4.3 Section summary The modulus function provides us with a measure of distance that turns the set of complex numbers into a metric space in much the same way as does the modulus function defined on R. From the point of view of analysis the importance of this is that we can talk of the closeness of two complex numbers. We can then define the limit of a sequence of complex numbers in a way which is almost identical to the definition of the limit of a real sequence. Another analogue of real analysis arises
4.2 Defining useful subsets of the complex number system, and proving the Nested Rectangles Theorem You will no doubt recall that in real analysis extensive use is made of the modulus function
. It gives us a way of measuring the “closeness” of two numbers, which we exploit in writing expressi
3.3 Section summary In this section we have seen a correspondence between complex numbers and points in the plane using Cartesian coordinates; the real part of the complex number is represented on the real axis (“horizontal”) and the imaginary part on the imaginary axis (“vertical”). We can also use polar coordinates (r,θ) in which case r, the modulus of a non-zero complex number z is positive and θ is an argument of z, defined only to within an additive intege
3.2 Relationship between complex numbers and points in the plane We have seen in Section 2.2 that the complex number system is obtained by defining arithmetic operations on the set R × R. We also know that elements of R × R can be represented as points in a plane. It seems reasonable to ask what insight can be obtained by representing complex numbers as
Keep on learning There are more than 800 courses on OpenLearn for you to 6 Curves from parameters In Section 5 we show how functions may be used to sketch curves in the plane, even when these curves are not necessarily the graphs of functions. Click the link below to open Section 5 (8 pages, 151KB). 1 Overview A fundamental concept in mathematics is that of a function. Consider, for example, the function f defined by
This is an example of a real function, because it associates with a given real number 3.3.1 Increasing by a percentage Our everyday experience of percentages includes percentage increases (like VAT at For example, £8 plus 2.1 Introduction Proportion is another way of expressing notions of part and whole. You might say that the proportion of village inhabitants who are children is a quarter, or that the proportion of fruit juice in the punch is two thirds, or that the proportion of sand in the concrete is three quarters. All these examples involve the fractions 1.6 Do, talk and record triad The do–talk–record triad (DTR) is a description of what is likely to take place in collaborative mathematics classrooms. It is concerned with observable events, and with the learner rather than the teacher, though many teaching insights flow from it. Although the order of the triad suggests that it should be followed in a particular sequence, this is not necessarily the case. Sometimes talking comes before doing or recording before talking. It also takes time for a learner to move 1.2.1 To know or to do? The so-called ‘content/process’ debate in mathematics involves discussion of the relative importance of content and process in mathematics. It originated as part of a discussion about the nature of mathematics, particularly of school mathematics, and of the purposes for which mathematics is learned. Identifying content and process in mathematics draws attention to the idea that mathematics is a human activity. As a teacher of mathematics in the UK, you are faced with a national curr 5 Approaches to problem solving You should not expect always to be able to read a problem and then just write down the answer. When you are faced with a written mathematical question or problem to solve, read it carefully. It is important that you get to grips with the question in two ways: first, that you absorb the information given; and second, that you find out what the question is really asking. Your solution will link the two. This method can be summarised by the following questions. World in transition: Migration and Trade World in transition: Managing Resources Reducing your ecological footprint Mediating Change: Culture and Climate Change Studying mammals: Food for thought Introduction This course will facilitate your exploration of a viable way of life by identifying actions that individuals can take to reduce environmental impacts while sustaining community well-being. You will be required to develop a worked example of how to go about doing this based on your own experience. In particular, you will explore how identifying a balance between reducing environmental impacts and sustaining community well-being should be an ongoing Author(s): Introduction Nature Matters considers environmental responsibility and what may matter from a caring perspective and an accountability perspective. A reading by Andrew Light reflects on four key debates in environmental ethics regarding the way in which nature is valued, and prompts the question on how such debates might inform environmental responsibility. Section 2 examines the formal processes involved in developing accountability in the context of sustainable development. The persuasiveness of t Introducing the environment: Ecology and ecosystems
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