4.4 Obesity and cardiovascular diseases Obesity and being overweight are well-known as risk factors for cardiovascular diseases. Carrying excess body fat predisposes individuals to developing elevated blood cholesterol and diabetes. You will begin to appreciate that many of the modifiable risk factors for cardiovascular diseases are interlinked. This means that influencing one, such as reducing the amount of stored lipids in the body, may have a positive effect in reducing the risk associated with high blood cholesterol levels and
Introduction You may be studying this unit because you – or a member of your family or a friend – have been personally affected by cardiovascular diseases in some way. You may be professionally involved in looking after people with one of these diseases. Perhaps you are interested in health issues in general. Whatever your motivation or underlying reasons for studying this unit, you will gain valuable insights into the extent of cardiovascular diseases and their treatment in the early twenty-first cen
2.12 How likely are particular results? In real experiments, as opposed to hypothetical ones, it is very rare that scientists make a sufficiently large number of measurements to obtain a smooth continuous distribution like that shown in Figure 7d. However, it is often convenient to assume a particular mathematical form for typically distributed measurements, and the form that is usually
2.11 Using a calculator for statistical calculations
Table 3 shows all the values for each step in the process of calculating a standard deviation, so that you can see what the operations encapsulated by Equation 7 actually entail, but you will probably be relieved to hear that it is not usually necessary to carry out such detailed calculations. Scientific and graphics calculators (or computer sp
2.8 Descriptive statistics Scientists collect many different types of information, but sets of data may be very loosely classified into two different types. In the first type, so-called ‘repeated measurement’, an individual quantity is measured a number of times. An astronomer wanting to determine the light output of a star would take many measurements on a number of different nights to even out the effects of the various possible fluctuations in the atmosphere that are a cause of stars ‘twinkling’. In the seco
2.6 Combining probabilities The probabilities described in Section 2.3 and Section 2.4 related to the outcomes of a single process, such as repeatedly tossing one coin. Now suppose you were to toss three separate coins simultaneously. What is the prob
1.4 How precise are the measurements? Scientists are always trying to get better and more reliable data. One way of getting a more precise measurement might be to switch to an instrument with a more finely divided scale. Figure 4 shows parts of two thermometers placed side by side to record the air temperature in a room. Introduction Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply ‘joining up the dots’ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of t Introduction When we try to use ordinary language to explore mathematics, the words involved may not have a precise meaning, or may have more than one meaning. Many words have meanings that evolve as people adapt their understanding of them to accord with new experiences and new ideas. At any given time, one person's interpretation of language may differ from another person's interpretation, and this can lead to misunderstandings and confusion. In mathematics we try to avoid these difficulties by ex Numbers Introduction This course looks at Babylonian mathematics. You will learn how a series of discoveries have enabled historians to decipher stone tablets and study the various techniques the Babylonians used for problem-solving and teaching. The Babylonian problem-solving skills have been described as remarkable and scribes of the time received a training far in advance of anything available in medieval Christian Europe 3000 years later. This OpenLearn course provides a sample of Level 2 study in Introduction This course provides an overview of John Napier and his work on logarithms. It discusses his approach to this lasting invention and looks at the key players who worked with him, including Briggs, Wright and Kepler. This OpenLearn course provides a sample of Level 2 study in Mathematics Learning outcomes After studying this unit you should be able to: create simple models, given a clear statement of the problem; write down the simplifying assumptions that underpin a model; identify the key variables and the parameters of a model; apply the input–output principle to obtain a mathematical model, where appropriate; obtain mathematical relationships between variables, based on or linking back to the simplifying assumption Introduction This unit explores a real-world system – the Great Lakes – where mathematical modelling has been used to understand what is happening and to predict what will happen if changes are made. The system concerned is extremely complex but, by keeping things as simple as possible, sufficient information will be extracted to allow a mathematical model of the system to be obtained. This unit is an adapted extract from the course Author(s): 5.3 Powers Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent). If a > 0, m 5.1 Arithmetic with real numbers At the end of Section 1, we discussed the decimals
4.4 Further exercises In this exercise, take
4.2 Least upper and greatest lower bounds We have seen that the set [0, 2) has no maximum element. However, [0, 2) has many upper bounds, for example, 2, 3, 3.5 and 157.1. Among all these upper bounds, the number 2 is the least upper bound because any number less than 2 is not an upper bound of [0, 2).
2.3 Inequalities involving modulus signs Now we consider inequalities involving the modulus of a real number. Recall that if a 1.1 Rational numbers The set of natural numbers is
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This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound Property. First published on Wed, 2
Definition
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and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least Upper Bound Property of Author(s):
Exercise 29
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, then its modulus, or abso
the set of integers is
and the set of rational numbers is
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