Numbers, Graphics and Algebra

“Can you do addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?” “I don’t know,” said Alice. “I lost count.” Lewis Carroll: Through the Looking Glass

1.1 Numbers, Graphics and Algebra

The purpose of these first few chapters is to get you to look with fresh eyes at some basic mathematical ideas that you might think you already know all about. You will see how technology can help you to do that and can help to develop a deeper understanding of those ideas. You need this understanding and “feel” for what numbers and pictures can tell you about the world. You will almost certainly see “doing maths” differently from the way you saw it earlier in your life.

This chapter does not give a complete treatment for beginners in its topics. It provides a fresh look at, and reminder of, a mixture of ideas that you will find useful. You will think again about numbers and how they behave and how you can very simply represent them by letters – the idea of algebra. You will revisit simple geometry, as well as the ideas of trigonometry, describing the links between distances and angles. You will also develop ideas about the strong links between numbers, algebra and pictures – the pictures are what bring the numbers to life.

As you work through this chapter, and those which follow, remember that mathematics is a “doing”, not a “watching” activity, so make sure you keep on doing the exercises and activities as you go.

1.2 WHAT NUMBERS ARE

First you need to remind yourself of the basic terminology of numbers – bearing in mind that in mathematics some words have a precise meaning which is more specialised than in standard English.

Natural numbers are the positive whole numbers 1, 2, 3 … (0 may be included).

Integers are all whole (including negative) numbers … -2, -1, 0, 1, 2, 3 … .

Rational numbers are those which can be written as fractions in the form b where a and b are integers.

Some important and useful numbers cannot be written in rational form − e.g. the special numbers π ≈ 3.14159‥, e ≈ 2.71828‥ and ≈1.41421‥, which you will meet later, if you have not already done so. Such numbers are called irrational and occur with infinitely many non-repeating decimal places.

When the rational and irrational numbers are combined, they are called real numbers and these are what you will normally think of as decimal numbers.

It is useful to think of numbers graphically being represented by their position on a horizontal straight line, with distance away from a fixed point О governed by their value, and with negative numbers to the left and positive to the right (see Figure 1.1).

1.3 HOW NUMBERS (AND LETTERS) BEHAVE

You will be familiar with the arithmetic operations – add, subtract, multiply, divide, exponentiation (raise to a power), and the use of brackets. You should of course be used to using your calculator (graphic or otherwise) to do such things, but you will find that even with technology, if we do not all agree to use the same rules and language, then you can have serious problems. You must communicate properly, not only with other people, but also with machines.

First let us ask you a question in words – answer it without using your calculator: “What is one plus two times three?” DO IT NOW!

Some of you may get the answer 7, and others may get 9. Can you explain why two answers are possible? Check it on your calculator. You see that we need to agree the order of operations: if you do the addition first, you get 9; doing the multiplication first gives 7.

Thus, in this section, we give you the chance to remind yourself of the rules. We must all agree to use BODMAS, which some remember as Please Excuse My Dear Aunt Sally, to tell us the order in which to carry out operations:

BODMAS (Brackets/Order/Division/Multiplication/Addition/Substraction)

PEMDAS (Please Excuse My Dear Aunt Sally -Parentheses/Exponents/Multiplication/Division/Addition/Subtraction).

This means that brackets can override any other order of operation by forcing calculation of quantities in brackets first; then powers are done; then multiply and divide at the same level of priority working from left to right, and, finally, add and subtract working from left to right.

Notice if you have a graphic calculator, it probably has two different keys. One will be the unary, or leading minus, often made with three pixels, as in (-1). The other is the take-away minus sign – which often has five pixels. See Figure 1.2 to see the unary minus in action. Sort this out on your personal technology NOW.

1.4 FRACTIONS, DECIMALS AND SCIENTIFIC NOTATION

Although you probably studied fractions years ago, we find that many people have forgotten the rules, or worse still, think they remember them but do not. It is true that you can do fractional number calculations on your calculator, and get your answer as either a fraction or a decimal, and that you can deal with algebraic fractions using a CAS (Computer Algebra System) such as Derive. However, it is part of getting a feel for what is happening, to be able to do some simple manipulations with algebraic expressions involving fractions, so here is an opportunity to do just that.

First, here is a quick reminder of the rules:

To add or subtract two fractions, put them over a common denominator (bottom) so that you are adding or subtracting like with like.

Example 1.4.1 2+23=36+46=76and23−12=46−36=16

To multiply two fractions, multiply the numerators (tops) and multiply the denominators (bottoms). To divide two fractions, just turn the second one upside down and multiply!

Example 1.4.2 2×23=1×22×3=26=13and12÷23=12×32=1×32×2=34

Note you can simplify a fraction by multiplying or dividing top and bottom by the same number:

Example 1.4.3 9=13, dividing top and bottom by 3. This is commonly called

cancelling. However remember that you cannot simply add or subtract numbers on top and bottom – a quick example should convince you of that:

Example 1.4.4 +19+1=510=12 is not the same as 9. That is to say you cannot cancel the + 1 terms if they are added (or subtracted).