## Further concepts

This chapter discusses some useful mathematical concepts including inequalities and the laws of logarithms. It helps the reader to apply the laws of indices and logarithms and solve simple inequalities. The chapter also assists in resolving a rational expression into partial fractions, constructing examples of direct proofs and indirect proofs, and using the method of proof known as ‘mathematical induction'. The chapter also demonstrates solving of a practical problem concerning a gas cylinder. From the laws of indices, the authors obtain the laws of logarithms. Logarithms are important because they provide a transformation which enables the arithmetical processes of multiplication and division to be replaced by those of addition and subtraction. An inequality involving the modulus sign which we shall encounter occasionally is the triangle inequality. A short cut is available for obtaining the partial fractions corresponding to factors which are linear (that is, polynomials of degree 1). This method is known as the cover-up rule and is simple to apply.