In this subsection we consider, briefly, some problems that can arise with certain ways of drawing bar charts and pie charts.

Figure 5 shows what is essentially the same bar chart as Author(s): The Open University

In several different parts of the world, footprints from prehistoric human civilisations have been found preserved in either sand or volcanic ash. From these tracks it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running by using the following three formulas:

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Now try the quiz, and see if there are any areas you need to work on.

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Discount can be calculated in the same way as an increase by a percentage. For example, Â£8 with 15% discount means you actually pay

Â Â Â£8 less (15% of Â£8)

Â Â 15% of 8 = Ã— 8 = Author(s): The Open University

1 Which is greater, 1.2 minutes or 70 seconds?

There are at least three ways of answering this:

• (a) 70 seconds is
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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:

Â Â Â 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

• Â
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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
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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
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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;
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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
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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 +
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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<
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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
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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
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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.

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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
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