^x = 1, and for nonzero x, x⁰ = 1 (see exponentiation) :x↑↑1 = x and 1↑↑x = 1 (see tetration). Using ordinary addition, we have 1 + 1 = 2; depending on the interpretation of the symbol "+" and the numeral system used, the expression can have many different meanings, listed at one plus one. One cannot be used as the base of a positional numeral system in the ordinary way. Sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this doesn't work in the same way as other positional numeral systems. Related to this, one cannot take logarithms with base 1, since the "exponential function" with base 1 is the constant function 1. In the Von Neumann representation of natural numbers, 1 is defined as the set . This set has cardinality 1 and hereditary rank 1. Sets like this with a single element are called singletons. In a multiplicative group or monoid, the identity element is sometimes denoted "1", but "e" (from the German Einheit, unity) is more traditional. However, "1" is especially common for the multiplicative identity of a ring. (Note that this multiplicative identity is also often called "unity".) One is its own factorial, and its own square and cube (and so on, as 1 × 1 × ... × 1 = 1). As a consequence of its being its own square, one is also a Kaprekar number. One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few. It is also the first and second numbers in the Fibonacci sequence, and is the first number in a lot of mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that didn't already have it, and considered these initial 1's in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases. One is the empty product. One is a harmonic divisor number. One is most often used for representing 'true' as a Boolean datatype in computer science. One is currently considered neither a prime number, nor a composite number - although it used to be considered prime. Defining a prime as a number that is only divisible by one and itself, one is a prime. However, for purposes of factorization and especially the fundamental theorem of arithmetic, it is more convenient to not think of one as a prime factor, or to think of it as an implicit factor that's always there but need not be written down. To exclude the number one from the list of prime numbers, primality is defined as a number having exactly two distinct divisors, one and itself. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899, although Carl Sagan included one in a list of prime numbers in his book Contact in 1985. One is one of three possible return values of the Möbius function. Passed an integer that is square-free with an even number of distinct prime factors, the Möbius function returns one. One is the only odd number that is in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2. One is the only 1-perfect number (see multiply perfect number). One is equal to the sum of its digits in any place-based numbering system, making it an all-Harshad number. One is the number of n × n magic squares for n = 1, 3. One is the number of n-queens problem solutions for n = 1. One is a meandric number, a semi-meandric number, and an open meandric number. By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix. One is the value of the sine and cosine at π/2 and 0 radians, respectively. One is the most common leading digit in many sets of data, a consequence of Benford's law. See also -1.

The Arabic glyph

Image:Evolution1glyph.png The glyph used today in the Western world to represent the number 1, a vertical line, often with a little serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmin Indians, who wrote 1 as a horizontal line (in Chinese today this is the way it is written). The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right, but kept the circle small. This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. In fonts with text figures, 1 is typically the same height as a lowercase X, for example, Image:TextFigs148.png.

In science

One is:

- set equal to celerity (c), the speed of light, in Heaviside notation to simplify calculations.
- the factor in ratios for unit conversions.
- the total density ratio for a flat universe.
- The atomic number of hydrogen In astronomy, : Messier object M1, a magnitude 7.0 supernova remnant in the constellation Taurus, also known as the Crab Nebula. : The New General Catalogue [http://www.ngcic.org/ object] NGC 1, a 13th magnitude 13 spiral galaxy in the constellation Pegasus :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the solar eclipse series which began on -2872 June 4 and ended on -1592 July 11 . The duration of Saros series 1 was 1280.1 years, and it contained 73 solar eclipses. :The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the lunar eclipse series which began on -2588 March 2 and ended on -1272 April 30. The duration of Saros series 1 was 1316.2 years, and it contained 74 lunar eclipses.

