N Number — notes on one page

N1 · Order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥

• Integers, decimals and fractions can be ordered once they are written in a comparable form.
• Convert fractions to decimals, or use a common denominator, then place the values on a number line: values increase from left to right.
• This is especially important for negatives, because the number with the greater distance below zero is smaller; for example, $-1.7<-\dfrac{3}{2}$ since $-1.7<-1.5$.
• Use $=$ for equal values, $\neq$ for unequal values, and read $\leq$ or $\geq$ as allowing equality.
• In an exam, show the conversion that justifies each comparison.

Exam tip. For a 2-mark ordering question, show decimal or common-denominator conversions before the final list.

N2 · Apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers, positive and negative; understand and use place value

Definition. Specification N2: apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper) and mixed numbers, all both positive and negative, and understand and use place value.

• Written methods score for the method, then for the answer. In a decimal division such as $818.4\div1.2$, multiply both numbers by the same power of $10$ so the divisor is an integer, divide, then place the decimal point by comparing the size of the numbers: $818.4\div1.2$ is a little less than $818$, so $682$ is sensible and $68.2$ is not.
• For fraction addition or subtraction, write both fractions over a common denominator and keep the answer in the form requested, usually a mixed number. For multiplication or division of mixed numbers, convert every mixed number to an improper fraction first; to divide, multiply by the reciprocal of the second fraction only.
• A show-that question must display each stage, including the unsimplified product.
• Track signs separately: like signs give a positive result, unlike signs a negative one.
• Cancelling common factors before multiplying keeps the arithmetic small.

What the examiner wants. Sensible place value and a visible reciprocal step. The November 2024 Paper 1H report notes that after correctly finding $682$ many wrote $68.2$ or $6.82$, and that in the division of mixed numbers some flipped the wrong fraction, flipped both, or multiplied whole numbers and fractions separately, scoring nothing.

Exam tip. Keep an exact fraction until the final line and show the reciprocal explicitly for the division method mark.

N3 · Recognise and use relationships between operations, including inverse operations; use conventional notation for priority of operations, including brackets, powers, roots and reciprocals

• Inverse operations undo one another: addition pairs with subtraction, multiplication with division, and squaring with square root when the required value is non-negative. Use these relationships to check calculations or reverse a sequence of operations.
• For a written calculation, follow conventional priority: brackets first; then powers and roots; then multiplication and division; then addition and subtraction.
• Operations at the same priority are completed from left to right.
• A reciprocal is also an inverse operation for multiplication: the reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$.
• Examiners expect each priority stage to be visible when the question carries more than one mark.

Exam tip. In a reverse-process question, undo the operations in the opposite order and write one inverse operation per line.

N4 · Prime numbers, factors (divisors), multiples, common factors and multiples, highest common factor, lowest common multiple, prime factorisation with product notation and unique factorisation theorem

Definition. Specification N4: use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple and prime factorisation, including using product notation and the unique factorisation theorem.

• A prime has exactly two factors, so $1$ is not prime. Every integer above $1$ is a product of primes in exactly one way, which is why the prime factorisation is a reliable route to the highest common factor and lowest common multiple.
• Find it with a factor tree or repeated division by primes, then write the answer as a product using index notation: $500=2^2\times5^3$. A list of primes without multiplication signs is not a product and drops the accuracy mark.
• For two numbers, a Venn diagram of their prime factors puts the shared primes in the overlap: the HCF is the product of the overlap and the LCM is the product of everything in the diagram.
• Given numbers already in index form, the HCF takes the lower power of each prime and the LCM the higher.
• Read the question twice, because HCF and LCM are routinely swapped.

What the examiner wants. A product, and the right one of HCF or LCM. The June 2024 Paper 2H report says students who listed the prime factors rather than writing them as a product scored one mark, and that asked for an LCM from index form many gave the HCF of $6$; the Paper 3H report notes a significant proportion gave the LCM of $315$ when the HCF of $21$ was asked for.

Exam tip. For HCF or LCM questions, write both prime-power decompositions first so the method marks remain available.

N5 · Apply systematic listing strategies, including use of the product rule for counting (m ways of doing one task and n ways of doing another gives m × n ways in total)

• A systematic listing strategy fixes one choice and cycles through every permitted second choice before moving on. Use a clear order, such as increasing digits or alphabetical letters, so you can see that no result is missing or repeated.
• When order matters, AB and BA are different outcomes; when order does not matter, list each pair only once.
• A table or tree can organise several stages.
• Higher tier: if one task can be completed in $m$ ways and a following independent choice in $n$ ways, the product rule gives $m\times n$ combined outcomes.
• Examiners award completeness only when the structure of the list is evident.

Exam tip. Group a list by one fixed position or case; an unstructured collection may not earn the method mark for systematic listing.

