Types of Number and Sets

The families of number Cambridge expects you to name

Every number you meet in the 0580 papers belongs to a family, and questions often ask you to name the family rather than calculate anything. Natural numbers are the counting numbers: 1, 2, 3, 4 and so on. Integers are the whole numbers, positive, negative and zero: -3, -2, -1, 0, 1, 2, 3. Zero is an integer but it is not a natural number, and that single fact is worth a mark somewhere in the paper.

A prime number has exactly two factors, itself and 1. So 2, 3, 5, 7, 11 and 13 are prime, while 1 is not prime (it has only one factor) and 9 is not prime (1, 3 and 9 divide it). A square number is the product of an integer with itself: 1, 4, 9, 16, 25, 36. A cube number is an integer multiplied by itself twice: 1, 8, 27, 64, 125. Cambridge expects you to recall squares and their roots from 1 to 15, and cubes and their roots for 1, 2, 3, 4, 5 and 10.

A reciprocal is 1 divided by the number. The reciprocal of 4 is $\frac{1}{4}$, the reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$, and the reciprocal of $x$ is $\frac{1}{x}$. Multiplying a number by its reciprocal always gives 1, which is the quickest way to check you have written the right one.

Factors, multiples, HCF and LCM

A factor of a number divides into it exactly, with nothing left over. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. A multiple of a number is what you get when you multiply it by an integer: the multiples of 6 are 6, 12, 18, 24, 30 and so on. Every number is a factor of itself, and every number has infinitely many multiples but only finitely many factors - which is why listing factors is a method and listing multiples is not.

The highest common factor (HCF) of two numbers is the largest number that divides into both of them. The lowest common multiple (LCM) is the smallest number that both of them divide into. For 24 and 36, the common factors are 1, 2, 3, 4, 6 and 12, so the HCF is 12; the LCM is 72. Listing factors works for small numbers, but prime factorisation is the reliable method for anything larger.

• HCF of 24 and 36: the shared prime factors are 2 x 2 x 3, so the HCF is 12
• LCM of 24 and 36: take every factor once, 2 x 2 x 2 x 3 x 3, so the LCM is 72
• For two numbers a and b, HCF x LCM = a x b. Check: 12 x 72 = 864 = 24 x 36

Prime factorisation: the tool that does the work

Any integer greater than 1 can be written as a product of prime numbers, and the answer is unique apart from the order. Cambridge asks for this as 'express 72 as a product of its prime factors'. Work down by dividing: 72 = 8 x 9, then 8 = 2 x 2 x 2 and 9 = 3 x 3, so 72 = 2 x 2 x 2 x 3 x 3 = $2^3\times 3^2$. Writing it with index notation is the tidier answer and the one the mark scheme expects.

The same method gives the HCF and the LCM without any guesswork. Write both numbers as products of primes, then for the HCF take the primes they have in common with the lower index, and for the LCM take every prime with the higher index. For 72 and 108: $72=2^3\times 3^2$ and $108=2^2\times 3^3$, so the HCF is $2^2\times 3^2=36$ and the LCM is $2^3\times 3^3=216$.

Prime factorisation is also how you decide whether a number is a perfect square or a perfect cube. A square number has every index even in its prime factorisation; a cube number has every index a multiple of 3. So $2^3\times 3^2$ is not a square and not a cube, while $2^4\times 3^2$ is a square because 4 and 2 are both even.

Rational and irrational numbers

A number is rational if it can be written as a fraction $\frac{p}{q}$ where p and q are integers and q is not zero. That covers more than you might expect: 7 is rational because it is $\frac{7}{1}$, 0.25 is rational because it is $\frac{1}{4}$, and even a decimal that goes on for ever is rational provided it repeats a block of digits. $0.333\dots =\frac{1}{3}$ and $0.454545\dots =\frac{5}{11}$.

A number is irrational if no such fraction exists. Irrational numbers have decimal expansions that never end and never fall into a repeating pattern: $\sqrt{2}=1.41421356\dots$, $\pi =3.14159265\dots$ and $\sqrt{7}$ are the ones you meet most often. A surd such as $\sqrt{9}$ is not irrational, because 9 is a perfect square and $\sqrt{9}=3$.

The distinction matters because rational numbers can be written down exactly and irrational ones cannot. When a question asks for $\sqrt{50}$ to three significant figures, the answer 7.07 is a rounded rational approximation - the exact value is still irrational.

Set language and the symbols

A set is a collection of things, usually written inside curly brackets. $A=\{2,4,6,8\}$ is the set of even numbers below 10. The universal set, written $\xi$, is everything under discussion - for a question about dice it might be $\{1,2,3,4,5,6\}$. The complement of A, written $A^{\prime }$, is everything in the universal set that is not in A.

The union $A\cup B$ is everything in A or in B or in both. The intersection $A\cap B$ is only the elements that are in both. If $A=\{1,2,3,4\}$ and $B=\{3,4,5,6\}$ then $A\cup B=\{1,2,3,4,5,6\}$ and $A\cap B=\{3,4\}$. The symbol $n(A)$ means the number of elements in A, so $n(A)=4$ for that set.

Extended candidates also need $\in$ and $\notin$, which say whether an element belongs to a set; $\varnothing$ for the empty set, the set with no elements; and $\subseteq$ and $⊈$ for subset and not a subset. A set is a subset of B when every element of A is also in B. Note that $\{3,4\}$ is a subset of $\{1,2,3,4\}$, but 3 on its own is an element of it, not a subset - $3\in A$ but $\{3\}\subseteq A$.

Venn diagrams: two sets, then three

A Venn diagram draws the sets as circles inside a rectangle, with the rectangle standing for the universal set. Two overlapping circles make four regions: in A only, in both, in B only, and in neither. Labelling those four regions rather than the whole circles is what makes Venn diagram questions quick - almost every one asks for the number in a named region.

The Core paper uses two sets. The Extended paper also uses three, where the diagram has eight regions and you have to work outwards from the middle. The technique is the same in both cases: put the number that belongs to every set in the centre first, then subtract from the totals as you move outwards. Never write a number into a region without checking it against the total given for that set.

Venn diagrams also do the HCF and LCM work for you. Put the prime factors of one number in the left circle and of the other in the right, with the shared primes in the middle. The intersection is the HCF and everything in the diagram, counted once each, is the LCM.

Numbers in words, and words in numbers

The syllabus asks you to convert between numbers and words, and it is an easy mark that students lose through carelessness. Six billion is 6 000 000 000: six followed by nine zeros. Ten thousand and seven is 10 007, not 10 007 with the wrong number of zeros and not 1 007.

Write large numbers in groups of three from the right, and the words follow the groups: thousand, million, billion. So 4 250 000 is four million, two hundred and fifty thousand. The group of three is worth the habit because it is also how you avoid dropping a zero when you copy a number into a calculation.

• Six billion = 6 000 000 000 (nine zeros after the 6)
• Ten thousand and seven = 10 007
• Four million, two hundred and fifty thousand = 4 250 000
• Three hundred and six thousand and forty = 306 040