Number Systems

Number Sets and Arithmetic Operations

Numbers belong to nested sets: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

• $\mathbb{N}$: Positive counting numbers $\{1, 2, 3, \dots\}$, excluding $0$.
• $\mathbb{Z}$: Integers $\{\dots, -1, 0, 1, \dots\}$.
• $\mathbb{Q}$: Fractions $\frac{a}{b}$ ($a, b \in \mathbb{Z}, b \neq 0$); terminating or recurring decimals.
• $\mathbb{R}\setminus\mathbb{Q}$: Irrational numbers with infinite, non-repeating decimals (e.g. $\sqrt{2}, \pi$).
• $\mathbb{R}$: All rational and irrational numbers.

Represent operations using models:

• Number line: shows addition and subtraction as directed shifts ($3 - 5 = -2$).
• Decomposition and equal groups: split numbers ($8 + 7 = 8 + 2 + 5$) or accumulate groups ($4 \times 3 = 12$).

Fractions require common denominators to add ($\frac{1}{2} - \frac{3}{4} = -\frac{1}{4}$), multiply across ($\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$), and invert to divide ($\frac{2}{3} \div \frac{4}{5} = \frac{5}{6}$).

Convert forms via decimals: $\frac{3}{4} = 0.75 = 75\%$. To order numbers, compare decimals: $3.14 < \pi < \frac{22}{7} < \sqrt{10}$.

Addition and multiplication are commutative: order does not matter. They are associative: grouping does not matter. Multiplication distributes over addition: a(b+c)=ab+ac. Subtraction and division are neither commutative nor associative.

Surds

A surd is a root of a rational number whose value is irrational, such as $\sqrt{3}$. Leaving an answer as a surd keeps it in exact form without rounding.

• Simplifying: extract square factors: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.
• Multiplication and division for non-negative radicands ($a, b \ge 0$, and $b > 0$ to avoid division by zero):

Examples: $\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6$, and $\frac{\sqrt{75}}{\sqrt{3}} = \sqrt{25} = 5$.

• Addition and subtraction: combine like roots only: $5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}$, but $\sqrt{2} + \sqrt{3} \neq \sqrt{5}$.

Index Laws and Order of Operations

A positive whole-number index counts repeated factors: $2^3 = 2 \times 2 \times 2$. Fractional indices represent roots. For positive bases and rational powers, these index laws apply:

• $a^p \times a^q = a^{p+q}$ and $\frac{a^p}{a^q} = a^{p-q}$
• $(a^p)^q = a^{pq}$
• $a^0 = 1$ ($a \neq 0$)
• $a^{-r} = \frac{1}{a^r}$ (e.g. $2^{-3} = \frac{1}{8}$)
• $a^{\frac{p}{q}} = \sqrt[q]{a^p} = (\sqrt[q]{a})^p$ (e.g. $27^{\frac{2}{3}} = 3^2 = 9$)
• $(ab)^r = a^r b^r$ and $\left(\frac{a}{b}\right)^r = \frac{a^r}{b^r}$

Equating powers: for $a > 0, a \neq 1$, if $a^x = a^y$, then $x = y$.

Order of operations follows BIMDAS: Brackets, Indices, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Factors, Accuracy, and Scientific Notation

• Prime factorisation writes a number as prime powers: $60 = 2^2 \times 3 \times 5$ and $80 = 2^4 \times 5$.
• HCF takes the lowest power of shared primes ($2^2 \times 5 = 20$).
• LCM takes the highest power of every prime ($2^4 \times 3 \times 5 = 240$).
• Rounding: round up if the next digit is $5$ or more; otherwise leave unchanged ($4.768 \approx 4.77$ to $2$ d.p.).
• Significant figures start at the first non-zero digit: $0.0407 \approx 0.041$ to $2$ significant figures.
• Scientific notation has form $a \times 10^n$, with $1 \le a < 10$ and $n \in \mathbb{Z}$ ($45\,000 = 4.5 \times 10^4$; $0.0032 = 3.2 \times 10^{-3}$).

Sets and Venn Diagrams

A set is a well-defined collection of elements. Equal sets contain exactly the same elements. For example, A={2,4,6} can also be defined by A={x∈ℕ: x is even and x<8}.

• Universal set ($U$): all items; null set ($\emptyset$): empty.
• Element ($\in$): belongs to; subset ($\subset$): contained within.
• Cardinal number ($\#A$): count of elements in $A$.
• Union ($A \cup B$): in $A$, $B$, or both.
• Intersection ($A \cap B$): in both $A$ and $B$.
• Set difference ($A \setminus B$): in $A$ but not $B$.
• Complement ($A'$): outside $A$ within $U$.

Union and intersection are commutative and associative; set difference is neither. For example, $\{1, 2\} \setminus \{2, 3\} = \{1\} \neq \{3\} = \{2, 3\} \setminus \{1, 2\}$.

In a 3-set Venn diagram:

• Centre: triple intersection $A \cap B \cap C$; full combined area: $A \cup B \cup C$.
• Working from the centre: subtract the triple count from pairwise intersections to find elements in exactly two sets.

Key terms

Rational Number ($\mathbb{Q}$)

A number that can be written as a fraction $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $b \neq 0$.

Irrational Number ($\mathbb{R}\setminus\mathbb{Q}$)

A real number that cannot be written as a fraction of integers; its decimal form is infinite and non-repeating.

Surd

A root of a rational number whose value is irrational, such as $\sqrt{5}$.

Scientific Notation

A number written in the form $a \times 10^n$, where $1 \le a < 10$ and $n \in \mathbb{Z}$.

Cardinal Number ($\#A$)

The total number of distinct elements in a set $A$.

HCF and LCM

The Highest Common Factor is the largest integer dividing each value; the Lowest Common Multiple is the smallest positive shared multiple.