Real Numbers: Important Questions (Part 1)

This lesson revises the most important short questions from the Real Numbers chapter: the main number sets, the Fundamental Theorem of Arithmetic, and using prime factorisation to find the LCM and HCF of two numbers.

The number sets

The lesson lists the basic sets of numbers in turn.

• Natural numbers: $1, 2, 3, 4, \dots$
• Whole numbers: $0, 1, 2, 3, 4, \dots$
• Integers: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

Rational and irrational numbers

A rational number is any number that can be written in the form $\tfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. The natural numbers, whole numbers, and integers are all rational, along with every number in $\tfrac{p}{q}$ form.

Terminating decimals such as $2.3$, $0.15$, and $5.222$, as well as repeating (non-terminating recurring) decimals such as $2.\overline{3}$, $1.\overline{09}$, and $4.\overline{5}$, are all rational numbers.

An irrational number is one that cannot be written as $\tfrac{p}{q}$. Non-terminating, non-recurring decimals such as $1.2134567914\dots$ and $0.2532101\dots$ are irrational, and so are roots like $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, and $\sqrt{7}$.

Together, the rational and irrational numbers make up the set of real numbers.

Prime, even prime, and composite numbers

Prime numbers are $2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$

The only even prime number is $2$.

Numbers that are not prime are called composite numbers: they have factors other than $1$ and the number itself.

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every composite number can be written as a product of prime factors, and this factorisation is unique apart from the order of the factors. By convention the prime factors are written in ascending order, from smallest to largest.

Worked example: factorising 24

Dividing $24$ repeatedly by primes gives

$24 = 2 \times 2 \times 2 \times 3 = 2^{3} \times 3.$

Finding LCM and HCF by prime factorisation

Once numbers are written as products of primes:

• The HCF is the product of the common prime factors, each taken to its lowest power.
• The LCM is the product of all prime factors, common and not common, each taken to its highest power.

Worked example: LCM and HCF of 10 and 20

First factorise each number:

$10 = 2 \times 5, \qquad 20 = 2^{2} \times 5.$

HCF: take the common primes $2$ and $5$ at their lowest powers:

$\text{HCF} = 2 \times 5 = 10.$

LCM: take each prime at its highest power:

$\text{LCM} = 2^{2} \times 5 = 4 \times 5 = 20.$

Relating LCM, HCF, and the product of two numbers

For any two numbers,

$a \times b = \text{LCM}(a,b) \times \text{HCF}(a,b).$

This relationship holds only for two numbers; it cannot be used for three. Rearranging gives

$\text{LCM} = \frac{a \times b}{\text{HCF}}, \qquad \text{HCF} = \frac{a \times b}{\text{LCM}}.$

If one number is unknown but the LCM and HCF are known, then

$\text{other number} = \frac{\text{LCM} \times \text{HCF}}{\text{given number}}.$

Worked example: find the HCF

Given the two numbers $10$ and $20$ with $\text{LCM} = 20$:

$\text{HCF} = \frac{10 \times 20}{20} = \frac{200}{20} = 10.$

Worked example: find the missing number

Given $\text{LCM} = 30$, $\text{HCF} = 15$, and one number equal to $15$:

$\text{other number} = \frac{30 \times 15}{15} = 30.$

Key takeaways

• The real numbers are made up of the rational numbers (including terminating and repeating decimals) and the irrational numbers (non-terminating, non-recurring decimals and roots like $\sqrt{2}$).
• By the Fundamental Theorem of Arithmetic, every composite number has a unique prime factorisation, written in ascending order.
• For two numbers, the HCF uses common primes at their lowest powers, the LCM uses all primes at their highest powers, and $a \times b = \text{LCM} \times \text{HCF}$.