The number system

When you first learnt to count, every num­ber you used was a count­ing num­ber. Later you met zero, then neg­a­tive num­bers, then frac­tions and dec­i­mals, and then num­bers such as $\sqrt{2}$ and $\pi$ whose dec­i­mals never end. Each new kind of num­ber was needed for a new job. Math­e­mati­cians sort all these num­bers into groups called sets. In this les­son you will learn the six sets, see how they fit inside one another, and learn the sym­bol that says a num­ber belongs to a set.

The six sets#

A set is writ­ten inside curly brack­ets $\{\ \}$. The num­bers inside are its mem­bers. Three dots mean "and so on", the list never ends.

1. Nat­ural num­bers, N#

The nat­ural num­bers are the count­ing num­bers. They start at $1$.

$$N = \{1, 2, 3, 4, \ldots\}$$

2. Whole num­bers, W#

The whole num­bers are the nat­ural num­bers with $0$ added.

$$W = \{0, 1, 2, 3, 4, \ldots\}$$

3. Inte­gers, Z#

The inte­gers are the whole num­bers together with their neg­a­tives. This set is writ­ten $Z$ or some­times $I$.

$$Z = \{\ldots, -2, -1, 0, 1, 2, 3, \ldots\}$$

The dots at both ends show that the list runs on for ever in both direc­tions.

4. Ratio­nal num­bers, Q#

A ratio­nal num­ber is a num­ber that can be writ­ten in the form $\displaystyle \frac{p}{q}$, where $p$ and $q$ are inte­gers and $q \neq 0$. The sign $\neq$ means "is not equal to".

$$\displaystyle Q = \left\{ \frac{p}{q} : p, q \text{ are integers and } q \neq 0 \right\}$$

Exam­ples are $\displaystyle \frac{1}{2}$, $\displaystyle \frac{3}{5}$, $\displaystyle -\frac{3}{2}$, $\displaystyle -\frac{1}{4}$, $0$, $1$, $5$ and $-7$.

Why are $0$, $1$ and $5$ in the list? Because every inte­ger can be writ­ten over $1$: $\displaystyle 0 = \frac{0}{1}$, $\displaystyle 1 = \frac{1}{1}$ and $\displaystyle -7 = \frac{-7}{1}$.

The bot­tom can never be $0$, because you can­not share some­thing into $0$ parts.

5. Irra­tional num­bers, Q*#

The irra­tional num­bers are the num­bers that can­not be writ­ten in the form $\displaystyle \frac{p}{q}$, where $p$ and $q$ are inte­gers and $q \neq 0$. We write this set as $Q^{*}$.

Exam­ples are $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{6}$. The num­bers $\pi = 3.14\ldots$ and $e = 2.71\ldots$ are irra­tional too.

$$\sqrt{2} = 1.41421\ldots$$

The dec­i­mal of an irra­tional num­ber is non-ter­mi­nat­ing and non-recur­ring: it never stops and it never repeats.

Dec­i­mals tell you which kind#

Remem­ber. Ter­mi­nat­ing dec­i­mals and non-ter­mi­nat­ing recur­ring dec­i­mals are ratio­nal num­bers, for exam­ple $1.2$ and $1.\overline{3}$. Non-ter­mi­nat­ing non-recur­ring dec­i­mals are irra­tional num­bers, for exam­ple $\pi$, $e$, $\sqrt{2}$ and $\sqrt{3}$.

6. Real num­bers, R#

The ratio­nal num­bers and the irra­tional num­bers together make the real num­bers, $R$.

Exam­ples are $1$, $0$, $-1$, $-5$, $\displaystyle \frac{1}{2}$, $\sqrt{3}$, $\pi$, $e$ and $\displaystyle -\frac{5}{7}$. Every num­ber you meet in this course is a real num­ber.

Ratio­nal and irra­tional num­bers sit side by side on the same num­ber line. Every point on the line is a real num­ber.