N6 · Use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5; estimate powers and roots of any given positive number

• A positive integer power represents repeated multiplication: $a^n$ contains $n$ factors equal to $a$.
• An associated root reverses that power, so $\sqrt{81}=9$, $\sqrt[3]{125}=5$, and $\sqrt[4]{16}=2$.
• Learn common powers of $2$, $3$, $4$ and $5$ because they make exact roots quickly recognisable.
• Higher tier: to estimate a root that is not exact, place its radicand between nearby known powers; for example, $3^3<40<4^3$ shows $3<\sqrt[3]{40}<4$.
• In an exam, distinguish an exact evaluation from an estimate and clearly state the bounding powers used.

Exam tip. For an estimate, write the two consecutive known powers that bound the radicand before choosing the nearer root.

N7 · Calculate with roots, and with integer and fractional indices

• Integer indices include positive, zero and negative powers. For non-zero $a$, $a^0=1$ and $a^{-n}=\dfrac{1}{a^n}$; the negative index creates a reciprocal, not a negative answer.
• Roots reverse powers, including odd roots of negative numbers such as $\sqrt[3]{-216}=-6$.
• Higher-tier questions also use fractional indices: $a^{1/n}=\sqrt[n]{a}$ and $a^{m/n}=\sqrt[n]{a^m}$.
• Foundation questions stay with roots and integer indices.
• Choose the form that makes evaluation simplest, keep brackets around a negative base, and show the reciprocal or root step very clearly before giving the final value.

Exam tip. On Foundation, show the reciprocal for a negative integer index; on Higher, rewrite a fractional index as a root before evaluating.

N8 · Calculate exactly with fractions, surds and multiples of π; simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators

• An exact answer keeps fractions, multiples of $\pi$ and, on Higher tier, surds rather than replacing them with rounded decimals. Combine exact fractions with a common denominator and leave circle results as a coefficient of $\pi$.
• Higher-tier surds are simplified by extracting square factors: $\sqrt{12}=\sqrt{4\times3}=2\sqrt3$.
• Like surds can then be collected, and a denominator such as $\sqrt5$ is rationalised by multiplying numerator and denominator by $\sqrt5$.
• Foundation questions do not require surd manipulation, but do require exact fractions and multiples of $\pi$.
• When the command says “exact”, no rounded decimal should appear in the final answer.

Exam tip. If the question says “exact”, retain $\pi$, fractions or surds and simplify them fully rather than using a calculator decimal.

N9 · Calculate with and interpret standard form A × 10^n, where 1 ≤ A < 10 and n is an integer

Definition. Specification N9: calculate with and interpret standard form $A\times10^n$, where $1\leq A<10$ and $n$ is an integer, with and without a calculator.

• A positive index moves the decimal point right, so $5.3\times10^4=53\,000$; a negative index moves it left, so $7.4\times10^{-5}=0.000\,074$. To multiply, multiply the $A$ values and add the indices; to divide, divide the $A$ values and subtract the indices; then normalise, because $24\times10^{-2}$ is not standard form until it becomes $2.4\times10^{-1}$.
• To add or subtract, write both numbers with the same power of ten first, or convert to ordinary numbers: $9.7\times10^6+2.45\times10^7=0.97\times10^7+2.45\times10^7=3.42\times10^7$.
• Adding the $A$ values when the powers differ is the standard error and scores nothing.
• A method mark is available for a correct value in a non-standard form such as $34.2\times10^6$, and a final answer left as an ordinary number when standard form was asked for loses the last mark.
• To order numbers, put them all in the same form first.

What the examiner wants. The powers dealt with before the coefficients. The June 2024 Paper 3H report says most reached $34\,200\,000$ on a calculator but many could not then write it in standard form, and that those scoring nothing had ignored the differing powers of ten, adding $9.7$ to $2.45$ to get $12.15$ and attaching a power of ten to that.

Exam tip. Circle the final coefficient and check $1\leq A<10$ before submitting any standard-form answer.

N10 · Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 or 3/8); change recurring decimals into their corresponding fractions and vice versa

• A terminating decimal has finitely many decimal places, so write it over the matching power of $10$ and simplify: $0.375=\dfrac{375}{1000}=\dfrac38$. Convert a fraction to a decimal by dividing its numerator by its denominator.
• On Higher tier, recurring decimals are also converted algebraically: name the decimal $x$, multiply by a power of $10$ so the repeating blocks align, subtract to remove the recurring part, and solve for $x$.
• A recurring dot or bar must cover the whole repeating block.
• Foundation questions use terminating decimal–fraction equivalents only.
• Examiners expect the resulting fraction in its simplest form.

Exam tip. For a recurring decimal, align identical recurring tails before subtracting so they cancel exactly.

N11 · Identify and work with fractions in ratio problems

• Fractions and ratios describe the same partition in different forms. If a group is $\dfrac35$ of the whole, the remainder is $\dfrac25$, so the part-to-part ratio is $3:2$.
• Conversely, in a ratio $a:b$, the first share is $\dfrac{a}{a+b}$ of the whole and the second is $\dfrac{b}{a+b}$.
• In multi-step problems, first use the ratio to find each share, then apply any fraction operator to the relevant share.
• Keep the named order of the groups throughout.
• Examiners commonly award one mark for finding one ratio part and a later accuracy mark for the required fractional amount.

