1.2 Chemical elements Atoms of the same atomic number behave virtually identically in chemical reactions. They are therefore given the same chemical name and chemical symbol. For example, the atom of atomic number 6, which is shown in Figure 1, is a carbon atom, whose symbol is C. All materials are made of atoms, but there is a special class of substan
6.2 Specific difficulties Some students contend with physical difficulties in reading. Here is one:
And here is another being offered advice by a friend:
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3.1 Overview Scattering calculations using wave packets are so laborious that they are generally done numerically, using a computer. However, in many cases, scattering phenomena can be adequately treated using a procedure based on stationary states. This approach can give valuable insight into the scattering process without the need for computer simulations. Session 3 introduces the stationary-state approach to scattering. The discussion is mainly confined to one dimension, so a stationary-state sol
2.1 Overview Session 2 discusses the scattering of a particle using wave packets. We shall restrict attention to one dimension and suppose that the incident particle is initially free, described by a wave packet of the form This is a superposition of de Broglie waves, with the function
Acknowledgements The content acknowledged below is Proprietary (see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence The author of this unit is Peter Sheldon. Grateful acknowledgement is made to the following sources for permission to reproduce material
4.4 Other Wenlock Limestone fossils Among the other fossils common in the Wenlock Limestone are brachiopods (Figure 12a and b), gastropods (Figure 12c) and bryozoans (Figure 12d). You may need to reread Section 1.3 to remind yourself about various aspects of these groups. Figure 13 (the unit image) is a reconstruction of a typical scene from a Wenlock Limestone environment. See
4.1 Trilobites As we've seen, the Cambrian explosion left the seas teeming with a huge variety of animals. In the following activity you will study some of the marine life at one particular time in the Palaeozoic Era – the middle part of the Silurian Period, 430 Ma ago. You'll look in detail at some fossils which come from a deposit in the UK called the Wenlock Limestone, famous for its many beautiful fossils. The Wenlock Limestone crops out mainly around Birmingham and the borders of Wales. Figure
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
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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
