Introduction When we try to use ordinary language to explore mathematics, the words involved may not have a precise meaning, or may have more than one meaning. Many words have meanings that evolve as people adapt their understanding of them to accord with new experiences and new ideas. At any given time, one person's interpretation of language may differ from another person's interpretation, and this can lead to misunderstandings and confusion. In mathematics we try to avoid these difficulties by ex
Numbers
This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound Property. First published on Wed, 2
Introduction This course looks at Babylonian mathematics. You will learn how a series of discoveries have enabled historians to decipher stone tablets and study the various techniques the Babylonians used for problem-solving and teaching. The Babylonian problem-solving skills have been described as remarkable and scribes of the time received a training far in advance of anything available in medieval Christian Europe 3000 years later. This OpenLearn course provides a sample of Level 2 study in
Introduction This course provides an overview of John Napier and his work on logarithms. It discusses his approach to this lasting invention and looks at the key players who worked with him, including Briggs, Wright and Kepler. This OpenLearn course provides a sample of Level 2 study in Mathematics
Learning outcomes After studying this unit you should be able to: create simple models, given a clear statement of the problem; write down the simplifying assumptions that underpin a model; identify the key variables and the parameters of a model; apply the input–output principle to obtain a mathematical model, where appropriate; obtain mathematical relationships between variables, based on or linking back to the simplifying assumption
Introduction This unit explores a real-world system – the Great Lakes – where mathematical modelling has been used to understand what is happening and to predict what will happen if changes are made. The system concerned is extremely complex but, by keeping things as simple as possible, sufficient information will be extracted to allow a mathematical model of the system to be obtained. This unit is an adapted extract from the course Author(s):
5.3 Powers Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent). If a > 0, m 5.1 Arithmetic with real numbers At the end of Section 1, we discussed the decimals
4.4 Further exercises In this exercise, take
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).
2.3 Inequalities involving modulus signs Now we consider inequalities involving the modulus of a real number. Recall that if a 1.1 Rational numbers The set of natural numbers is
Learning outcomes By the end of this unit you should be able to: Section 1 – Real numbers explain the relationship between rational numbers and recurring decimals; explain the term irrational number and describe how such a number can be represented on a number line; find a rational and an irrational number between any two distinct real numbers; Section 2 – Inequalities solve inequalities by re 8.2.2 The screen You can see the calculations that you have entered as well as the answers. This means you can easily check whether you have made any mistakes. 6.2 Getting the feel of big and small numbers Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 1099 are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big? First express 1 000 000 000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109 6.1 Scientific notation Understanding how your calculator displays and handles very large and very small numbers is important if you are to interpret the results of calculations correctly. This section focuses on a way of representing numbers known as scientific notation. Before you start put your calculator into the float mode, so it will display up to about 10 digits and return to the home screen ready to do some calculations. What answer would you expect if you square 20 million? How man 2.1 The four rules of arithmetic You are now going to use the four operation keys (on the bottom right-hand side of the TI-84 keyboard): 3 Aims The aim of this section is to help you to think about how you study mathematics and consider ways in which you can make your study more effective. Pressing onwards Work through Sections 1.6 and 1.7 of the Calculator Book, using the method suggested above of glancing ahead-pressing on-glancing back, if you find it useful. A num Does it make sense?
Definition
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and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least Upper Bound Property of Author(s):
Exercise 29
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, then its modulus, or abso
the set of integers is
and the set of rational numbers is
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Activity 15
Example 3
