If a process is repeated in identical fashion a very large number of times, the **probability** of a given outcome is defined as the fraction of the results corresponding to that particular outcome.

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

In the OpenLearn unit on Developing modelling skills (MSXR209_3), the idea of revising a model was introduced. In this unit you will be taken through the whole modelling process in detail, from creating a first simple model, through evaluating it, to the subsequent revision of the model by changing one of the assumptions. The new aspect here is the emphasis on a revised model, which comes in Section 2. The problem that will be examined is one based on heat transfer.

This unit, the four

All written material contained within this unit originated at the Open University.

## Audio Materials

The audio extract is taken from M208 Â© Copyright 2006 The Open University.

## Unit image

http://www.flickr.com/photos/re_birf/69485963/ [Details correct as of 9th June 2008]

To practise using the techniques described in the audio, we suggest that you now try the following exercises.

## Exercise 18

Use the Binomial Theorem to prove that

Here are two short investigations involving large numbers for you to try. Please do not turn to the comments on these exercises until you have made some notes and had a go yourself.

## Exercise 13: Where did I come from?

Family tree

Now look at what happens when the power is negative. What does 10^{âˆ’3} mean? What is the result of the following calculation?

100 Ã· 100 000

What you are actually being asked to find is:

But look at the calculation again. Using the rule for the division

**1** Write the following as one number to a single power:

(a) 2

^{3}Ã— 2^{4}(b) 3

^{2}Ã— 3^{4}(c) 4

^{2}Ã— 4^{3}Ã— 4^{4<Author(s): The Open University}

**1** Find the following powers by hand, as estimates for calculator work.

(a) 10

^{7}(b) 10

^{8}(c) 3

^{4}(d) (

^{âˆ’}2)^{2Author(s): The Open University}

**1** The size of a population of micro-organisms doubles every hour. If there are two of these creatures to start with, how many will there be after five hours?

## Answer

Af

**1** The new home owners from Example 4 above want to price grass seed, as well as the turf (transport only). The best buy seems to be loose seed, which says â€˜1 kilo covers 80 m^{2}â€™. They wonder what length the side of an 80 m^{2Author(s): The Open University}

**1** Evaluate the following:

(a) 6

^{2}(b) 0.5

^{2}(c) 1.5

^{2}

## Answer<

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

evaluate the squares, cubes and other powers of positive and negative numbers with or without your calculator;

estimate square roots and calculate them using your calculator;

describe the power notation for expressing numbers;

use your calculator to find powers of numbers;

multiply and divide powers of the same number;

understand and apply negative powers, t

**1** Look at the diagram below and answer the following questions:

(a) Write down the coordinates of the points

*P*,*Q*,*R*,*S*and*T*.(b) On this diagram,

**1** Write down the coordinates of *A* and *B*.

**1** This table categorises Tom's activities for the day.