1 What are scattering and tunnelling?

The phenomenon of **scattering** was an important topic in physics long before the development of wave mechanics. In its most general sense, scattering is a process in which incident particles (or waves) are affected by interaction with some kind of target, quite possibly another particle (Figure 1). The interac

6.2 An outline of vertebrate evolution

Let's now place the early evolution of tetrapods in perspective by taking an overview of the whole of vertebrate evolution.

Corals are especially abundant in the Wenlock Limestone.

## SAQ 12

According to Author(s):

The amount of lipid stored within the body – an individual's adiposity – can be indirectly measured. Body shape (e.g. ‘apple’ or ‘pear’ shapes), waist-to-hip ratio, waist circumference and body mass index (BMI) are all used to classify obesity and being overweight, although BMI is the most common. Everyone should aim to have a body weight within the normal range for their height. Slightly different ranges apply between populations due to different body shapes. It is worth noting t

It is no accident that the examples used to illustrate the statistics for repeated measurements of individual quantities were drawn from chemistry and physics. Experiments involving repeated measurements of some quantity are typical of the physical sciences. There are, however, many other types of scientific work in which a typical procedure is to collect data by measuring or counting the members of a sub-set of things which form part of a larger group, and

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

2.10.1 Mean and standard deviation for repeated measurements

In everyday terms, everybody is familiar with the word ‘average’, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation exemplified by Table 2, the sort to use is the **mean**
(or more strictly the ‘arithmetic mean’) For a set of measurements, this is de

Scientists are always concerned with the reliability and precision of their data, and this is the prime reason for them to repeat measurements many times. Consider the photograph shown in Figure 6, which is the result of X-ray diffraction (see Box 5). To determine the atomic structure of the substance that produced this pattern, it would be necessary to measure the diameters of the fuzzy rings. At one time, this would probably have been done with an instrument called a travelling microscope;

Statistical information is a familiar aspect of modern life, which features routinely in, for example, news reports, sports commentaries and advertising. Scientists who have collected large amounts of data by either counting or measuring quantities also rely on statistical techniques to help them make sense of these data. Suppose you had information collected from, say, three thousand patients, all with the same medical condition but undergoing a variety of treatments. First you would need te

In this course we look at some different systems of numbers, and the rules for combining numbers in these systems. For each system we consider the question of which elements have additive and/or multiplicative inverses in the system. We look at solving certain equations in the system, such as linear, quadratic and other polynomial equations.

In Section 1 we start by revising the notation used for the **rational numbers** and the **real numbers**, and we list their arithmetical pr

For many centuries, ancient Egypt was seen as the source of wisdom and knowledge, about mathematics as well as other things. There was a long classical Greek tradition to this effect, and in later centuries the indecipherability of the hieroglyphs did nothing to dispel this belief. But since the early nineteenth century, when the deciphering of the Rosetta Stone by Young and Champollion enabled rapid progress to be made in translating extant Egyptian texts, the picture has changed to reveal a

This course is devoted solely to complex numbers.

In Section 1, we define complex numbers and show you how to manipulate them, stressing the similarities with the manipulation of real numbers.

Section 2 is devoted to the geometric representation of complex numbers. You will find that this is very useful in understanding the arithmetic properties introduced in Section 1.

In Section 3 we discuss methods of finding *n*th roots of complex numbers and the solutions of simple

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

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 0.4 MB).

4.1 Upper bounds and lower bounds

Any finite set of real numbers has a greatest element (and a least element), but this property does not necessarily hold for infinite sets. For example, neither of the sets = {1, 2, 3, … } and [0, 2) has a greatest element.

However, the set [0, 2) is *bounded above* by 2, since all points of [0, 2)

1.4 Real numbers and their properties

Together, the rational numbers (recurring decimals) and irrational numbers (non-recurring decimals) form the set of **real numbers**, denoted by .

As with rational numbers, we can determine which of two real numbers is greater by comparing their decimals and noticing the first pair of corresponding digits

1.2 Decimal representation of rational numbers

The decimal system enables us to represent all the natural numbers using only the ten integers
which are called *digits*. We now remind you of the basic facts about the representation of *rational* numbers by decimals.

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This unit has introduced you to some aspects of using a scientific or graphics calculator. However, in many ways, it has only scratched the surface. Hopefully your calculator will be your friend throughout your study of mathematics and beyond. Like any friend, you will get to know it better and appreciate its advantages as you become more familiar with it. Don't expect to know everything at the beginning. You may find the instruction booklet, or other help facility, a bit hard going to begin

Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase ‘garbage in, garbage out’ applies just as much to calculators as to computers. Your calculator is just that – a calculator!<