In human society

Many human cultures have given the concept of one-ness symbolic meanings. Many religions consider God to be a perfect example of one-ness. See monad for a detailed discussion of other types of one-ness. One represents unity, togetherness, and absence of separation or discrimination, e.g. "We are all one" or "everyone". Something is unique if it is the only one of its kind. More loosely and exaggeratingly (especially in advertising) the term is used for something very special. One is also an (archaic) expression of the first person singular ("one is not amused") and of the second person singular ("does one take sugar?"). In Western culture, it is believed by many that the maximum number of girlfriends or boyfriends one may have at one time is 1. Also, it is strongly believed that you can be married to only 1 person at any time - this is called monogamy. Being married to more than one person at any time is called bigamy or polygamy. This is illegal in many Western societies. Among children, or when otherwise calling for subtlety, the phrase "number 1" can refer to the act of urination. This can derive from a traditional U.S. elementary school practice of holding up one or two fingers to indicate the approximate time of a requested absence. "Number 1" can also refer to oneself, or that something is first in its class, the latter being used often as a cheer in sports games. On a clock, 1:00 signals that one full hour has passed since the last change of the "AM" or "PM" meridian.

In music

In harmonic analysis of tonal music, the tonic chord is referred to as I. "One is the loneliest number that you'll ever do", according to the first line of "One" by Three Dog Night. The number appears in the title of songs by Metallica, U2, Creed, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, Harry Nilsson, Three Dog Night, and the Bee Gees. Also the title of a best-selling compilation album of all Beatles songs that reached number 1 in the UK or US charts (see List of Number 1 Hits (USA)). The song Green grow the rushes, O has the line "One and all alone and evermore shall be so."

In religion

There is one god according to monotheism (see also: tawhid). There is one surat al-Fatiha in the Qur'an.

In sports

In some sports, one is the number of a specific position: in rugby union, the number of the loosehead prop; in baseball, the number representing the pitcher's position; in football, the number of the goalkeeper. In 2004, fans of the Philadelphia Eagles NFL team used the phrase "One" to show support for the team as they inched closer to the Super Bowl. The full text of the phrase was "One Team. One City. One Dream."

In technology

One is the DVD region of the United States and Canada. In the DOS Shell and many Windows programs, the function key F1 calls up online help. On most standard phones, the 1 key is not associated with any letters the way other number keys are, but on the BlackBerry, 1 is also the key for the letters E and R. Some cellular phones associate the "1" key with various symbols (i.e. the pound sign, the ampersand, etc.) when users engage in text messaging. In the Rich Text Format specification, 1 is the language code for the Arabic language. All codes for dialects of Arabic are congruent to 1 mod 256. 1 is a punctuation mark indicating exclamation, or the letter "l" in "leetspeak". 1, in binary, stands for 'yes'.

In other fields

One is:
- the denomination of U.S. dollar bill with George Washington's portrait, and the denomination of coin with Sacagawea's portrait. It is also the denomination of the older Eisenhower and Susan B. Anthony tender coins and the American Silver Eagle bullion coin.
- the denomination of Canadian dollar coin with a swimming loon on the reverse, hence its universally employed nickname, the "loonie"
- in cents of a U.S. dollar, the denomination of coin with Abraham Lincoln's portrait, commonly known as a penny.
- in cents of a Canadian dollar, the denomination of coin with two maple leaves on the reverse, also known as a penny.
- the code for international direct dial phone calls to countries participating in the North American Numbering Plan, such as the United States and Canada.
- the designation of many roads, listed at Route 1.
- as Air Force One is the callsign of any United States Air Force aircraft carrying the President of the United States.
- the house number of Number One Observatory Circle, the US Vice-President's residence.
- the address of Apsley House, known simply as Number 1, London.
- a subway service in New York City. See 1 (New York City Subway service).
- the name of a train operating company in East Anglia, England. See: 'one'
- an enneagram personality type.
- part of a nickname for the U. S. Army's First Infantry Division, "The Big Red One".
- as a word in all capitals, ONE, a trademark of Nestlé Purina PetCare for a line of pet food products (acronym for the "optimum nutritional effectiveness" they are claimed to possess).
- as ONE, the self-referential acronym for ONE North East, the regional development agency in North East England.
- used as a pronoun in English grammar to refer to an unspecified person: see generic you.
- in Astrology, Aries is the 1st astrological sign of the Zodiac.
- the number of syllables in a monosyllabic word.
- the number of humps on a dromedary.
- the number of wheels on a unicycle.
- the number of players in solitaire, Tetris, and many video games.
- the year A.D. 1 or the immediately previous year 1 B.C., or the year 1 in a century.
- January 1, the first day of the year in the Gregorian calendar, is New Year's Day.
- April 1 is commonly known as April Fool's Day.
- To Roman Catholics, November 1 is All Saint's Day. Category:Famous numbers 01 ko:1 ja:1 simple:One th:1 (จำนวน)

