Natural numbers and integers

Highlights only0 highlighted

Every number topic in this unit starts with knowing what kind of number you are dealing with.

This section covers the two most basic families: the natural numbers you count with, and the integers, which add zero and the negative numbers.

It also covers reading and writing large numbers in words and in figures.

Natural numbers

The natural numbers are the counting numbers: 1, 2, 3, 4, 5, and so on for ever.

Some books also count 0 as a natural number, and Cambridge does not say which convention it follows.

These notes take the natural numbers to start at 1.

If a set in a question depends on whether 0 is included, read how the question defines it.

Natural numbers have no fractional or decimal part, and none of them is negative.

So 7 and 250 are natural numbers, but 2.5, $\frac{3}{4}$ and $-6$ are not.

Place value and large numbers

Each digit in a number has a value that depends on its place.

In 4 735 the 7 stands for 7 hundreds, or 700.

Large numbers are written in groups of three digits, counting from the right, with a small space between the groups: 6 400 000, not 6400000.

The groups are units, thousands, millions and billions.

Word

Figures

Zeros after the 1

one thousand

1 000

one million

1 000 000

one billion

1 000 000 000

A billion here means a thousand million, so it has nine zeros.

Four billion is written 4 000 000 000.

Writing numbers in words and in figures

To write a number in words, read it one group of three at a time, from the left, and say the name of each group.

Use "and" before the last part if it is less than a hundred, the usual British style.

2 306 015 is two million, three hundred and six thousand and fifteen.

70 040 is seventy thousand and forty.

To write a number in figures, set out the groups first, then fill in each group with three digits.

A group that is missing becomes three zeros, and a short group gets zeros in front so that every group after the first has exactly three digits.

That is where most mistakes come from: "five million and twenty" is 5 000 020, with two whole groups of zeros in the middle.

The place-value table shows the number in Worked example 1, eight million, sixty thousand and nine, set out group by group.

A place-value table with three groups, millions, thousands and units, each split into hundreds, tens and units columns (H, T, U). Eight million, sixty thousand and nine fills it as 8 in the millions group, 0 6 0 in the thousands group and 0 0 9 in the units group, giving 8 060 009

Integers

The integers are the whole numbers: the positive whole numbers, zero, and the negative whole numbers.

\ldots,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ \ldots

Zero is an integer, but it is neither positive nor negative.

A negative integer is less than zero and is written with a minus sign, such as $-4$ .

Negative numbers describe things below a starting level: temperatures below freezing, depths below sea level, an overdrawn bank balance or a floor below the ground floor.

The number line shows how the families fit together.

A number line from −5 to 5. A bracket over the whole line marks the integers; underneath, −5 to −1 are marked negative integers, 0 is marked zero, and 1 to 5 are marked natural numbers

Every natural number is an integer, but not every integer is a natural number.

So $-8$ and 0 are integers but not counting numbers, and $3.5$ is neither.

You will meet the other families (primes, squares, rational numbers) later in this unit; each is a set of numbers with a special property.

Key terms

Natural numbers: the counting numbers 1, 2, 3, 4, and so on.

Integer: a whole number that may be positive, negative or zero.

Place value: the value a digit has because of its position in the number.

Billion: one thousand million, written with nine zeros after the 1.

Worked example 1: words to figures

Write eight million, sixty thousand and nine in figures.

1. The largest group named is millions, so the number has two more groups after the millions: thousands, then units.

2. Millions group: 8.

3. Thousands group: sixty thousand, so the group is 060.

4. Units group: nine, so the group is 009.

5. Put them together with spaces: 8 060 009.

Answer: 8 060 009

Worked example 2: sorting numbers into families

From the list $-7$ , $0$ , $12$ , $3.5$ , $-\frac{1}{2}$ , $1\,000\,000$ , write down (a) the natural numbers, (b) the integers.

1. Natural numbers are the counting numbers, so pick out the positive whole numbers: 12 and 1 000 000.

2. Integers are all whole numbers: positive, zero or negative. Add $-7$ and $0$ to the list from step 1.

3. $3.5$ and $-\frac{1}{2}$ have a fractional part, so they are in neither list.

Answer: (a) 12 and 1 000 000; (b) $-7$ , $0$ , 12 and 1 000 000

Exam tip

When you write a large number in figures, count the digits after you finish: a number in the millions has 7, 8 or 9 digits, and one in the billions has 10, 11 or 12.

A missing group of zeros is the usual slip, and the digit count catches it.

Common mistakes

Leaving out a group of zeros: "three million and forty" is 3 000 040, not 3 040 or 300 040.

Saying zero is positive. Zero is an integer, but it is neither positive nor negative.

Calling a negative whole number a natural number. $-5$ is an integer, not a natural number.

Writing a billion with six zeros. Six zeros is a million; a billion has nine.

Check your understanding

1. (no calculator) Write 40 300 512 in words. [1]

2. (no calculator) Write three billion, two hundred million in figures. [1]

3. (no calculator) Write down the integers in this list: $5.2$ , $-11$ , $0$ , $\frac{9}{3}$ , $-0.6$ . [2]

4. (no calculator) Write down the value of the digit 6 in 2 615 004. [1]

Answers

1. Forty million, three hundred thousand, five hundred and twelve [1]

2. 3 200 000 000 [1]

3. $-11$ , $0$ and $\frac{9}{3}$ (which equals 3) [2: 1 for $-11$ and $0$ , 1 for spotting that $\frac{9}{3}$ is the integer 3]

4. 600 000 (six hundred thousand) [1]

Natural numbers and integers

Your course level