E1.1 Types of number

Learning objectives

E1.1.1Identify and use: • natural numbers • integers (positive, zero and• Identify and use: • natural numbers • integers (positive, zero and negative) • prime numbers • square numbers • cube numbers • common factors • common multiples • rational and irrational numbers • reciprocals. Example tasks include: • convert between numbers and words, e.g. six billion is 6 000 000 000 10 007 is ten thousand and seven • express 72 as a product of its prime factors • find the highest common factor (HCF) of two numbers • find the lowest common multiple (LCM) of two numbers.

Number types describe properties, and one number may belong to several types. Prime factorisation then exposes the building blocks used to find common factors and multiples.

Type	Meaning	Examples
natural number	positive whole number	1, 2, 3, …
integer	whole number, including zero and negatives	−4, 0, 19
prime	integer greater than 1 with exactly two positive factors	2, 3, 5, 7
square / cube	$n^2$ / $n^3$ for an integer $n$	49 / 64
rational	can be written $a/b$ for integers $a,b$ , $b\ne0$	$-3$ , $2/5$ , 0.125, $0.\overline{3}$
irrational	cannot be written as an integer fraction	$\sqrt3$ , $\pi$
reciprocal of $x$	number whose product with $x$ is 1	reciprocal of $2/3$ is $3/2$

Place value controls number words. Six billion is 6 000 000 000, while 10 007 is ten thousand and seven: zero placeholders preserve the missing hundreds and tens.

To express an integer as a product of prime factors, divide repeatedly by primes until every factor is prime, then collect repeats with indices. For example, $180=2\times2\times3\times3\times5=2^2\times3^2\times5$ .

Job from prime factors	Selection rule
HCF	take only primes common to every number, using the lowest power
LCM	take every prime that appears, using the highest power

$126=2\times3^2\times7$ and $180=2^2\times3^2\times5$ . Their HCF is $2\times3^2=18$ . Their LCM is $2^2\times3^2\times5\times7=1260$ .

The number 1 is not prime, and 0 has no reciprocal. A square root is not automatically irrational: $\sqrt9=3$ is rational, whereas $\sqrt3$ is irrational. HCF selects shared factors; LCM builds the smallest number containing both factorisations.

Next Topic

—E1.2 SetsRead