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
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Introduction This unit is from our archive and it is an adapted extract from Open mathematics (MU120) which is no longer in presentation. If you wish to study formally at The Open University, you may wish to explore the courses we offer in this curriculum area
The topics in this unit, ratios, proportion and percentages, are concerned with dividing something into parts. Fo
4.3 Further exercises Let Examples 1. The relation ‘is equal to’ on It is reflexive since, for all x Relations We shall use the symbol Some texts use ρ, rather than 1.1 Rational numbers In OpenLearn unit M208_5 Mathematical language you met the sets Learning outcomes After studying this unit you should be able to: understand the arithmetical properties of the rational and real numbers; understand the definition of a complex number; perform arithmetical operations with complex numbers; represent complex numbers as points in the complex plane; determine the polar form of a complex number; use de Moivre's Theorem to find the nth roots o Introduction In this unit 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 prop Acknowledgements Except for third party materials and otherwise stated (see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence 1. Join the 200,000 students currently studying with The Open Unive 5.3.3 Torus with 1 hole In our last example, we consider a pentagon with two pairs of edges identified. As we saw in Section 2.3, identification of the edges produces a torus with a hole. In this case there are five vertex-neighbourhoods to fit together, as shown in Author(s): 5.3.2 Klein bottle If x lies on an edge, then each of the two points in [x] has a half-disc-like neighbourhood (see Figure 107). When we identify edges, these neighbourhoods fit together to form disc-like neighbourhoods in the Klein bottle. 5.3.1 Torus If x lies on an edge, then each of the two points in [x] has a half-disc-like neighbourhood. When we identify edges, these neighbourhoods fit together to form disc-like neighbourhoods on the torus, as Figure 105 shows. 5.3 Neighbourhoods We know that a polygon X is a surface and so each point x in X has a disc-like or half-disc-like neighbourhood. We shall show that a map f that identifies edges of a polygon to create an object Y automatically creates corresponding disc-like or half-disc-like neighbourhoods of each point y = f(x) of Y. If x is in the interior of X, there is no difficulty: the point x has a disc-like neighbourhood U 5.2.1 Proof We check that Tf satisfies conditions (T1)–(T3) for a topology.
Since (T1)–(T3) are satisfied, Tf is a topology on I(X). Thus (I(X),Tf) is a topological space. We give the topology Tf a sp 5.2 The identification topology Our aim is to show that the object that we produce when we identify some or all the edges of a polygon is a surface. Therefore, by the definition of a surface given in Section 2.5, we must show how it can be given the structure of a topological space, and that this space is Hausdorff. Furthermore, we must show that every point has 5.1 Identifying edges of a polygon In this section, we revisit the construction of surfaces by identifying edges of polygons, as described in Section 2. Recall that, if we take any polygon in the plane and identify some of its edges in pairs, then we obtain a surface. When specifying how a given pair of edges is to be identified, we choose one of the two possible re 4.6.1 Remarks We already know that the characteristic numbers are topological invariants, that is, any two homeomorphic surfaces have the same values for the characteristic numbers. Thus it is solely the converse, namely if two surfaces have the same values for the characteristic numbers then they are homeomorphic, that we have to prove. It follows from the Author(s): 4.6 The Classification Theorem In this subsection we state the Classification Theorem for surfaces, which classifies a surface in terms of its boundary number β, its orientability number ω and its Euler characteristic χ, each of which is a topological invariant – it is preserved under homeomorphisms. Let us remind ourselves of these three numbers. A surface may or may not have a boundary, and, if it does, then the boundary has finitely many disjoint pieces. The nu 4.5.2 n-fold toruses We can use a similar technique to find the Euler characteristic of a 2-fold torus. If we cut the surface into two, as shown in Figure 95, and separate the pieces, we obtain two copies of a 1-fold torus with 1 hole, each with Euler characteristic −1. 4.5.1 Surfaces with holes Using this result, we can obtain the Euler characteristic of a surface with any number of holes by successively inserting the holes one at a time. For example, since a closed disc has Euler characteristic 1, it follows that a closed disc with 1 hole has Euler characteristic 0, a disc with 2 holes has Euler characteristic −1, and so on.
Exercise 59
is reflexive, symmetric and transitive.
(known as tilde or twiddle) to represent a relation between two elements of a set.Author(s):
= {1, 2, 3, …}, the natural numbers;
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