The Imaginary Number

As we saw above, the solutions to algebraic equations do not always yield real numbers. For example, the solution of the equation x 1 = 0 yields the apparently meaningless result  

A The formula given in equation (2.2) for the roots of a quadratic equation yields  

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This equation is exact and constitutes an iterative equation equivalent to the time-dependent Schrodinger equation [185,186]. The iterative process itself does not involve the imaginary number i therefore, if h(f) and )( — x) were the real parts of the wavepacket, then (f + x) would also be real and would be the real part of the exact wavepacket at time (f + x). Thus, if )( ) is complex, we can use Eq. (4.68) to propagate the real part of 4>(f) forward in time without reference to the imaginary part. 

In order to illuminate both the phase problem and its solution, I will represent structure factors as vectors on a two-dimensional plane of complex numbers of the form a + ib, where i is the imaginary number (—1)1/2. This allows me to show geometrically how to compute phases. I will begin by introducing complex numbers and their representation as points having coordinates (a,b) on the complex plane. Then I will show how to represent structure factors as vectors on the same plane. Because we will now start thinking of the structure factor as a vector, I will hereafter write it in boldface (FM,Z) instead of the italics used for simple variables and functions. Finally, I will use the vector representation of structure factors to explain a few common methods of obtaining phases. 

The horizontal axis in the figure represents the real-number line. Any real number a is a point on this line, which stretches from a = - to a = + . The vertical axis is the imaginary-number line, on which lie all imaginary numbers... 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

Recall that, when the imaginary number i = V—1 is used, an exponential ew becomes... 

The imaginary number j = is file solution to tibe algebraic equation... 

As can be found in any field, the notation used in the impedance spectroscopy literature is inconsistent. In treatments of diffusion impedance, for example, the symbol 6 is used to denote the dimensionless oscillating concentration variable whereas, the symbol 9 used in kinetic studies denotes the fractional surface coverage by a reaction intermediate. Compromises were necessary to create a consistent notation for this book. For example, the dimensionless oscillating concentration variable was given the symbol 6, and 7 was used to denote the the fractional surface coverage by a reaction intermediate. As discussed in Section 1.2.3, the book deviates from the lUPAC convention for the notation used to denote the imaginary number and the real and imaginary parts of impedance. 

The mechanics of fracture along bimaterial interfaces have been studied extensively. Excellent reviews have been published [18]. The stress and deformation field near the tip of a crack lying along a bimaterial interface can be uniquely characterized by means of the complex stress intensity factor K = Kl + iK2. K and K2 have the dimension (Pa m112 " ) and are functions of the sample geometry, applied loading and material properties, i = is the imaginary number and is a dimensionless material constant defined below. 

This chapter extends the familiar number system to include complex numbers containing the imaginary number i. By the end of this chapter, you should be able to ... 

It may seem odd to think of the exponential function, z = e , as periodic because it is clearly not so when the exponent is real. However, the presence of the imaginary number i in the exponent allows us to define a modulus and argument as 1 and 6, respectively. If we represent the values of the function on an Argand diagram, we see that they lie on a circle of radius, r= 1, in the complex plane (see Figure 2.4). Different values of 6 then define the location of complex numbers of modulus unity on the circumference of the circle. We can also see that the function is periodic, with period 2% ... 

An introduction of the imaginary number i = Vm as a means to finding all roots of polynomial equations. 

The frequency response characteristics of a process element or a group of elements can be computed readily from the corresponding transfer function merely by substituting ju for s, where j is the imaginary number, /— 1, and is the angular velocity. Thus the frequency response characteristics of a simple first order lag are given by... 

Let us mention that biquaternionic solutions of the central potential problem had been already achieved by Sommerfeld (see [54]), but in the frame of the complex formalism, the Dirac spinor being expressed by means of the Dirac matrices. Nevertheless, though, in particular, some ambiguities related to the role of the imaginary number t/—1 in the complex formalism had been removed in this work, the use of the pure real formalism of Hestenes brings noticeable simplifications and above all the entire geometrical clarification of the theory of the electron. 

The links of the Pauli and the Dirac spinors with Cl+(E3) and Cl+(M) are less evident. These spinors are eachone the decompostion into two and four complex numbers of an element of Cl+(E3) and Cl+(M). These numbers are written in the form a + ib in the complex formalism, but the imaginary number i = a/—T is in fact 103 = 7271 (see Sect. 3.1.3) which becomes real by the above identifications. The presence of 7271, that is, the bivector of M, e2ei = e2 Aei (whose square in Cl(M) is equal to —1) in place of the imaginary number i is closely related to the existency of the spin of the electron. 

The following operations of identification are complicated by the ambiguities of the complex formalism in which the imaginary number y/ 1 may correspond to two different real objects. 

What does that mean That means that the imaginary number / T is a symbol which hides a geometrical object. And, this object may be different following the use of this symbol different bivectors, but furthermore, objects of different geometrical nature as shown in (2.15). 

Note that 1/j = —j. We see that the imaginary number j is useful, because in effect, it also carries with it information about the 90° phase shift existing between the voltage and current in reactive components. So, the previous equations for impedance indicate that in an inductor, the current lags behind the voltage by 90°, whereas in a capacitor the current leads by the same amount. Note that resistance only has a real part, and it will therefore always be aligned with the x-axis of the complex plane (i.e. zero phase angle). 

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