Types of number

Natural numbers and integers

The natural numbers are the counting numbers: $1,2,3,4,\dots$ They go on for ever, and each one is a positive whole number. (Some books also count 0 as a natural number. If a question asks for natural numbers, positive whole numbers are always safe answers, so do not rely on 0.)

The integers are all the whole numbers: positive, zero and negative. $\dots ,-3,-2,-1,0,1,2,3,\dots$ So $-12$ and $0$ are integers but not natural numbers, and $2.5$ and $\frac{1}{2}$ are not integers at all.

One number can belong to several types at once. 9 is a natural number, an integer, a square number and a rational number. Questions often give a list and ask for all the numbers of one type, so test every number in the list against the definition, one at a time. Work out anything that is not yet a plain number first: $\frac{12}{4}=3$ is an integer and $-\sqrt{16}=-4$ is an integer, even though neither looks like one.

Numbers in words and figures

Large numbers are read in groups of three digits, counting from the right: units, thousands, millions, billions. Cambridge prints the groups with a small space between them, $6000000000$, rather than commas.

• one thousand $=1000$
• one million $=1000000$, a thousand thousands
• one billion $=1000000000$, a thousand millions

So six billion is $6000000000$, and $10007$ is ten thousand and seven.

Words to figures. Fill each group of three in turn, and use 0 as a placeholder in any place that is empty. Four million, twenty thousand and five has 4 in the millions group, 020 in the thousands group and 005 in the units group: $4020005$. Count the digits as a check: any number in the millions has 7 or 8 or 9 digits.

Figures to words. Read each group as a number from 1 to 999, then say the group's name. $2305060$ is two million, three hundred and five thousand and sixty. In British English the word and comes before the tens and units of a group.

News reports mix decimals and words: 3.2 million means $3.2\times 1000000=3200000$.

Factors, multiples and primes

A factor of a number divides into it exactly, with no remainder. Find factors in pairs that multiply to the number. For 18: $1\times 18$, $2\times 9$, $3\times 6$. When the pairs start to repeat you have them all, so the factors of 18 are 1, 2, 3, 6, 9 and 18.

A multiple of a number is in its times table. The multiples of 6 are $6,12,18,24,30,\dots$

A common factor of two numbers is a factor of both of them, and a common multiple is a multiple of both. 6 is a common factor of 18 and 24, and 72 is a common multiple of 18 and 24.

A prime number has exactly two factors: 1 and itself. The primes below 50 are $2,3,5,7,11,13,17,19,23,29,31,37,41,43,47$.

• 1 is not prime. It has only one factor.
• 2 is the only even prime, because every other even number has 2 as a factor.

To test whether a number is prime, divide by the primes $2,3,5,7,11,\dots$ in turn. You only need to go as far as the square root of the number. If $n$ had a factor bigger than $\sqrt{n}$, it would pair with a factor smaller than $\sqrt{n}$, and you would already have found that one. To test 221, try the primes up to $\sqrt{221}\approx 14.9$: 13 works, because $221=13\times 17$, so 221 is not prime.

Square and cube numbers

A square number is an integer multiplied by itself: $1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,\dots$ The name comes from the picture: 16 dots make a 4 by 4 square.

A cube number is an integer used three times in a multiplication: $1^3=1$, $2^3=8$, $3^3=27$, $4^3=64$, $5^3=125$ and ${10}^3=1000$. 64 is both a square and a cube: $8^2=64$ and $4^3=64$.

In prime factors, a square number has every index even, such as $144=2^4\times 3^2$, and a cube number has every index a multiple of 3, such as $216=2^3\times 3^3$. That fact answers questions like: what is the smallest number you can multiply by to make a square?

Writing a number as a product of prime factors

Every whole number bigger than 1 is either prime or can be written as a product of primes, and in only one way, apart from the order. That product is what you use to find HCFs and LCMs.

There are two methods, and both give the same primes:

1. Factor tree. Split the number into any factor pair, then split every branch that is not prime, until each branch ends in a prime.
2. Repeated division. Divide by the smallest prime that goes in exactly, again and again, until you reach 1.

