1.4 Real numbers and their properties
Together, the rational numbers (recurring decimals) and irrational numbers (nonrecurring 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.
The set of natural numbers is
the set of integers is
and the set of rational numbers is
Author(s):
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.
Activity 1 Carl Jung's school days
Read
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 TI83 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
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).
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 twoline symbol to represent a symmetry;
describe geometrically th
Section 4 introduces some important mathematical theorems.
Click 'View document' below to open Section 4 (7 pages, 237KB).
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 oneone
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
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.
Audio Materials
These extracts are from M208 © 2006 The Open University.
All material contained within this unit originated at The Open University.
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).
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
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 AttributionNonCommercialShareAlike 2.0 Licence
1. Join the 200,000 students currently studyi
1 Use the method outlined in Example 9 to estimate each of the following, and then use yo
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
3.1 Expressing numbers in scientific notation
Earlier you looked at place values for numbers, and why they were called powers of ten.
Place value  10 000  1000  100  10  1  Author(s): 1 On the plan of the bathroom in Example 1, what is the width of the window and 1.6.3 Mailing lists and newsgroups Mailing or discussion lists are emailbased discussion groups. When you send an email to a mailing list address, it is sent automatically to all the other members of the list. The majority of academicrelated mailing lists in the UK are maintained by Jiscmail. You will find details of joining these mailing lists on the Jiscmail website. Mailing lists are useful for getting in touch with likeminded colleagues. They are also handy for keeping up to date with current thinking and research Pages
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