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.

Num­bers are sorted into groups. In this les­son, each group is called a set.

This les­son names six sets. You will use these six names all through the course.

So it is worth learn­ing them well.

The six sets

1. Nat­ural num­bers $N$

Nat­ural num­bers are the count­ing num­bers. You begin at $1$ and keep going.

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

The curly brack­ets hold every­thing in the set. The num­bers inside are called its mem­bers.

The three dots mean the list never ends.

You count real things with them. 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. You need it to say you have none of some­thing.

3. Inte­gers $Z$

Inte­gers are the whole num­bers and their neg­a­tives.

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

Dots sit at both ends now. The list runs on for ever in both direc­tions.

The chap­ter also writes this set as $I$. Both let­ters mean the same set.

Neg­a­tive num­bers let you count below zero. Money owed and cold weather both need them.

4. Ratio­nal num­bers $Q$

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

Write the top num­ber as $p$ and the bot­tom num­ber as $q$. Both are inte­gers.

The bot­tom num­ber is never $0$. You can­not share a cake into $0$ parts.

In sym­bols you write $q \neq 0$. The sign $\neq$ means is not equal to.

Every inte­ger is ratio­nal too. Any inte­ger sits over a $1$.

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

Why the answer $40$ is a ratio­nal num­ber

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

A ratio­nal num­ber in dec­i­mal form either stops or repeats.

$\displaystyle \frac{1}{2} = 0.5$ stops. $\displaystyle \frac{1}{3} = 0.333\ldots$ repeats for ever.

5. Irra­tional num­bers $Q^{*}$

Some num­bers can­not be writ­ten as a frac­tion at all. $\sqrt{2}$ is one of them.

No pair of inte­gers $p$ and $q$ will do it. Those num­bers 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. You get $\pi$.

$\pi$ is irra­tional too.

An irra­tional num­ber has a dec­i­mal that never stops. It never set­tles into a repeat­ing pat­tern either.

Worked exam­ple: ratio­nal or irra­tional?

$0.75$ stops, so it is ratio­nal: $\displaystyle 0.75 = \frac{75}{100} = \frac{3}{4}$.

$0.4444\ldots$ repeats, so it is ratio­nal: $\displaystyle 0.4444\ldots = \frac{4}{9}$.

$\sqrt{9}$ looks like a square root, but $3 \times 3 = 9$, so $\sqrt{9} = 3$, which is ratio­nal (and nat­ural too).

$\sqrt{5}$ is irra­tional, because no whole num­ber or frac­tion mul­ti­plies by itself to give $5$. Its dec­i­mal, $2.2360679\ldots$, never stops and never repeats.

6. Real num­bers $R$

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

$R$ is the biggest set here. Every num­ber you meet in this course is real.

$e = 2.71\ldots$ is a real num­ber. You will meet it later in the course.

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 boxes sit­ting 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. They stay out­side $Q$ for good, because no frac­tion can ever equal them.

So $3$ sits in every set except $Q^{*}$. And $\sqrt{2}$ 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 to write belongs to.

$5 \in R$ 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. $-3 \in N$ is false, because nat­ural num­bers start at $1$.

One num­ber and one set sit either side of the sign. So you write the sign again for each set.

The sym­bol saves you writ­ing the same words again and again.

Worked exam­ple: find every set for each num­ber

$\displaystyle -\frac{12}{4}$ looks like a frac­tion, but it sim­pli­fies to $-3$. So $\displaystyle -\frac{12}{4} \in Z$, $\displaystyle -\frac{12}{4} \in Q$ and $\displaystyle -\frac{12}{4} \in R$. It is not in $N$ or $W$, because it is below zero.

$\sqrt{16} = 4$, so $\sqrt{16}$ belongs to $N$, $W$, $Z$, $Q$ and $R$.

$2.5 = \displaystyle \frac{5}{2}$, so $2.5 \in Q$ and $2.5 \in R$, but it is not an inte­ger.

Always sim­plify a num­ber before you decide which sets it belongs to.

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. $\displaystyle \frac{1}{4} \in Q$: $T$.