The number system

When you first learnt to count, every num­ber you used was a count­ing num­ber. Later you met zero, then neg­a­tive num­bers on a ther­mome­ter, then frac­tions and dec­i­mals, and then strange num­bers such as $\sqrt{2}$ and $\pi$. Each new kind of num­ber was needed for a new job. Math­e­mati­cians sort all these num­bers into sets, and the names of those sets are used in every chap­ter of alge­bra and geom­e­try that fol­lows.

So num­bers get sorted into groups, and each group is called a set. There are six sets to get to know here, and you'll hear their names again and again all through the course, so it really pays to learn them well now.

The six sets#

1. Nat­ural num­bers $N$#

Nat­ural num­bers are the count­ing num­bers, the very first num­bers you ever used. You begin at $1$ and keep going.

$N = \{1, 2, 3, \dots\}$

The curly brack­ets hold every­thing in the set, and the num­bers inside are called its mem­bers. The three dots mean the list never ends.

These are the num­bers you count real things with: three apples, ten steps, one door.

2. Whole num­bers $W$#

Whole num­bers are the nat­ural num­bers with $0$ added.

$W = \{0, 1, 2, 3, \dots\}$

Zero earns its place, because you need it to say you have none of some­thing. Ask any stu­dent with an empty tif­fin box after lunch!

3. Inte­gers $Z$#

Inte­gers are the whole num­bers and their neg­a­tives.

$Z = \{\dots, -2, -1, 0, 1, 2, \dots\}$

Notice the dots at both ends now: the list runs on for ever in both direc­tions. The chap­ter some­times writes this set as $I$; both let­ters mean the same set.

Neg­a­tive num­bers let you count below zero, and you need them for money owed or a cold win­ter night in the hills.

4. Ratio­nal num­bers $Q$#

A ratio­nal num­ber is one you can write as a frac­tion. $\displaystyle \frac{3}{4}$ is ratio­nal, and so is $\displaystyle \frac{-7}{2}$.

Call the top num­ber $p$ and the bot­tom num­ber $q$; both are inte­gers. The bot­tom num­ber is never $0$, because you can­not share a cake into $0$ parts.

In sym­bols you write $q \neq 0$, where the sign $\neq$ means "is not equal to".

Every inte­ger is ratio­nal too, since any inte­ger can sit over a $1$.

$\displaystyle 6 = \frac{6}{1}$, and $\displaystyle -5 = \frac{-5}{1}$.

Why the answer $40$ is a ratio­nal num­ber

$$\displaystyle \begin{aligned}10 \times 4 &= 40 \\ \text{and } 40 &= \frac{40}{1} \\ &\text{so } 40 \text{ is rational}\end{aligned}$$

Here's a handy test: a ratio­nal num­ber in dec­i­mal form either stops or repeats. $\displaystyle \frac{1}{2} = 0.5$ stops, while $\displaystyle \frac{1}{3} = 0.333\ldots$ repeats for ever.

5. Irra­tional num­bers $Q^{*}$#

Sur­pris­ingly, some num­bers can­not be writ­ten as a frac­tion at all, and $\sqrt{2}$ is one of them. No pair of inte­gers $p$ and $q$ will ever do it. Num­bers like this are the irra­tional num­bers.

$\sqrt{2}$ means the num­ber that mul­ti­plies by itself to give $2$.

$\pi$ comes from a cir­cle: divide the dis­tance round a cir­cle by the dis­tance across it and you get $\pi$. And $\pi$ is irra­tional too.

An irra­tional num­ber has a dec­i­mal that never stops, and it never set­tles into a repeat­ing pat­tern either.

6. Real num­bers $R$#

All the ratio­nal and irra­tional num­bers together form the real num­bers.

$R$ is the biggest set here, and every num­ber you meet in this course is real. For instance, $e = 2.71\ldots$ is a real num­ber that you will meet later on.

The num­ber line in the pic­ture holds every real num­ber. The ratio­nal num­bers and the irra­tional num­bers are mixed together along it: between any two ratio­nal num­bers you can always find an irra­tional one, and between any two irra­tional num­bers you can always find a ratio­nal one.

