1.4.1 Try some yourself 1 Round a measurement of 1.059 metres: (a) to the nearest whole number of metres; (b) to two decimal places; 1.4 Rounding decimals Using a calculator often gives a long string of digits. For example, 1 ÷ 3 might give .333333333. But very often, for practical purposes, this level o 1.3 Rounding in general Numbers are often approximated to make them easier to handle, but sometimes it doesn’t help very much to round to the nearest 10 or the nearest 100 if the number is very large. For example, suppose the monthly balance of payments deficit was actually £24 695 481. Rounded to the nearest 10, it's £24 695 480; and to the nearest 100, it's £24 695 500. But £24 695 500 is still a complicated number to deal with in your head. That's why it was rounded to £25 000 000 in the newspaper 5.1: What are the CPI and RPI? The Consumer Prices Index (CPI) and the Retail Prices Index (RPI) are published each month by the UK Office for National Statistics. These are the main measures used in the UK to record changes in the level of the prices most people pay for the goods and services they buy. The RPI is intended to reflect the average spending pattern of the great majority of private households. Only two classes of private households are excluded, on the grounds that their spending patterns differ greatly from t 1.3: A statistical interlude—averages
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The main aim of this section is to discuss several ways of finding averages and to introduce you to the statistical facilities of your calculator. A single number which is typical or representative of a collection (or batch - statistical term for a set of collected data.) of numbers is commonly referred to as an average. There are several different ways of defining such a number. Two are discussed briefly in Author(s): Learning outcomes After studying this Unit you should be able to: calculate the mean, weighted mean and the median of a batch of numbers; calculate a percentage and a percentage price increase; use a weighted mean to find an average percentage price increase (given the weights); use the statistical facilities of your calculator to find the mean, the median and (given the weights) the weighted mean of a batch of numbers; calculate a perce 6.5.1 Another ‘making a lawn’ solution Suppose you have some friends who are planning to put a new lawn in their garden. The lawn is to be 12 m by 14 m and they have a choice of either laying turf or sowing grass seed. You have been asked to help them decide between the two. 3.2 Using formulas Formulas are important because they describe general relationships, rather than specific numerical ones. For example, the tins of paint formula applies to every wall. To use such a formula you need to substitute specific values for the general terms, as the following examples show. Introduction Do you want to improve your ability to subtract one number from another, especially if decimals are involved, without having to rely on a calculator? This unit will help you get to grips with subtraction and give you some practice in doing it. You can start with some practice in subtracting small numbers in your head if you want to. Then we will show you how to subtract bigger numbers on paper. Finally we look at how to subtract decimal numbers. You don’t need to complete the 15 Appendix: multiplication tables If you want to be able to do division without using a calculator, you need to know by heart what you get if you multiply any two numbers up to 10. All the possible combinations can be shown in a multiplication table (also called a times table), like the one below. 8 Dealing with remainders How do you deal with divisions where there is a remainder and there are no more digits you can carry to? In most cases you will need to express your answer as a decimal number rather than as a whole number plus a remainder. Take the example of 518 divided by 8. 3.5 Maths in archaeology 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: 4.6 Hyperbola (e > 1) A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0; the directrix d is the line with equation x = a/e.
4.2 Circles Recall that a circle in 3.1 Definition, properties and some applications In the previous section we saw how to add two vectors and how to multiply a vector by a scalar, but we did not consider how to multiply two vectors. There are two different ways in which we can multiply two vectors, known as the dot product (or scalar product) and the vector product. They are given these names because the result of the first is a scalar and the result of the second is a vector. (We shall not consider vector products in this course.) In the audio sec 2.4 Components and the arithmetic of vectors We introduce now a different method of representing vectors, which will make the manipulation of vectors much easier. Thus we shall avoid having to solve problems involving vectors by drawing the vectors and making measurements, which is very time-consuming and never very accurate. We can think of a vector as a translation, that is, as representing a movement by a certain amount in a given direction. Then we can use the Cartesian axes in the plane or in Author(s): 2.2 Multiplication by a scalar In the collection of vectors sketched in Section 2.1, although v is not equal to c, the vectors v and c are closely related: c is a vector in the same direction as v, but it is twice as long as v. Thus it is natural to write 2v for c, since we can think of a journey repre 1.8 Intersection of two planes We saw earlier that two arbitrary lines in 1.3 Parallel and perpendicular lines We often wish to know whether two lines are parallel (that is, they never meet) or perpendicular (that is, they meet at right angles). Two distinct lines, y = m1x + c1 and y = m2x + c2, are parallel if and only if they have the same gradient; that is, if and only if m1 = m2. For example, the lines y = −2x + 7 and Learning outcomes By the end of this unit you should be able to: Section 1 recognise the equation of a line in the plane; determine the point of intersection of two lines in the plane, if it exists; recognise the one-one correspondence between the set of points in three-dimensional space and the set of ordered triples of real numbers; recognise the equation of a plane in three dimensions.
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Example 18 Making a lawn
Example 8
2 is the set of points (x, y) that lie at a fixed distance, called the radius, from a fixed point, called the centre of the circle. We can use the techniques of coordinate geometry to find the equation of a circle with a given centre and radius.
2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in
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