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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 $3,00\times {10}^{8}\phantom{\rule{3pt}{0ex}}m\xb7s{}^{-1}$
As another example, the size of the HI virus is around $1,2\times {10}^{-7}$ m. This is equal to $1,2\times 0,0000001\phantom{\rule{3pt}{0ex}}m$ , which is $\mathrm{0,00000012}\phantom{\rule{3pt}{0ex}}m$ .
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] .
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.
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.
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.
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:
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.
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
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