1 Convert each of the following fraction ratios to decimal ratios.

• (a) Author(s): The Open University

1 A local supermarket sells a popular breakfast cereal in a â€˜Large Packâ€™ and â€˜New Extra Large Packâ€™. They are both being sold at â€˜knock downâ€™ prices. The large pack contains 450 g of cereal priced at Â£1.85. The new extra large pac
Author(s): The Open University

Ratios crop up often in official statistics. The government wants the teacherâ€“pupil ratio in schools to be increased to one teacher to thirty pupils or less. The birth rate has fallen: the ratio of children to women of child bearing age has gone down. It used to be 2.4 to 1, and now it is 1.9 to 1. Predictions for the ratio of working adults to retired adults is disturbing. Predictions are, that by 2030 the ratio will be two working adults to every retired person, instead of three to one no
Author(s): The Open University

By the end of this unit you should be able to:

• work with simple ratios;

• convert between fractions, decimals and percentages;

• explain the meaning of ratio, proportion and percentage;

• find percentages of different quantities;

• calculate percentage increases and decreases;

• calculate average speeds in given units and find speeds, distances and times for travel at constant speed;

• convert units;
Author(s): The Open University

Our formal definition of an equivalence relation involves three key properties. A relation that has these three properties partitions the set on which the relation is defined, as we show later in this subsection.

In this final section we look at a method of classifying the elements of a set by sorting them into subsets. We shall require that the set is sorted into disjoint subsets â€“ so each element of the set belongs to exactly one subset. Such a classification is known as a partition of a set. In order to achieve a partition, we need to have a method which enables us to decide whether or not one element belongs to the same subset as another. We look first at the general idea of a r
Author(s): The Open University

3.5 Further exercises

Exercise 51

Evaluate the following sums and products in modular arithmetic.

• (a)Â Â 21 +26 15, Â Â Â Â 21 Ã—26 15.

• (b)Â Â 19 +
Author(s): The Open University

3.4 Modular multiplication

In the last subsection we stated that, for any integer n â‰¥ 2, the set n satisfies the same rules for addition modulo n as the real numbers satisfy for ordinary addition. When it comes to multiplication in Author(s): The Open University

2.11 Further exercises

Exercise 28

Let z1 = 2 + 3i and z2 = 1 âˆ’ 4i. Find z1 + z2, z1 âˆ’ z2, z1<
Author(s): The Open University

2.10 The complex exponential function

Consider the real exponential function f (x) = ex (that is, f (x) = exp x). We now extend the definition of this function to define a function f(z) = ez whose domain and codomain are .

We expect complex powers
Author(s): The Open University

2.9 Roots of polynomials

We begin by reminding you of what we mean by the word â€˜rootâ€™. In this unit we use this term in two different, but related, senses, as given below.

Definition

If p(z) is a polynomial, then the solutions of the polynomial equation p(z) = 0 are called the roots o
Author(s): The Open University

2.8 Polar form

You have seen that the complex number x + iy corresponds to the point (x, y) in the complex plane. This correspondence enables us to give an alternative description of complex numbers, using so-called polar form. This form is particularly useful when we discuss properties related to multiplication and division of complex numbers.

Author(s): The Open University

2.7 Arithmetical properties of complex numbers

The set of complex numbers satisfies all the properties previously given for arithmetic in . We state (but do not prove) these prope
Author(s): The Open University

2.6 Division of complex numbers

The second of the conjugateâ€“modulus properties enables us to find reciprocals of complex numbers and to divide one complex number by another, as shown in the next example. As for real numbers, we cannot find a reciprocal of zero, nor divide any complex number by zero.

Example 2

Author(s): The Open University

2.5 Modulus of a complex number

We also need the idea of the modulus of a complex number. Recall that the modulus of a real number x is defined by

For example, |7| = 7 and |âˆ’6| = 6.

In other words, |x| is the distance from the point x on the real line to the origin. We
Author(s): The Open University

2.4 Complex conjugate

Many manipulations involving complex numbers, such as division, can be simplified by using the idea of a complex conjugate, which we now introduce.

Definition

The complex conjugate Author(s): The Open University

2.3 Complex arithmetic

Arithmetical operations on complex numbers are carried out as for real numbers, except that we replace i2 by âˆ’1 wherever it occurs.

Example 1

Let z1 = 1 + 2i and z2 = 3 âˆ
Author(s): The Open University

2.1 What is a complex number?

We will now discuss complex numbers and their properties. We will show how they can be represented as points in the plane and state the Fundamental Theorem of Algebra: that any polynomial equation with complex coefficients has a solution which is a complex number. We will also define the function exp of a complex variable.

Earlier we mentioned several sets of numbers, including Author(s): The Open University

1.3 Further exercises

Exercise 4

Solve the following linear equations.

• (a)Â Â  5x + 8 = âˆ’2

• (b)Â Â
Author(s): The Open University

1.2 Real numbers

The rational and irrational numbers together make up the real numbers. The set of real numbers is denoted by . Like rationals, irrational numbers can be represented by decimals, but unlike the decimals for rational numbers, those for irrationals are neither finite nor recurring. All such infinite non-recurr
Author(s): The Open University