How the sets fit together#

1. Every nat­ural num­ber is a whole num­ber. $7$ is in $N$, and $7$ is in $W$.
2. Every whole num­ber is an inte­ger. $0$ is in $W$, and $0$ is in $Z$.
3. Every inte­ger is a ratio­nal num­ber. $\displaystyle -4 = \frac{-4}{1}$.
4. Every ratio­nal num­ber is a real num­ber. $\displaystyle \frac{2}{5}$ is in $Q$, and $\displaystyle \frac{2}{5}$ is in $R$.

Pic­ture a set of steel dab­bas nested one inside the next. $N$ sits inside $W$, and $W$ sits inside $Z$. Then $Z$ sits inside $Q$, and $Q$ sits inside $R$.

The irra­tional num­bers live inside $R$ too, but they stay out­side $Q$ for good, because no frac­tion can ever equal them.

So $3$ sits in every set except $Q^{*}$, while $\sqrt{2}$ sits only in $Q^{*}$ and $R$.

Remem­ber. The sets grow in this order: $N$, $W$, $Z$, $Q$, $R$. Each one holds all the sets before it.

The sym­bol $\in$#

$\in$ is a short way of writ­ing "belongs to". So $5 \in R$ is read as "$5$ belongs to $R$".

You can use it with any set: $5 \in N$ says $5$ is a nat­ural num­ber. $-3 \in Z$ is true, but $-3 \in N$ is false, because nat­ural num­bers start at $1$.

Only one num­ber and one set sit either side of the sign, so you write the sign again for each set. It's a small sym­bol that saves you writ­ing the same words over and over.

Prac­tice#

Write down every set each num­ber below belongs to.

Now write $T$ for true or $F$ for false beside each line.

Com­mon mis­takes#

• Putting 0 in $N$. The nat­ural num­bers start at $1$; $0$ first appears in $W$.
• 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. $\sqrt{9} = 3$ and $\sqrt{16} = 4$ are ratio­nal.
• Call­ing a long dec­i­mal irra­tional. $0.333\ldots$ goes on for ever but repeats, so it is $\displaystyle \frac{1}{3}$, a ratio­nal num­ber.
• Think­ing $\displaystyle \frac{22}{7}$ is $\pi$. It is only an approx­i­ma­tion. $\displaystyle \frac{22}{7}$ is ratio­nal; $\pi$ is irra­tional.
• Writ­ing two sets after one $\in$. Write $5 \in N$ and $5 \in R$ sep­a­rately.

Key terms#

Set

A col­lec­tion of num­bers, writ­ten inside curly brack­ets.

Mem­ber

A num­ber that belongs to a set.

Nat­ural num­bers $N$

The count­ing num­bers $1, 2, 3, \dots$

Whole num­bers $W$

The nat­ural num­bers together with $0$.

Inte­gers $Z$

The whole num­bers and their neg­a­tives.

Ratio­nal num­bers $Q$

Num­bers that can be writ­ten as $\displaystyle \frac{p}{q}$ with $p$, $q$ inte­gers and $q \neq 0$.

Irra­tional num­bers $Q^{*}$

Real num­bers that can­not be writ­ten as such a frac­tion; their dec­i­mals never stop or repeat.

Real num­bers $R$

All the ratio­nal and irra­tional num­bers together.

Answers#

Every set each num­ber belongs to#

1. $7$: $N$, $W$, $Z$, $Q$ and $R$.
2. $0$: $W$, $Z$, $Q$ and $R$.
3. $-4$: $Z$, $Q$ and $R$.
4. $\displaystyle \frac{2}{5}$: $Q$ and $R$.
5. $\sqrt{2}$: $Q^{*}$ and $R$.
6. $\pi$: $Q^{*}$ and $R$.

True or false#

1. $0 \in N$: $F$, nat­ural num­bers start at $1$.
2. $-5 \in W$: $F$, whole num­bers are never neg­a­tive.
3. $\pi \in R$: $T$.
4. $\displaystyle \frac{1}{4} \in Q$: $T$.