Natural numbers, whole numbers, even and odd

You learned to count before you learned any other maths. One, two, three, four. This les­son gives those num­bers their proper name. Then it adds one more num­ber to make a sec­ond fam­ily. Then it sorts every one of those num­bers into two halves.

Nat­ural num­bers

Count­ing num­bers $1, 2, 3, 4, \dots$ are called nat­ural num­bers. The set of nat­ural num­bers has a short name. That name is $N$. You write the set like this: $N = \{1, 2, 3, \dots\}$.

They are called count­ing num­bers because count­ing is what they do. Tip out a bag of mar­bles. You want to know how many there are. So you touch them one at a time. You say one, two, three. You never start at zero. Before you touch the first mar­ble, you have not counted any­thing yet. So the nat­ural num­bers start at $1$.

There is no last nat­ural num­ber. Pick any one you like. You can add $1$ to it. What you get is a count­ing num­ber too. So the list is writ­ten with dots on the end. The dots are not lazi­ness. They are the only hon­est way to write a list that never stops.

Read that line aloud like this. $N$ is the set whose mem­bers are $1$, $2$, $3$, and so on. The three dots sit inside the brack­ets. They mean carry on the same way, for ever. You need them here. You could write all day and never reach the end of $N$.

Remem­ber. The nat­ural num­bers are the count­ing num­bers. They start at $1$. They never end.

Whole num­bers

The nat­ural num­bers along with $0$ are called whole num­bers. The set of whole num­bers has the short name $W$. You write it like this: $W = \{0, 1, 2, 3, \dots\}$.

Zero needed a home some­where. Some­times you ask how many and there is noth­ing there at all. Zero is the answer you give then. You can­not count up to zero. So it is not a count­ing num­ber. But you still need to write it down. Maths keeps a sec­ond fam­ily with room for it.

Why zero is the only dif­fer­ence

Put the two rows side by side. The whole story is there. Every nat­ural num­ber is a whole num­ber too. Here is why. After its $0$, the set $W$ car­ries on with $1, 2, 3, \dots$. That is just what $N$ does. But one whole num­ber is not a nat­ural num­ber. That num­ber is $0$. It is the only dif­fer­ence between the two sets.

Think of where you meet each fam­ily. The num­ber of pages in a book, the play­ers in a team, the steps to your door: these are nat­ural num­bers. The goals scored in a match, the marks lost on a test, the sweets left in a tin: these can be $0$, so they are whole num­bers.

Remem­ber. $W$ is $N$ with $0$ put in front. Noth­ing else is added. Noth­ing is taken away.

Even num­bers

Share a num­ber out into pairs. What is left at the end is called the remain­der. Some num­bers leave noth­ing at all. Num­bers which give zero as remain­der when divided by $2$ are called even num­bers. For exam­ple $0, 2, 4, 6, \dots$ are even num­bers.

Divid­ing by $2$ asks how many pairs a num­ber makes. Take $6$ coun­ters. Pair them up. You get three pairs. Your hands are empty. Noth­ing is left over. The remain­der is $0$. So $6$ is even. An even num­ber can be shared between two peo­ple. Noth­ing is left in the mid­dle.

Every one of those divi­sions comes out exactly. Noth­ing is left after it. That is what makes each of those num­bers even.

46

$$\begin{aligned}46 &= 2 \times 23 + 0 \\ &\text{remainder } 0\end{aligned}$$

There are $23$ pairs. The remain­der is $0$. So $46$ is even. Noth­ing stands out­side the pairs.

Odd num­bers

Other num­bers leave one over. Num­bers which give $1$ as remain­der when divided by $2$ are called odd num­bers. For exam­ple $1, 3, 5, 7, \dots$ are odd num­bers.

Try to pair up an odd num­ber of coun­ters. One is always left stand­ing on its own. It has nobody to go with. That lonely counter is the remain­der $1$.

53

$$\begin{aligned}53 &= 2 \times 26 + 1 \\ &\text{remainder } 1\end{aligned}$$

There are $26$ pairs. $1$ is left over. So $53$ is odd. Now look at how the work­ing is writ­ten. You can­not put $53 \div 2$ on one side of an equals sign and $26$ on the other. Why not? Because $26$ is not the whole answer. The $1$ that is left belongs to it as well. Writ­ing $53 = 2 \times 26 + 1$ keeps both parts together. It says just what hap­pened.

Every whole num­ber is one or the other

Divide by $2$ and take away all the pairs you can. What is left is the remain­der. You could not take one more pair away. So what is left has to be smaller than $2$. So it can only be $0$ or $1$. A remain­der of $0$ means the num­ber is even. A remain­der of $1$ means it is odd. There is no third answer. So no whole num­ber can be both. And no whole num­ber can be nei­ther.

Read the table two rows at a time. $46$ and $53$ both make a heap of pairs. $46$ makes $23$ of them. $53$ makes $26$. The only dif­fer­ence is what is left over. $46$ leaves noth­ing. $53$ leaves one counter. Every row works the same way. So the left-over col­umn is the one that decides.

