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 2.3 Section summary In this section we have seen that the complex number system is the set R × R together with the operations + and × defined by From this, one can justify the performance of ordinary algebraic operations on expressions of the form a + ib 2.1 Introduction In this section we shall define the complex number system as the set R × R (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define the real and imaginary parts of a complex number and compare the properties of the complex number system with those of the real number system, particularly from the point of view of analysis. 4 New graphs from old In Section 3 we consider how to sketch the graphs of more complicated functions, sometimes involving trigonometric functions. We look at graphs which are sums, quotients and composites of different functions, and at those which are defined by a different rule for different values of x. Click 'View document' below to open Section 3 (7 pages, 133KB). 3.2 Using scientific notation Scientific notation can be very useful when estimating the answers to calculations involving very large and/or small decimal numbers. A lottery winner won £7851 000. He put the money straight into a deposit account which earns 7.5% interest per annum (i.e. each year). If he wanted to 3.1 Expressing numbers in scientific notation Earlier you looked at place values for numbers, and why they were called powers of ten.
Activity 1 Carl Jung's school days
Example 9
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