Magnitude of a complex number

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 product of any complex number and its complex conjugate is equal to the square of the magnitude of that complex number, denoted by z p, and is always a real quantity. The magnitude or absolute value of z is the positive square root of z z. 

To analyse the response of this circuit to an alternating voltage, it turns out to be rather easier to replace the simple sinusoidal form of the voltage used above by V = V e ", where the complex number e,w = cos wt + t sin wt and i = J — 1 Any component of a circuit such as that shown can be defined as having an impedance Z, which can also be thought of as a complex number, containing both phase and magnitude information. For a resistor, Z is entirely real and simply equal to the resistance R, but for a capacitor ... 

This is a complex number with magnitude of one and argument equal to —ciiD. 

This example illustrates a very important property of complex numbers. The magnitude of the product of two complex numbers is the product of the magnitudes of each. The argument of the product of two complex numbers is the sum of arguments of each. 

Before any computational study on molecular properties can be carried out, a molecular model needs to be established. It can be based on an appropriate crystal structure or derived using any technique that can produce a valid model for a given compound, whether or not it has been prepared. Molecular mechanics is one such technique and, primarily for reasons of computational simplicity and efficiency, it is one of the most widely used technique. Quantum-mechanical modeling is far more computationally intensive and until recently has been used only rarely for metal complexes. However, the development of effective-core potentials (ECP) and density-functional-theory methods (DFT) has made the use of quantum mechanics a practical alternative. This is particularly so when the electronic structures of a small number of compounds or isomers are required or when transition states or excited states, which are not usually available in molecular mechanics, are to be investigated. However, molecular mechanics is still orders of magnitude faster than ab-initio quantum mechanics and therefore, when large numbers of... 

---