Notice too that $0$ sits at the top of the even col­umn. Zero coun­ters make no pairs. They leave noth­ing over. So the remain­der is $0$. That makes zero even. It is why the list of even num­bers starts with $0$.

Remem­ber. Divide by $2$ and look at what is left. Noth­ing left means even. One left means odd.

Telling even from odd at a glance

You do not have to divide every time. Just look at the last digit. It is the one at the unit place. A num­ber is even if its unit digit is $0$, $2$, $4$, $6$ or $8$. Is the unit digit any­thing else? Then the num­ber is not even. And you know from the sec­tion above what that means. A whole num­ber that is not even is odd.

Look at the tens first. Every ten is $2 \times 5$. So one ten splits into two fives with noth­ing over. Any num­ber of tens splits the same way. So the tens can never leave any­thing over. That leaves only the unit digit to leave any­thing over. That is why $74$ is even and $89$ is odd. You do not have to divide at all. This short­cut has a name. It is the Divis­i­bil­ity Rule for 2. The name means a rule for telling when one num­ber divides by another with noth­ing left over. The les­son Divis­i­bil­ity rules for 2, 3, 4 and 5 states it and works it through. It gives the rules for $3$, $4$ and $5$ as well.

Remem­ber. Look only at the last digit. A num­ber end­ing in $0, 2, 4, 6$ or $8$ is even.

Worked exam­ples with big­ger num­bers

Exam­ple 1. Is $738$ even or odd? Step 1: the unit digit is $8$. Step 2: $8$ is in the list $0, 2, 4, 6, 8$. So $738$ is even. Check by divid­ing:

$$738 = 2 \times 369 + 0$$

Exam­ple 2. Is $4\,105$ even or odd? Step 1: the unit digit is $5$. Step 2: $5$ is not in the even list. So $4\,105$ is odd. Check:

$$4105 = 2 \times 2052 + 1$$

Exam­ple 3. A class has $27$ chil­dren. Can they all stand in pairs? The unit digit of $27$ is $7$, so $27$ is odd. There will be $13$ pairs and one child left over, because $27 = 2 \times 13 + 1$.

Your turn

Say whether each num­ber is even or odd

Answer these in words

1. Write down the first six nat­ural num­bers.
2. Write down the set $W$, show­ing its first five mem­bers and the dots.
3. Which num­ber is a whole num­ber but is not a nat­ural num­ber?
4. Is every nat­ural num­ber also a whole num­ber? Say why.
5. The dots in $N = \{1, 2, 3, \dots\}$ stand for some­thing. What?
6. A num­ber leaves a remain­der of $0$ when it is divided by $2$. What is such a num­ber called?
7. A num­ber leaves a remain­der of $1$ when it is divided by $2$. What is such a num­ber called?
8. Explain why no whole num­ber can be even and odd at the same time.

Com­mon mis­takes

• Start­ing the nat­ural num­bers at $0$. Count­ing starts at $1$, so $0$ is not in $N$.
• Think­ing zero is nei­ther even nor odd. Zero leaves remain­der $0$ when divided by $2$, so it is even.
• Leav­ing out the dots when writ­ing $N$ or $W$. With­out them the list looks as if it stops.
• Look­ing at the first digit to decide even or odd. Only the unit digit mat­ters: $377$ is odd, $250$ is even.
• Writ­ing $53 \div 2 = 26$. The remain­der must be kept: $53 = 2 \times 26 + 1$.

Key terms

Nat­ural num­bers

The count­ing num­bers $1, 2, 3, \dots$, writ­ten as the set $N$.

Whole num­bers

The nat­ural num­bers together with $0$, writ­ten as the set $W$.

Set

A col­lec­tion of mem­bers, writ­ten inside curly brack­ets.

Remain­der

What is left over after shar­ing out as many equal groups as you can.

Even num­ber

A whole num­ber that leaves remain­der $0$ when divided by $2$.

Odd num­ber

A whole num­ber that leaves remain­der $1$ when divided by $2$.

Unit digit

The last digit of a num­ber, in the ones place.

Answers

Even or odd

1. $7$: odd
2. $12$: even
3. $0$: even
4. $35$: odd
5. $48$: even
6. $91$: odd
7. $100$: even
8. $63$: odd
9. $6$: even
10. $1$: odd
11. $250$: even
12. $377$: odd

Answers in words (model answers)

1. $1, 2, 3, 4, 5, 6$.
2. $W = \{0, 1, 2, 3, 4, \dots\}$.
3. $0$.
4. Yes. $W$ con­tains $0$ and then $1, 2, 3, \dots$, which is every nat­ural num­ber.
5. They mean the list car­ries on the same way for ever; there is no last nat­ural num­ber.
6. An even num­ber.
7. An odd num­ber.
8. When a whole num­ber is divided by $2$, the remain­der is either $0$ or $1$, never both. Remain­der $0$ means even and remain­der $1$ means odd, so a num­ber can­not be both.