Complex number absolute value

As we have seen, if is a root of a polynomial equation, then z is also a root. Recall that for real numbers, absolute value refers to the magnitude of a number, independent of its sign. Thus, 3.14 = — 3.14 = 3.14. We can also write - 3.14 = - 3.14. The absolute value of a complex number z, also called its magnitude or modulus, is likewise written as z - It is defined by... 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

In the real-number system a greater than h a > b) and b less than c(b < c) define an order relation. These relations have no meaning for complex numbers. The absolute value is used for ordering. Some important relations follow bl > x bl > y z Z9 z- + bgl bi - Zol Ibil — zo z- > (bl -I- lyl)/V2. Parts of the complex plane, commonly called regions or domains, are described by using inequalities. 

When expressed in polar coordinates, the quantity r is the magnitude or absolute value or modulus and (f> is the argument or phase of the complex number. It follows immediately that... 

We use common (but not universal) mathematical notation and terminology for functions. When we define a function, we indicate its domain (the objects it can accept as arguments), the target space (the kind of objects it puts out as values) and a rule for calculating the value from the argument. For example, if we wish to introduce a function f that takes a complex number to its absolute value squared, we write... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

V — 1 and x and y are real numbers. We can represent z by a point in the complex xy plane by associating the complex number z = x + iy with the point whose Cartesian coordinates are (x,y). The distance r of the point z from the origin is the absolute value z of z the angle 0 that the radius vector from the origin to z makes with the positive axis is the phase of z. We have... 

It is frequently helpful to deal with a number 2 in the complex plane, where the x axis represents the real part of the number and the y axis represents the imaginary part of the number, designated by the coefficient i = V—1. A point in the complex plane may be represented by a pair of numbers, x and y, or by absolute value (modulus) r and an angle 6. The quantities are related as follows ... 

R is the intrachain separation between neighbouring [Pt(CN)4]2- complexes, k is the absolute value of the wavevector in chain direction restricted to the first Brillouin zone (- Jt/R Periodic boundary conditions are used. 

In Table VII are recorded mean values for in a number of ethylene-metal carbonyl complexes and parent metal carbonyls as well as values for the double-bond infrared stretching frequency rc c the magnetic shielding parameter t for ethylene in those transition metal complexes for which data are available. Although with the metal carbonyl complexes, differences of geometry, oxidation state, etc., do not permit a correlation to be drawn between the absolute values of rco and for the various complexes, it is quite apparent from the tabulated data for the Mo, Mn, and Fe complexes that for a given metal. 

Introducing the modulus (= absolute value in complex number language) of the surface dilational modulus K° as... 

F is, in general, a complex number, and it expresses both the amplitude and phase of the resultant wave. Its absolute value i gives the amplitude of the resultant wave in terms of the amplitude of the wave scattered by a single electron. Like the atomic scattering factor/, Ifj is defined as a ratio of amplitudes ... 

Figure 13. Dependence of the absolute value of the open-circuit photopotential [(pp l couple equilibrium potential WSe trode (solid line, p-type dashed line, n-type) in acetonitrile solutions. Redox couple (figures in parentheses are charge numbers of the ox and red components) (1) anthracene (0/-1) (2) phtalonitrile (0/ - 1) (3) nitrobenzene (0/ - 1) (4) 2,2 -bipyridyl complex of ruthenium (+2/+1) (5) azobenzene (0/ -1) (6)...

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