Review of past work  (Page 7/8)

In this example, the decimal point must go after the first 2, but since the number after the 9 is a 7, $a=3,00$ .

So the number is $3,00\times {10}^m$ , where $m=8$ , because there are 8 digits left after the decimal point. So, the speed of light in scientific notation totwo decimal places is $3,00\times {10}^{8}m·s{}^{-1}$

As another example, the size of the HI virus is around $1,2\times {10}^{-7}$  m. This is equal to $1,2\times 0,0000001m$ , which is $\mathrm{0,00000012}m$ .

Real numbers

Now that we have learnt about the basics of mathematics, we can look at what real numbers are in a little more detail. The following are examples of realnumbers and it is seen that each number is written in a different way.

Depending on how the real number is written, it can be further labelled as either rational, irrational, integer or natural. A set diagram of the differentnumber types is shown in [link] .

Non-real numbers

All numbers that are not real numbers have imaginary components. We will not see imaginary numbers in this book but they come from $\sqrt{-1}$ . Since we won't be looking at numbers which are not real, if you see a number you can be sure it is a realone.

Natural numbers

The first type of numbers that are learnt about are the numbers that are used for counting. These numbers are called natural numbers and are the simplest numbers in mathematics:

Mathematicians use the symbol ${\mathbb{N}}_0$ to mean the set of all natural numbers . These are also sometimes called whole numbers . The natural numbers are a subset of the real numbers since every natural number is also a real number.

Integers

The integers are all of the natural numbers and their negatives:

Mathematicians use the symbol $\mathbb{Z}$ to mean the set of all integers . The integers are a subset of the real numbers, since every integer is a real number.

Rational numbers

The natural numbers and the integers are only able to describe quantities that are whole or complete. For example, you can have 4 apples, but what happens whenyou divide one apple into 4 equal pieces and share it among your friends? Then it is not a whole apple anymore and a different type of number is needed todescribe the apples. This type of number is known as a rational number.

A rational number is any number which can be written as:

where $a$ and $b$ are integers and $b\ne 0$ .

The following are examples of rational numbers:

Notation tip

Rational numbers are any number that can be expressed in the form $\frac{a}{b};a,b\in \mathbb{Z};b\ne 0$ which means “the set of numbers $\frac{a}{b}$ when $a$ and $b$ are integers”.

Mathematicians use the symbol $\mathbb{Q}$ to mean the set of all rational numbers . The set of rational numbers contains all numbers which can be written as terminating or repeating decimals.

Rational numbers

All integers are rational numbers with a denominator of 1.

You can add and multiply rational numbers and still get a rational number at the end, which is very useful. If we have 4 integers $a,b,c$ and $d$ , then the rules for adding and multiplying rational numbers are