E (mathematical constant)

The mathematical constant e is the base of the natural logarithm function. Its value to the 29th decimal digit is: :e = 2.71828 18284 59045 23536 02874 7135... Alongside the number π and the imaginary unit i, e is one of the most important numbers in mathematics. It has a number of equivalent definitions; some of them are given below. e is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms.

Definitions

The three most common definitions of e are listed below. :1. Define e as the limit ::

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

:2. Define e as the sum of the infinite series ::

e = \sum_^\infty  =  + 
  +  + 
  +  + \cdots

: where n! is the factorial of n. :3. Define e to be the unique real number x > 0 such that ::

\int_^ \frac \, dt = .

These different definitions have been proven to be equivalent.

Properties

The exponential function e^x is important because it is the unique function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive: :

\frace^x=e^x

and :

\int e^x\,dx=e^x + C

, where C is a constant. e is known to be both irrational (proof) and transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare Liouville number); the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features in Euler's formula, one of the most important identities in mathematics: :

e^ = \cos(x) + i\sin(x) \,\!

The special case with x = π is known as Euler's identity: :

e^ + 1 = 0 \,\!

described by Richard Feynman as Euler's jewel. The infinite continued fraction expansion of e contains an interesting pattern (sequence A005131 in OEIS) that can be written as follows: :

e = [1; 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12,\ldots] \,

History

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of natural logarithms calculated from the constant. It is assumed that the table was written by William Oughtred. The first indication of e as a constant was discovered by Jacob Bernoulli, trying to find the value of the following expression. :

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

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler started to use the letter e for the constant in 1727, and the first use of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard. The exact reasons for the use of the letter e are unknown, but it may be because it is the first letter of the word exponential. Another possibility is that Euler used it because it was the first vowel after a, which he was already using for another number, but his reason for using vowels is unknown. It is unlikely that Euler choose the letter because it is his first initial, since he was a very modest man, always trying to give proper credit to the work of others.

Non-mathematical uses of e

One of the most famous mathematical constants, e is also frequently referenced outside of mathematics. Some examples are:
- In the IPO filing for Google Inc., in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is, of course, e billion dollars to the nearest dollar.
- Google was also responsible for a mysterious billboard [http://mattwalsh.com/twiki/pub/Main/GoogleBillboardContestFindingPrimesInE/IMG_0742.JPG] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts, which read .com. Solving this problem and visiting the web site advertised led to an even more difficult problem to solve. (The first 10-digit prime in e is 7427466391, which surprisingly starts as late as at the 101st digit.) [http://www.mkaz.com/math/google/]
- The famous computer scientist Donald Knuth let the version numbers of his book METAFONT approach e (the versions are 2, 2.7, 2.71, 2.718, etc.).

References

- Maor, Eli; e: The Story of a Number, ISBN 0691058547
- O'Connor, J.J., and Roberson, E.F.; The MacTutor History of Mathematics archive: [http://www-history.mcs.st-andrews.ac.uk/HistTopics/e.html "The number e"]; University of St Andrews Scotland (2001)

Notes

O'Connor, "The number e"

External links

- [http://www.gutenberg.org/etext/127 The number e to 1 million places]
- [http://members.aol.com/jeff570/constants.html Earliest Uses of Symbols for Constants]
- [http://members.optusnet.com.au/exponentialist/The_Scales_Of_e.htm 'The Scales Of e' demonstrates that fixed rate and variable rate compound growth are both exponential in nature.] Category:Transcendental numbers Category:Mathematical constants Category:Exponentials 2.71828 ko:E (수학상수) ja:ネイピア数

