In this section we show you how to *prove* inequalities of various types. We use the rules for rearranging inequalities given in Section 2, and also other rules which enable us to deduce â€˜new inequalities from oldâ€™. We met the first such rule in Author(s):

2.3 Inequalities involving modulus signs

Now we consider inequalities involving the *modulus* of a real number. Recall that if *a* , then its **modulus**, or **abso**

**The set of natural numbers is
the set of integers is
and the set of rational numbers is
Author(s): **

**Earlier you met the square function and on most calculators the square root is the second function on the same key. Look to see if this is the case for your calculator and check the calculator handbook on how to use this function. In many cases you will need to press the square root key before the number, instead of afterwards, as for the square key. This is the case on the TI-84. Check that you can find the square root of 25 and of 0.49 (you should get 5 and .7 respectively).**

**Now find **

**Section 6 contains solutions to the exercises that appear throughout sections 1-5.**

**Click 'View document' below to open the solutions (15 pages, 468KB).**

**Section 4 introduces some important mathematical theorems.**

**Click 'View document' below to open Section 4 (7 pages, 237KB).**

**3.2 Relationship between complex numbers and points in the plane **

**We have seen in Section 2.2 that the complex number system is obtained by defining arithmetic operations on the set RÂ Ã—Â R. We also know that elements of RÂ Ã—Â R can be represented as points in a plane. It seems reasonable to ask what insight can be obtained by representing complex numbers as p**

**A fundamental concept in mathematics is that of a function.**

**Consider, for example, the function f defined by
**

**This is an example of a real function, because it associates with a given real number x the real number 2x^{2} âˆ’ 1: it maps real numbers to real n**

**1** On the plan of the bathroom in Example 1, what is the width of the window and

**The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.**

All materials included in this unit are derived from content originated at the Open University.

**After studying this unit you should:**

**know some basic definitions and terminology associated with scalars and vectors and how to represent vectors in two dimensions;****understand how vectors can be represented in three (or more) dimensions and know both plane polar and Cartesian representations;****know ways to operate on and combine vectors.**

**This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.**

**This is an adapted extract from the Open University course Mathematical methods and models (MST209)**

**The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.**

**Click 'View document' to open the workbook (PDF, 1.4 MB).**

**After studying this unit you should:**

**be able to solve homogeneous second-order equations;****know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;****know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.**

**The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.**

All materials included in this unit are derived from content originated at the Open University.

**1 Modelling with Fourier series **

**The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.**

**Click 'View document' to open the workbook (PDF, 0.6 MB).**

**After studying this unit you should be able to:**

**understand how the wave and diffusion partial differential equations can be used to model certain systems;****determine appropriate simple boundary and initial conditions for such models;****find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the meth**

**During this unit you will:**

**learn some basic definitions and terminology associated with differential equations and their solutions;****be able to visualize the direction field associated with a first-order differential equation and be able to use a numerical method of solution known as***Euler's method*;**be able to use analytical methods of solution by direct integration; separation of variables; and the integrating factor method.**

**First-order differential equations **

**This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration.**

**This unit is an adapted extract from the course Mathematical methods and **

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