A note on pi and 22 over 7#

$\pi$ is irra­tional, but $\displaystyle \frac{22}{7}$ is ratio­nal. They are close, but not equal: $\displaystyle \frac{22}{7} = 3.\overline{142857}$, while $\pi = 3.14159\ldots$. We say $\pi$ is approx­i­mately equal to $\displaystyle \frac{22}{7}$ and write

$$\displaystyle \pi \approx \frac{22}{7}$$

How the sets fit together#

• Every nat­ural num­ber is a whole num­ber.
• Every whole num­ber is an inte­ger.
• Every inte­ger is a ratio­nal num­ber.
• Every ratio­nal num­ber is a real num­ber.

Pic­ture steel dab­bas packed one inside the next. $N$ sits inside $W$, $W$ inside $Z$, $Z$ inside $Q$, and $Q$ inside $R$. The irra­tional num­bers are also inside $R$, but they stay out­side $Q$, because no frac­tion can equal them.

So $7$ is in every set except $Q^{*}$, while $\sqrt{2}$ is only in $Q^{*}$ and $R$.

The sym­bols for "belongs to"#

The sym­bol $\in$ means belongs to. The sym­bol $\notin$ means does not belong to.

• $\sqrt{5} \in R$ is read as "root 5 belongs to R". It is true, because $\sqrt{5}$ is a real num­ber.
• $\displaystyle \frac{2}{0} \notin R$ is read as "2 over 0 does not belong to R". Divid­ing by $0$ has no mean­ing, so $\displaystyle \frac{2}{0}$ is not a num­ber at all.

Exam­ple 1#

Is $0 \in N$ true?

No. The nat­ural num­bers start at $1$. So $0 \in N$ is false. But $0 \in W$ is true.

Exam­ple 2#

Is $-2 \in W$ true?

No. Whole num­bers are never neg­a­tive. So $-2 \in W$ is false. But $-2 \in Z$ is true.

Exam­ple 3#

Is $\displaystyle \frac{1}{2} \in Z$ true?

No. $\displaystyle \frac{1}{2}$ is not an inte­ger. It is a ratio­nal num­ber, so $\displaystyle \frac{1}{2} \in Q$ is true.

Exam­ple 4#

Is $\sqrt{2} \in Q$ true?

No. $\sqrt{2}$ is irra­tional, so $\sqrt{2} \in Q$ is false and $\sqrt{2} \in Q^{*}$ is true.

All four num­bers, $0$, $-2$, $\displaystyle \frac{1}{2}$ and $\sqrt{2}$, are real num­bers.

Exam­ple 5: sim­plify first#

Which sets does $\sqrt{16}$ belong to?

$\sqrt{16} = 4$. So it belongs to $N$, $W$, $Z$, $Q$ and $R$. Always sim­plify a num­ber before you decide its sets.

Com­mon mis­takes#

• Putting $0$ in $N$. The nat­ural num­bers start at $1$.
• Think­ing a num­ber belongs to only one set. $7$ is nat­ural, whole, an inte­ger, ratio­nal and real, all at once.
• Call­ing every square root irra­tional. $\sqrt{9} = 3$ is ratio­nal.
• Call­ing every long dec­i­mal irra­tional. $0.333\ldots$ repeats, so it is $\displaystyle \frac{1}{3}$, a ratio­nal num­ber.
• Writ­ing $\displaystyle \pi = \frac{22}{7}$. They are only approx­i­mately equal.

Prac­tice#

1. Write every set that $7$ belongs to.
2. Write every set that $0$ belongs to.
3. Write every set that $-4$ belongs to.
4. Write every set that $\displaystyle \frac{2}{5}$ belongs to.
5. Write every set that $\pi$ belongs to.
6. True or false: $0 \in N$.
7. True or false: $-5 \in W$.
8. True or false: $\displaystyle \frac{1}{4} \in Q$.
9. True or false: $\sqrt{3} \in R$.
10. Is $1.\overline{3}$ ratio­nal or irra­tional?
11. Which sets does $\displaystyle -\frac{12}{4}$ belong to?

Answers#