Modulus of a complex number

Ey is called the modulus of a complex number, z. Unlike real numbers. 

This expression is similar to that for the modulus of a complex number. Indeed, waves can be expressed conveniently by complex numbers, and thus the structure factor can be written as such ... 

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... 

The distance from the origin 0 to the point Pisr (a2 + A1)A, which is called the norm or modulus of the complex number [a, A], The product of two complex numbers is as follows ... 

Determine the modulus and argument of a complex number, and denote its location on an Argand diagram... 

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... 

The modulus of the complex number lei represents the amplitude Cq, while the argument Arg e represents the phase angle cat. The mathematical shorthand does not imply that strain is a complex quantity. E has the fixed values... 

Modulus ma-j3-l3s -li [NL, fr. L, small measure] (1753) n, pi. (1) A modulus is a measure of a mechanical property of a material, most frequently a stiffness property. (2) The absolute value of a complex number or quantity, equal to the square root of the sum of the squares of the real and imaginary parts. (3) Modulus at 300% n The tensile stress required to elongate a specimen to three times its original length (200% elongation) divided by 2. Although other elongations are used, 300% is the one most often employed for rubbers and flexible plastics. 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

Most of the graphs we deal with are trees- that is, they do not have any self tracing loops. Choose a vertex which will be the root . In the example we consider here we shall take vertex 5 as the root (see figure l.(c)). A wave which propagates from the root along a certain branch is reflected, and this reflection can be expressed by a reflection from the vertex which is next to the root. Once we know the reflection coefficient, which because of unitarity is a complex number with unit modulus, we can construct the SB(k) matrix. In the present cases, when the valency of the root is 3, we get... 

An alternative set of terms is best introduced by noting that a complex number can be represented as in Fig. 11-15 by a point P (with coordinates x and y) or by a vector OP in a plane. Since dynamic mechanical behavior can be represented by a rotating vector in Fig. 11-13, this vector and hence the dynamic mechanical response is equivalent to a single complex quantity such as G (complex modulus)... 

The second restriction of this type of smdy is the determination of the stmcture factor. This is because this term is a complex number, whose phase has a random value in the general case. The determination of these stmcture factors is achieved based on the measurements of the intensities diffracted by each of the families of planes, but, as we showed in Chapter 1, the diffracted intensity is proportional to the square modulus of the stmcture factor. Therefore, the measurements do not make it... 

At the quantum mechanical level, collision information takes the form of a scattering amplitude /, a complex number whose square modulus f is proportional to the collision cross section a, which is a measure of the collision probability. Specifically, for an electron incident with initial momentum Mj, the cross section for a collision that leads to an electron departing with final momentum ftkj is... 

A complex number is an ordered pair of real numbers, for instance G and B. Introducing the imaginary unit j = the complex number Y = G + jB. G is the real part and can be written Y, and B the imaginary part written Y". Y or Y is called the absolute value, magnitude, or modulus, and the phase angle is cp = arctan B/G. 

Impedance is a complex number that is described by the frequency-dependent modulus, IZI, and the phase angle, 9, or, alternatively, by the real component, Z, and the imaginary component, Z". The mathematical convention for separating the real and imaginary components is to multiply the magnitude of the imaginary component by j and report the real and imaginary... 

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