Absolute value

In mathematics, the absolute value (or modulus¹) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3. In computers, the mathematical function used to perform this calculation is usually given the name abs(). Generalizations of the absolute value for real numbers, occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. norm

Real numbers

For any real number

a,

the absolute value or modulus of

a,

is denoted ²

|a|,

and is defined as :

|a| := \begin a, & \mbox  a \ge 0  \\ -a,  & \mbox a < 0. \end

As can be seen from the above definition, the absolute value of

a

is always either positive or zero, never negative. From a geometric point of view, the absolute value of a real number is the distance along the real number line of that number from zero, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed the notion of an abstract distance function in mathematics can be seen to be a generalization of the properties of the absolute value (see "Distance" below). The following proposition, gives an identity which is sometimes used as an alternative (and equivalent) definition of the absolute value: PROPOSITION 1: :

|a| = \sqrt

The absolute value has the following four fundamental properties: PROPOSITION 2: : Other important properties of the absolute value include: PROPOSITION 3: : Two other useful inequalities are: :

|a| \le b \iff -b \le a \le b

|a| \ge b \iff a \le -b \mbox b \le a

The above are often used in solving inequalities; for example: :

Complex numbers

image:complex.png

Since the complex numbers are not ordered, the definition given above for the real absolute value, can not be directly generalized for a complex number. However the identity given in Proposition 1: :

|a| = \sqrt

can be seen as motivating the following definition. For any complex number :

z = x + iy\,

the absolute value or modulus of

z

is denoted

|z|,

and is defined as :

|z| :=  \sqrt.

It follows that the absolute value of a real number x is equal to its absolute value considered as a complex number since: :

|x + i0| = \sqrt = \sqrt = |x|.

Similar to the geometric interpretation of the absolute value for real numbers, it follows from the Pythagorean theorem that the absolute value of a complex number is the distance in the complex plane of that complex number from the origin, and more generally, that the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers. The complex absolute value shares all the properties of the real absolute value given in Propositions 2 and 3 above. In addition, If :

z = x + \mathrmy = r (\cos \phi + \mathrm\sin \phi ) \,

and :

\bar = x - iy

is the complex conjugate of

z

, then it is easily seen that :

|z| = r\,

|z|=|\bar|

|z| = \sqrt

Absolute value functions

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (-∞, 0] and monotonically increasing on the interval [0, ∞). Since a real number and its negative have the same absolute value, it is an even function, and is hence not invertible. The complex absolute value function is continuous everywhere but differentiable nowhere (One way to see this is to show that it does not obey the Cauchy-Riemann equations). Both the real and complex functions are idempotent.

Ordered rings

The definition of absolute value given for real numbers above can easily be extended to any ordered ring. That is, if

a

is an element of an ordered ring

R

, then the absolute value of

a

, denoted by

|a|

, is defined to be: :

|a| := \begin a, & \mbox  a \ge 0  \\ -a,  & \mbox a < 0, \end

where

-a

is the additive inverse of

a

, and

0

is the additive identity element.

Distance

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. The standard Euclidean distance between two points :

a = (a_1, a_2, \cdots , a_n)

and :

b = (b_1, b_2, \cdots , b_n)

in Euclidean n-space is defined as: :

\sqrt.

