Natural numbers, whole numbers, even and odd

You learned to count before you learned any other maths: one, two, three, four. Now those num­bers get their proper name. Then we add one more num­ber to make a sec­ond fam­ily. Last of all, we sort every num­ber into two groups, even and odd.

Nat­ural num­bers#

The count­ing num­bers $1, 2, 3, 4, \dots$ are called nat­ural num­bers. The set of nat­ural num­bers is writ­ten $N$:

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

They are called count­ing num­bers because count­ing is what they do. Tip out a bag of mar­bles and touch them one at a time: one, two, three. You never start at zero. So the nat­ural num­bers start at 1.

There is no last nat­ural num­ber. Pick any one you like and add 1 to it. You get another count­ing num­ber. That is why the list ends with three dots. The dots mean "carry on the same way for ever".

Whole num­bers#

The nat­ural num­bers together with 0 are called whole num­bers. The set of whole num­bers is writ­ten $W$:

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

Zero is the answer when there is noth­ing at all: the sweets left in an empty tin, or the goals in a match that nobody scored. You can­not count up to zero, so it is not a nat­ural num­ber. But it is a whole num­ber.

Every nat­ural num­ber is also a whole num­ber. The only whole num­ber that is not a nat­ural num­ber is 0.

Remem­ber. $W$ is $N$ with 0 put in front. Noth­ing else changes.

Even num­bers#

Num­bers which give zero as remain­der when divided by 2 are called even num­bers. For exam­ple: $0, 2, 4, 6, 8, 10, 12, \dots$

Divid­ing by 2 asks how many pairs a num­ber makes. Take 6 coun­ters and pair them up. You get three pairs and noth­ing is left over. The remain­der is 0, so 6 is even.

Zero is even too. Zero coun­ters make no pairs and leave noth­ing over, so the remain­der is 0.

Odd num­bers#

Num­bers which give 1 as remain­der when divided by 2 are called odd num­bers. For exam­ple: $1, 3, 5, 7, 9, 11, 13, \dots$

Exam­ple 1#

Is 5 even or odd?

Divide 5 by 2. Two twos are 4, and $5 - 4 = 1$ is left over.

$$\begin{array}{rl} 2\,\big) & 5\,\big(\,2 \\ - & \underline{4} \\ & 1 \end{array}$$

The remain­der is 1, so 5 is odd. With coun­ters: 5 coun­ters make two pairs, and one counter is left stand­ing on its own.

The pic­ture shows the dif­fer­ence between even and odd. Six coun­ters pair up exactly. Seven coun­ters leave one on its own.

Exam­ple 2#

Is 53 even or odd?

Two times 26 is 52, and one more makes 53. So 53 makes 26 pairs with 1 left over.

$$53 = 2 \times 26 + 1$$

The remain­der is 1, so 53 is odd.

Every whole num­ber is one or the other#

When you divide by 2, the remain­der must be smaller than 2. So it can only be 0 or 1. A remain­der of 0 means even. A remain­der of 1 means odd. There is no third choice. So every whole num­ber is either even or odd, and never both.

Even or odd at a glance#

You do not have to divide every time. Look at the last digit, the one in the units place. If it is 0, 2, 4, 6 or 8, the num­ber is even. If it is 1, 3, 5, 7 or 9, the num­ber is odd.

Why does the last digit decide? Every ten is $2 \times 5$, so the tens always pair off with noth­ing left. Only the units can leave one over. So 74 is even and 89 is odd, with­out any divid­ing.

Exam­ple 3#

Is 738 even or odd?

The units digit is 8, so 738 is even. Check: $738 = 2 \times 369$, with noth­ing left over.

Exam­ple 4#

A class has 27 chil­dren. Can they all stand in pairs?

The units digit of 27 is 7, so 27 is odd. There will be 13 pairs and one child left over, because $27 = 2 \times 13 + 1$.

Com­mon mis­takes#

• Start­ing the nat­ural num­bers at 0. Count­ing starts at 1, so 0 is not a nat­ural num­ber.
• Think­ing zero is nei­ther even nor odd. Zero leaves remain­der 0 when divided by 2, so it is even.
• Look­ing at the first digit. Only the units digit mat­ters: 377 is odd and 250 is even.
• Leav­ing out the dots when writ­ing $N$ or $W$. With­out them the list looks as if it stops.

Key terms#

• Nat­ural num­bers: the count­ing num­bers $1, 2, 3, \dots$, writ­ten $N$.
• Whole num­bers: the nat­ural num­bers together with 0, writ­ten $W$.
• Remain­der: what is left over after divid­ing.
• Even num­ber: a num­ber that leaves remain­der 0 when divided by 2.
• Odd num­ber: a num­ber that leaves remain­der 1 when divided by 2.

Prac­tice#

1. Write the first six nat­ural num­bers.
2. Write the first six whole num­bers.
3. Which whole num­ber is not a nat­ural num­ber?
4. Is 12 even or odd?
5. Is 35 even or odd?
6. Is 0 even or odd?
7. Is 250 even or odd?
8. Is 377 even or odd?
9. Divide 19 by 2. What is the remain­der, and is 19 even or odd?
10. Write all the even num­bers between 21 and 31.
11. Write all the odd num­bers between 40 and 50.
12. 45 chil­dren stand in pairs. How many pairs are there, and how many chil­dren are left over?

Answers#