Babylonian mathematics
This unit 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 trainng far in advance of anything available in medieval Christian Europe 3000 years later.Author(s): Creator not set

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All materials included in this unit are derived from content originated at the Open University.

Author(s): The Open University

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 inputâ€“output principle to obtain a mathematical model, where appropiate;

• obtain mathematical relationships between variables, based on or linking back to the simplifying assumptions
Author(s): The Open University

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).

Definition

If aÂ >Â 0, m Author(s): The Open University

Just as we usually take for granted the basic arithmetical operations with real numbers, so we usually assume that, given any positive real number a, there is a unique positive real number bÂ =Â  such that b2Â =Â a. We now discuss the justification
Author(s): The Open University

This unit is devoted to the real numbers and their properties. In particular, we discuss inequalities, which play a crucial role in analysis.

Author(s): 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

Grateful acknowledgement is made to the following sources for permission to reproduce material in this unit:

The content ackno
Author(s): The Open University

The calculator does not make mistakes in the way that human brains tend to. Human fingers do, however, make mistakes sometimes; and the calculator may not be doing what you think you have told it to do. So correcting errors and estimating the approximate size of answers are important skills in double-checking your calculator calculations. (Just as they are for checking calculations done in your head or on paper!)

Author(s): The Open University

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 1â€‰000â€‰000â€‰000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109
Author(s): The Open University

While reading a newspaper article, I noticed some examples of how prices changed in the 10 years from 1984 to 1994. The table below shows the typical prices that you would have expected to pay in 1984 and in 1994 for a pint of milk and a Ford Fiesta motor car.

Price in 1984 Price in 1994
Pint o
Author(s): The Open University

Many people see calculators only as a way of producing answersâ€”indeed some people see them almost as a means of cheating, of short-cutting procedures that can and should be carried out in one's head or on paper. However, the calculator can also be a means of learning mathematics more effectively, something you will come to appreciate more. Many previous mathematics students have found that their graphics calculator, used with understanding and intelligence, has become a most effective aid t
Author(s): The Open University

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.

Author(s): The Open University

In Section 5 we show how functions may be used to sketch curves in the plane, even when these curves are not necessarily the graphs of functions.

Click 'View document' below to open Section 5 (8 pages, 151KB).

In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different.

Click 'View document' below to open Section 4 (5 pages, 104KB).

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).

A fundamental concept in mathematics is that of a function.

Consider, for example, the function f defined by

This is an example of a real function, because it associates with a given real number x the real number 2x2 âˆ’ 1: it maps real numbers to real n
Author(s): The Open University

By the end of this section you should be able to:

Section 1: Real functions

• understand the definition of a real function;

• use the notation for intervals of the real line;

• recognise and use the graphs of the basic functions described in the audio section;

• understand the effect on a graph of translations, scalings, rotations and reflections;

• understand how the shape o
Author(s): The Open University

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 â€˜joining up the dotsâ€™ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of t
Author(s): The Open University

Now try the quizÂ  and see if there are any areas you need to work on.

Author(s): The Open University

1 Use the method outlined in Example 9 to estimate each of the following, and then use yo
Author(s): The Open University