This can be seen to be a generalization of

|a - b|,

since if

a,

b

are real, then by Proposition 1, :

|a - b| = \sqrt

while if :

a = a_1 + i a_2 \,

and :

b = b_1 + i b_2 \,

are complex numbers, then : The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclidean spaces respectively. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given in Propositions 2 and 3 above, can be seen to motivate the more general notion of a distance function as follows: A real valued function

d

on a set

X \times X

is called a distance function (or a metric) for

X

, if it satisfies the following four axioms: :

Fields

The fundamental properties of the absolute value for real numbers given in Proposition 2 above, can be used to generalize the notion of absolute value to an arbitrary field, as follows. A real-valued function

v

on a field

F

is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms: : It follows from the above that

v(1) = 1

, where

1

denotes the multiplicative identity element of

F

. The real and complex absolute values defined above are examples of absolute values for an arbitrary field. If

v

is an absolute value on

F

, then the function

d

F \times F

, defined by

d(a, b) = v(a - b)

, is a metric, and if

e

is the multiplicative identity in

F

, then the following are equivalent:
-

d

satisfies the ultrametric inequality

d(x, y) \le \mathrm\.

\big\

is bounded in R.
-

v\Big(\sum_^n e\Big) \le 1

for every

n \in \mathbb.

v(a + b) \le \mathrm\

for all

a, b \in F.

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.³

Vector spaces

Again the fundamental properties of the absolute value for real numbers, can be used, with a slight modification, to generalize the notion to an arbitrary vector space. A real valued function ||·|| on a vector space

V

a over a field

F

, is called an absolute value (or more usually a norm) if it satisfies the following axioms: For all

a

F

, and

\mathbf

\mathbf

V

, : The norm of a vector is also called its length or magnitude. In the case of Euclidean space Rⁿ, the function :

\|(x_1, x_2, \cdots , x_n) \| = \sqrt

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R¹, the absolute value is a norm, and is the p-norm for any p. In fact the absolute value is the "only" norm in R¹, in the sense that, for every norm ||·|| in R¹, ||x||=||1||·|x|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean plane R².

Algorithms

In the C programming language, the abs(), labs(), llabs() (in C99), fabs(), fabsf(), and fabsl() functions compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the largest negative integer is input: int abs(int i) The floating-point versions are trickier, as they have to contend with special codes for infinity and not-a-numbers. Using assembly language, it is possible to take the absolute value of a register in just three instructions (example shown for a 32-bit register on an x86 architecture, Intel syntax): cdq xor eax, edx sub eax, edx cdq extends the sign bit of eax into edx. If eax is nonnegative, then edx becomes zero, and the latter two instructions have no effect, leaving eax unchanged. If eax is negative, then edx becomes 0xFFFFFFFF, or -1. The next two instructions then become a two's complement inversion, giving the absolute value of the negative value in eax.

References

- Nahin, Paul J.; [http://www.amazon.com/gp/reader/0691027951/ref=sib_dp_pt/103-5443484-7306247#reader-link An Imaginary Tale]; Princeton University Press; (hardcover, 1998). ISBN 0691027951
- O'Connor, J.J. and Robertson, E.F.; [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html "Jean Robert Argand"]
- Schechter, Eric; Handbook of Analysis and Its Foundations, pp 259-263, [http://www.amazon.com/gp/reader/0126227608/103-5443484-7306247?v=search-inside&keywords=absolute%20value "Absolute Values"], Academic Press (1997) ISBN 0126227608
- Weisstein, Eric W.; MathWorld: [http://mathworld.wolfram.com/AbsoluteValue.html "Absolute Value"]

Notes

¹ Jean-Robert Argand, is credited with introducing the term "modulus" in 1806, see: [http://www.amazon.com/gp/reader/0691027951/ref=sib_vae_pg_73/103-5443484-7306247?%5Fencoding=UTF8&keywords=modulus&p=S02K&twc=4&checkSum=0BsRgLAMFNMXnqArYGxr33gLjR56d%2Bc2nsSoQnGOEKE%3D#reader-page Nahin], [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html O'Connor and Robertson], and [http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolram.com].

² [http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolram.com] credits Karl Weierstrass with introducing the notation |x| in 1841.

³ [http://www.amazon.com/gp/reader/0126227608/103-5443484-7306247?v=search-inside&keywords=absolute%20value Schechter, p 260-261].

Category:Numeration ja:絶対値 th:ค่าสัมบูรณ์

Real number

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247… The three dots indicate that there would still be more digits to come, no matter how many more might be added at the end. Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approxi

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E (mathematical constant)

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