Exam tip. Label each share and write the total number of ratio parts before applying any later fraction.

N12 · Interpret fractions and percentages as operators

Definition. Specification N12: interpret fractions and percentages as operators. A fraction or percentage “of” a quantity means multiply by it, and a sequence of operators is applied in order, each to the current amount.

• To find $\dfrac{a}{b}$ of an amount, divide by $b$ and multiply by $a$, or multiply by the fraction directly: $\dfrac{2}{7}$ of $420$ is $420\times2\div7=120$. A percentage becomes a decimal or fraction operator, $35\%$ of $420$ being $420\times0.35=147$.
• A percentage change is one multiplier: a $15\%$ decrease multiplies by $0.85$, an increase of $15\%$ by $1.15$.
• When several operators apply, each acts on the amount that remains, not on the original: “$\dfrac{3}{8}$ of the reduced price” means multiply the reduced price by $\dfrac{3}{8}$.
• Problems that mix a fraction, a percentage and a ratio are worked one step at a time, subtracting the shares already taken before splitting the remainder, and the scheme gives a process mark for each share.
• A correct answer with no working scores nothing on these questions.

What the examiner wants. Each operator applied to the right amount, in order. The November 2024 Paper 3H report describes a ratio and percentage problem where students who found the quantities correctly then worked out a percentage profit for each item separately and added them, giving $88.8\%$, and the Paper 3H report of the same series notes that increasing or decreasing $\pounds46.75$ by $15\%$ scored nothing when the operator had to be reversed.

Exam tip. Write each fraction or percentage as a multiplier beside the amount it operates on.

N13 · Use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate

Definition. Specification N13: use standard units of mass, length, time, money and other measures, including with decimal quantities.

• Work in one unit at a time and convert before calculating: $1$ kg $=1000$ g, $1$ km $=1000$ m, $1$ litre $=1000$ cm$^3$, and money in either pounds or pence throughout, converting at the end.
• Time is the trap: $1.25$ hours is $1$ hour $15$ minutes, not $1$ hour $25$ minutes, and $2.75$ hours is $2$ hours $45$ minutes, so a time from $\text{distance}\div\text{speed}$ must be converted to hours and minutes when the question asks for it.
• A rate written per $100$ km or per kilogram is scaled to the actual distance or mass first.
• Round money to the nearest penny only at the end.
• Check the size of the answer against common sense; a journey time of hundreds of minutes for a few miles signals a wrong operation.

What the examiner wants. A sensible size. The November 2024 Paper 1H report says answers for the time to travel about $5$ miles at $30$ miles per hour ranged from fractions of a minute up to $900$ minutes, and urges checking the reasonableness of an answer; those who wrote “$30$ miles in $60$ minutes” and divided both by $6$ found the $10$ minutes cleanly.

Exam tip. Write the target unit before starting; every conversion should move the data towards that unit.

N14 · Estimate answers; check calculations using approximation and estimation, including answers obtained using technology

• An estimate replaces the original inputs with nearby values that are easy to calculate, often by rounding each to one significant figure. Carry out the operation on those rounded inputs and use $\approx$, because the result is not exact.
• Estimation checks whether a calculator answer has a sensible size, sign and decimal position; it should be an independent calculation, not a rounding of the calculator display.
• Choose compatible approximations, especially for division, so mental arithmetic stays simple.
• If the exact answer differs by about a factor of $10$ or $100$, suspect a place-value entry error.
• Examiners require both the estimate and a clear comparison when asked to comment.

Exam tip. In a “check using estimation” question, state whether the given answer is reasonable and support the decision with your rounded calculation.

N15 · Round numbers and measures to an appropriate degree of accuracy (decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding

• Decimal places count digits after the decimal point; significant figures begin at the first non-zero digit. Identify the deciding digit immediately after the required place: $5$ or more rounds up, while $4$ or less leaves the retained digit unchanged.
• Choose a degree of accuracy appropriate to the context, such as money to the nearest penny.
• A rounded value also represents an error interval.
• For rounding to a unit $u$, subtract and add $\dfrac u2$; include the lower boundary but exclude the upper boundary because that endpoint rounds to the next value.
• Truncation instead keeps values from the stated number up to the next truncation step.

Exam tip. State the rounding or truncation unit first; it determines both interval endpoints and which endpoint is excluded.

N16 · Apply and interpret limits of accuracy, including upper and lower bounds

• A measured or rounded value stands for a range of possible true values. The lower and upper limits are usually half a rounding unit below and above the stated value, with the upper limit excluded.
• Use these limits to find a maximum possible error or to decide whether a claimed result is possible. For a sum, the smallest total uses all lower limits and the greatest possible total approaches all upper limits.
• Higher-tier bounds calculations also choose numerator and denominator limits deliberately, but Foundation questions can require interpreting accuracy and maximum error.
• Never treat rounded inputs as exact.
• Examiners expect the chosen limits to be written before the calculation.

Exam tip. Write “lower” or “upper” beside each substituted value so the examiner can see why it gives the required extreme.