1.5 Speeds Speed is the ratio of distance travelled to time taken. A runner's speed may be quoted in metres per second, miles per hour or kilometres per hour. The units are given as:    unit of distance per unit of time. When you have a distance covered (such as a mile) and a time taken (such as four minutes) the average speed is defined as 1.4.1 Try some yourself 1 Convert each of the following fraction ratios to decimal ratios. (a) 1.3.1 Try some yourself 1 A local supermarket sells a popular breakfast cereal in a ‘Large Pack’ and ‘New Extra Large Pack’. They are both being sold at ‘knock down’ prices. The large pack contains 450 g of cereal priced at £1.85. The new extra large pac 1.1 Introduction Ratios crop up often in official statistics. The government wants the teacher–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 Learning outcomes 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; Examples 1.   The relation ‘is equal to’ on It is reflexive since, for all x 4.2 Equivalence relations 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. Relations We shall use the symbol Some texts use Ï, rather than 4.1 What is a relation? 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 – 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 3.5 Further exercises Evaluate the following sums and products in modular arithmetic. (a)  21 +26 15,     21 ×26 15. (b)  19 + 3.4 Modular multiplication In the last subsection we stated that, for any integer n ≥ 2, the set 2.11 Further exercises Let z1 = 2 + 3i and z2 = 1 − 4i. Find z1 + z2, z1 − z2, z1< 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 2.9 Roots of polynomials We begin by reminding you of what we mean by the word ‘root’. In this unit we use this term in two different, but related, senses, as given below. If p(z) is a polynomial, then the solutions of the polynomial equation p(z) = 0 are called the roots o 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.
2.7 Arithmetical properties of complex numbers The set of complex numbers 2.6 Division of complex numbers The second of the conjugate–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. 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 |−6| = 6.
In other words, |x| is the distance from the point x on the real line to the origin. We 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. The complex conjugate
2.3 Complex arithmetic Arithmetical operations on complex numbers are carried out as for real numbers, except that we replace i2 by −1 wherever it occurs. Let z1 = 1 + 2i and z2 = 3 âˆ
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Exercise 51
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Exercise 28
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satisfies all the properties previously given for arithmetic in . We state (but do not prove) these prope
Example 2
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Example 1
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