Often it is the case that an equation involving complex numbers us satisfied by a whole range of points, often a continuum, called a locus. For exampleis satisfied by all points of the form We can often find a Cartesian equation for the set of points satisfying an equation with complex variables by substitutingand manipulating. […]

## Proof of Formulae Used to Solve Differential Equations Numerically

The Taylor series for a function expanded about a pointis We can derive two very useful series, evaluated atand (1) (2) (1)-(2) gives (3) (1)+(2) gives(4) (3) and (4) may be used to solve second order differential equations numerically at any point, given any two boundary conditionsandWrite and Then (3) becomes and (4) becomes We […]

## Abelian and Non Abelian Groups

If a groupis abelian thenfor all This has several consequences. All cyclic groups are abelian since a cyclic group is generated by a single element, somay be writtenwithThe group is abelian since ifthenAn abelian group may always be constructed in this way. The group table is symmetric about the main diagonal since the element in […]

## Complex Fractions, Argand Diagrams, Magnitudes, Arguments and Products of Complex Numbers and Polar Forms

Complex Fractions Typically we have to express a complex fractionin the formWe do this by multiplying top and bottom by the complex conjugate of the denominator, remembering thatThe complex conjugate of Example: Expressin the form Argand Diagrams We may also have to plot complex numbers on an Argand diagram. This is a normal set of […]

## Commutativity of Cyclic Groups

A cyclic group is a group generated by a single element. All the elements of the group are formed by repeated composition of some elementwith itself, including the identity element. If the grouphas orderwe may write This necessarily means that all elements of cyclic groups commute and and that cyclic groups abelian, since ifandfor someso […]

## Proof of Lagrange’s Theorem

Lagranges theorems says that the order of any subgroup of a finite groupdivides the order of (the order of a group is the number of distinct elements of). The idea of the proof is to take any subgroupand form all the subsetsof The subsetsare called the left cosets ofin(we could as well use the right […]

## Isomorphic Groups

Two groupsandare isomorphic if they have the same group structure, so the the elements of one group can be put in a one to one correspondence with the elements of the other. All the relationships between the elements of each group are preserved by the correspondence. If we can label the mapping of one group […]

## Classifying Groups of Order 4

Two groups are essentially the same group if they have the same group structure. If two groups have the same group structure which can find a mapping between the groups, called an isomorphism, which preserver the structure of the groups, including the order of each element, the relationships between the elements, inverses and identities. Classifying […]

## Cyclic Groups

Sometimes it is the case that a group is generated by repeatedly composing a single elementwith itself. In this case, the group is said to be a cyclic group generated by the element Every rotation group of orderconsisting of the rotations of the regular polygon withsides. The elementmay be taken to be the element that […]

## Using de Moivre’s Theorem to Find Roots of Polynomial Equations

De Moivre‘s theorem states that forwhere We can obtain polynomial expressions forandfor anyusing de Moivre’s theorem. For example We can use de Moivre’s theorem to find roots of some polynomial equations. Suppose we have the equationThis has the same coefficients on the right as the third and fourth equations above. We can setthen the equation […]