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).
1.6 Further exercises Arrange the following numbers in increasing order: (a) 7/36, 3/20, 1/6, 7/45, 11/60; (b) 1.5 Arithmetic with real numbers We can do arithmetic with recurring decimals by first converting the decimals to fractions. However, it is not obvious how to do arithmetic with non-recurring decimals. For example, assuming that we can represent 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
1.1 Rational numbers The set of natural numbers is
1.1 Mathematics and you Many people's ideas about what mathematics actually is are based upon their early experiences at school. The first two activities aim to help you recall formative experiences from childhood. Read Introduction This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics.
In order to complete this unit you will need 4 Proofs in group theory In Section 4 we prove that some of the properties of the groups appearing earlier in the unit are, in fact, general properties shared by all groups. In particular, we prove that in any group the identity element is unique, and that each element has a unique inverse. Click 'View document' below to open Section 4 (9 pages, 237KB). Learning outcomes By the end of this unit you should be able to: explain what is meant by a symmetry of a plane figure; specify symmetries of a bounded plane figure as rotations or reflections; describe some properties of the set of symmetries of a plane figure; explain the difference between direct and indirect symmetries; use a two-line symbol to represent a symmetry; describe geometrically th 4 Two identities Section 4 introduces some important mathematical theorems. Click 'View document' below to open Section 4 (7 pages, 237KB). Learning outcomes By the end of this unit you should be able to: Section 1: Sets use set notation; determine whether two given sets are equal and whether one given set is a subset of another; find the union, intersection and difference of two given sets. Section 2: Functions determine the image of a given function; determine whether a given function is one-one 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 Learning outcomes After studying this unit you should: be able to perform basic algebraic manipulation with complex numbers; understand the geometric interpretation of complex numbers; know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations. Acknowledgements These extracts are from M208 © 2006 The Open University. All material contained within this unit originated at The Open University. 2 Real functions In Section 1 we formally define real functions and describe how they may arise when we try to solve equations. We remind you of some basic real functions and their graphs, and describe how some of the properties of these functions are featured in their graphs. Click 'View document' below to open Section 1 (12 pages, 1.8MB). 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 Acknowledgements All written material contained within this unit originated at the Open University. 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 studyi 3.2.1 Try some yourself 1 Use the method outlined in Example 9 to estimate each of the following, and then use yo 3.1.1 Try some yourself 1 Express each of the following numbers in scientific notation. (a) Light travels 9460 700 000 000 km in a year. (b) The average distance from the centre of the Earth to the centre o
Exercise 7
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which are called digits. We now remind you of the basic facts about the representation of rational numbers by decimals.
the set of integers is
and the set of rational numbers is
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Activity 1 Carl Jung's school days
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