Write the answer as a product in index form, smallest prime first. For example $72=2\times 2\times 2\times 3\times 3=2^3\times 3^2$. Check by multiplying back: $8\times 9=72$.

An answer such as $8\times 9$, or $2^3\times 9$, is a product of factors but not of prime factors, so it does not earn the mark.

Highest common factor and lowest common multiple

The highest common factor (HCF) of two numbers is the largest number that divides both of them. The lowest common multiple (LCM) is the smallest number that both of them divide into.

For small numbers, list. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The HCF is 6. Multiples of 12: 12, 24, 36, 48. Multiples of 18: 18, 36. The LCM is 36.

For larger numbers, use the prime factors:

• HCF: take the primes that the numbers share, each to the lower power.
• LCM: take every prime that appears in either number, each to the higher power.

Why this works: a common factor can only be built from primes that are in both numbers, and it cannot use more copies of a prime than the number with fewer copies has. A common multiple has to contain everything that each number is made of, so it needs the larger number of copies of each prime; the lowest one uses no more than that.

A useful check for two numbers: $\text{HCF}\times \text{LCM}$ equals the two numbers multiplied together.

Prime factors in a Venn diagram

$84=2^2\times 3\times 7$ and $120=2^3\times 3\times 5$. The shared primes go in the overlap. The HCF is the product of the overlap, $2\times 2\times 3=12$. The LCM is the product of everything inside the circles, $7\times 2\times 2\times 3\times 2\times 5=840$.

HCF or LCM? Reading a problem

Word problems do not say HCF or LCM, so decide from the situation.

• HCF when you split things into the largest possible equal groups or pieces, with nothing left over: the longest equal lengths a ribbon can be cut into, the largest square tile that fits a floor exactly.
• LCM when things repeat and you want the first time they line up again: buses leaving together, lights flashing together, buying packs so the totals match.

A quick sense check: the HCF is never bigger than the smaller number, and the LCM is never smaller than the bigger number.

Rational and irrational numbers

A rational number can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b\ne 0$. Every integer is rational, since $7=\frac{7}{1}$, and so is every decimal that stops: $0.125=\frac{1}{8}$ and $2.3=\frac{23}{10}$. A decimal that repeats in a pattern for ever, such as $0.333\dots =\frac{1}{3}$, is rational too.

An irrational number cannot be written as a fraction of two integers. Its decimal goes on for ever without a repeating pattern. The ones you will meet are:

• square roots of positive integers that are not square numbers, such as $\sqrt{2}$, $\sqrt{10}$ and $\sqrt{50}$;
• cube roots of integers that are not cube numbers, such as $\sqrt[3]{4}$;
• $\pi$;
• numbers made by adding a rational number to one of these, such as $3+\sqrt{2}$, or multiplying one by a non-zero rational number, such as $2\pi$.

A root is not automatically irrational. $\sqrt{49}=7$ and $\sqrt[3]{27}=3$ are rational, and so is $\sqrt{0.09}=0.3$. Always work out the value first, then decide.

A calculator display cannot settle it either. $\frac{22}{7}=3.142857\dots$ looks very like $\pi =3.141592\dots$, but $\frac{22}{7}$ is a fraction of integers, so it is rational. It is only an approximation to $\pi$.

Reciprocals

The reciprocal of a number is 1 divided by that number. A number multiplied by its reciprocal always gives 1.

• The reciprocal of 5 is $\frac{1}{5}$, or 0.2. Check: $5\times \frac{1}{5}=1$.
• The reciprocal of a fraction turns it upside down: the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.
• For a mixed number, change to an improper fraction first: $2\frac{1}{2}=\frac{5}{2}$, so its reciprocal is $\frac{2}{5}$. Turning over only the fraction part is a common slip.
• For a decimal, divide 1 by it, or write it as a fraction first: $0.25=\frac{1}{4}$, so its reciprocal is 4.
• 0 has no reciprocal, because you cannot divide by 0.

Taking the reciprocal twice brings you back to the number you started with. So if the reciprocal of a number is 0.2, the number is $\frac{1}{0.2}=5$. On a calculator the $x^{-1}$ key gives the reciprocal, which links to negative indices in Topic 1.7.