Review of past work

Introduction

This chapter describes some basic concepts which you have seen in earlier grades and lays the foundation for the remainder of this book. You should feel confident with the content in this chapter, before moving on with the rest of the book.

You can try out your skills on exercises in this chapter and ask your teacher for more questions just like them. You can also try to make up your own questions, solve them and try them out on your classmates to see if you get the same answers.

Practice is the only way to get good at maths!

What is a number?

A number is a way to represent quantity. Numbers are not something that you can touch or hold, because they are not physical. But you can touch three apples, three pencils, three books. You can never just touch three, you can only touch three of something. However, you do not need to see three apples in front of you to know that if you take one apple away, there will be two apples left. You can just think about it. That is your brain representing the apples in numbers and then performing arithmetic on them.

A number represents quantity because we can look at the world around us and quantify it using numbers. How many minutes? How many kilometers? Howmany apples? How much money? How much medicine? These are all questions which can only be answered using numbers to tell us “how much” of something we want to measure.

A number can be written in many different ways and it is always best to choose the most appropriate way of writing the number. For example, “a half” may be spoken aloud or written in words, but that makes mathematics very difficult and also means that only people who speak the same language as you can understand what you mean. A better way of writing “a half” is as a fraction $\frac{1}{2}$ or as a decimal number $0,5$ . It is still the same number, no matter which way you write it.

In high school, all the numbers which you will see are called real numbers and mathematicians use the symbol $\mathbb{R}$ to represent the set of all real numbers , which simply means all of the real numbers. Some of these real numbers can be written in ways that others cannot. Different types of numbers are described in detail in Section 1.12.

Sets

A set is a group of objects with a well-defined criterion for membership. For example, the criterion for belonging to a set of apples, is that the object must be an apple. The set of apples can then be divided into red apples and green apples, but they are all still apples. All the red apples form another set which is a sub-set of the set of apples. A sub-set is part of a set. All the green apples form another sub-set.

Now we come to the idea of a union , which is used to combine things. The symbol for union is $\cup$ . Here, we use it to combine two or more intervals. For example, if $x$ is a real number such that $1<x\le 3$ or $6\le x<10$ , then the set of all the possible $x$ values is:

where the $\cup$ sign means the union (or combination) of the two intervals. We use the set and interval notation and the symbols described because it is easier than having to write everything out in words.