Number sets

Each number has an exact position on the number line (a straight line). Conversely, each position on the number line corresponds to a number.

We can divide the numbers into different groups:

The natural numbers

These are the numbers you use to count things.

The natural numbers can be further divided, e.g.:

• The even numbers ("gerade" Zahlen): $\{2,4,6,8,...\}$ All natural numbers that are divisible by 2 without a remainder ("Rest").
• The odd numbers ("ungerade Zahlen"): $\{1,3,5,7,...\}$ All natural numbers which are not even. Divided by $2$, the odd numbers have a remainder of $1$.
• The square numbers: $\{1,4,9,16,25,...\}$
• The multiples of $7$: $\{7, 14, 21, 28, ...\}$
• The divisors of $12$: $\{1,2,3,4,6,12\}$ (pronounce it as "divisors") The result of the division has to be a natural number!
• The prime numbers: $\{2,3,5,7,11, ...\}$ All natural numbers with two different divisors ($1$ and the number itself).

Note: $0$ is not counted as a natural number. However, we can use the notation

to include the $0$.

The integers

If $0$ and the "negative" natural numbers are added to the natural numbers, we get the integers:

Note that we say "negative 2" and not "minus 2".

The rational numbers

If we add to the integers all the fractions $\frac{1}{2}, -\frac{1}{3}, \frac{2}{3}, ...$, we get the rational numbers:

Any integer can of course also be represented as a fraction: $1=\frac{1}{1}, 2=\frac{2}{1}, -2=\frac{-2}{1}, ...$. The integers are therefore automatically included in the above definition.

Note that $p$ is called the numerator and $q$ the denominator, and the small horizontal line is called the fraction line. We say "$p$ over $q$". But for small denominators we also say, for example,

• $\frac{1}{2}$ "one-half"
• $\frac{1}{3}$ "one-third"
• $\frac{2}{3}$ "two-third"
• $\frac{1}{4}$ "one-fourth"
• $\frac{2}{4}$ "two-fourth"
• and so on.

Note that because of $1=\frac{1}{1}, 2=\frac{2}{1}, -2=\frac{-2}{1}, ...$ it follows that each integer is also a rational number.

The "Q" stands for "quotient".

The irrational numbers

All numbers which are not rational numbers, that is, all numbers which we cannot write as a fraction. We will not introduce a symbol for this set. It can be shown that $\pi$ (say "pai") and $\sqrt{2}$ are irrational numbers.

The real numbers

All numbers on the number line, that is, the rational and irrational numbers.

Exercise 1

Draw a Venn-diagram showing the relationship between the different number sets $\mathbb{N},\mathbb{Z},\mathbb{Q}$ and $\mathbb{R}$.

Solution

We have $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$:

The decimal representation of numbers

We can also write the numbers with the help of a decimal point. For example:

• $1=1.0=1.0$ (say "one point zero")
• $-2 = -2.0$ (say "negative two point zero")
• $\frac{5}{2}=2.5$ (say "two point 5")
• $\frac{2}{3}=0.666... = 0.\overline{6}$ (say "zero point 6 repeated")
• $\frac{1}{7}=0.142857142857142857... = 0.\overline{142857}$
• $\pi = 3.1415926535897932384626433832795028841971693993751058209749445923...$ (the three dots mean "and so on")
• $\sqrt{2}=1.4142135623730950488016887242096980785696718753769480731766797379...$

Observe that the rational numbers have a terminating or repeating decimal representation, while this does not seem to be the case for the irrational numbers like $\pi$ or $\sqrt{2}$. Indeed, this is a general rule, which we will not prove:

So, finding the decimal representation of an irrational number is impossible. Read also the following article about this topic.

In contrast, finding the decimal representation of rational numbers is straight forward using a division. Here is an example:

Example 1

Find the decimal representation of $\frac{2}{3}$:

It is also possible to find the fraction for a terminating or repeating decimal number.