Number — domain overview

Number is the foundation of AQA GCSE Maths and accounts for roughly 25% of marks. It spans 16 specific points (N1–N16) covering everything from basic arithmetic to complex surds and standard form.

The number strands

Strand	Key spec points	Core skill
Integers & structure	N1–N3	Place value, order, factors, multiples, prime decomposition
Fractions, decimals, %	N5–N8	Equivalence, operations, rounding, recurring decimals
Powers & roots	N6–N7	Index laws, surds, standard form
Estimation & bounds	N14–N16	Rounding, significant figures, error intervals

Why number matters across the paper

Number skills underpin every other topic. Probability fractions, geometry areas and algebra substitution all rely on accurate number work. A solid foundation here stops silly errors losing marks elsewhere.

Must-know number facts

Prime decomposition

Every integer > 1 is either prime or can be written uniquely as a product of primes:

• $360 = 2^3 imes 3^2 imes 5$
• Use a factor tree or successive division

Index laws (same base)

• $a^m imes a^n = a^{m+n}$
• $a^m div a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^0 = 1$ (anything to the power 0)
• $a^{-n} = dfrac{1}{a^n}$
• $a^{1/n} = sqrt[n]{a}$

Standard form

$A imes 10^n$ where $1 le A < 10$ and $n$ is an integer.

• $3,400,000 = 3.4 imes 10^6$ (large → positive $n$)
• $0.00056 = 5.6 imes 10^{-4}$ (small → negative $n$)

Surds

$sqrt{a}$ is a surd if $a$ has no perfect-square factor other than 1.

• Simplify: $sqrt{50} = sqrt{25 imes 2} = 5sqrt{2}$
• Rationalise: $dfrac{1}{sqrt{2}} = dfrac{sqrt{2}}{2}$

Recurring decimals → fractions

$0.overline{3} = dfrac{1}{3}$; for $0.overline{27}$: let $x = 0.overline{27}$, then $100x = 27.overline{27}$, subtract: $99x = 27$, $x = dfrac{27}{99} = dfrac{3}{11}$

Common exam mistakes

1. HCF vs LCM confusion — HCF = intersection of prime factor sets; LCM = union
2. Standard form: $A$ must satisfy $1 le A < 10$ — $15 imes 10^3$ is wrong; correct: $1.5 imes 10^4$
3. Negative indices — $2^{-3}$ means $dfrac{1}{8}$, not $-8$
4. Truncation vs rounding — truncation always goes down; rounding goes to nearest
5. Error intervals for truncation — $7.3$ truncated → $7.3 le x < 7.4$ (not $7.25 le x < 7.35$)