A summary of this Mathematics chapter — open a section to read it. The full notes, worked examples and practice questions are in the study modules above.

What is Number about?

Cambridge IGCSE Mathematics 0580 is examined by two routes. Core candidates take Papers 1 and 3 and can be awarded grades C to G. Extended candidates take Papers 2 and 4 and can be awarded grades A* to E. The Extended subject content repeats the Core foundation and then adds greater depth and two extra topics. This chapter is written as one course with an unmistakable boundary: every Core explanation is complete on its own, and Extended material is placed immediately beside the idea it extends, never hidden in an appendix at the end.

Topic 1 is not a list of unrelated skills — sixteen subtopics on the Core route, eighteen on the Extended route. It is one habit applied over and over: choose the representation that makes the question easy, work exactly for as long as you can, and round only once, at the end. A number can appear as a product of primes, a fraction, a decimal, a percentage, a power, a standard-form pair, a ratio, a rate or a bound — and, on the Extended route, a surd. Every calculation in Chapter 1 becomes short the moment you pick the right one. Every later chapter draws on this: algebra needs indices and exact fractions, mensuration needs ratio, rates and bounds, and statistics needs proportional reasoning.

The big picture. Number work controls almost every later topic. Algebra needs factors, indices and exact fractions; mensuration needs ratios, rates, units and bounds; probability and statistics need proportional reasoning. The aim is not merely to calculate but to choose an efficient representation and judge whether the result is sensible. Everything below is one of those representations.

Every number in this syllabus is a real number, and every real number is either rational (can be written as an integer fraction, and therefore has a terminating or recurring decimal) or irrational (cannot, and therefore has a non-terminating, non-recurring decimal). Inside the rationals sit the integers, and inside those the natural counting numbers. The words prime, square, cube, factor, multiple and reciprocal are descriptions applied within those systems, not extra systems of their own.

A Venn diagram is a counting device, not a picture. The reliable method is always the same: fill the most-overlapped region first and work outwards, because every other region is then a subtraction from a number you already know. For two sets that means the intersection, then the “only” regions, then the outside. The same rule extends to three sets on the Extended route.

Powers and roots are inverse operations, and on the non-calculator paper a large part of the work is recall rather than calculation. Knowing $13^2=169$ instantly turns $\sqrt{169}$ into a one-second answer, and recognising $288=144\times2$ is what makes $\sqrt{288}=12\sqrt2$ possible without a calculator. The recall table below is assumed knowledge; the rest of Topic 1 leans on it constantly.

Key ideas to remember

Every other subtopic in Topic 1 is identical in both routes. Subtopics 1.1, 1.3, 1.5, 1.6, 1.8, 1.9, 1.11, 1.12, 1.14, 1.15 and 1.16 have Core and Extended statements that ask for the same mathematics, so eleven of the eighteen sections below carry both tags and need no route decision from you at all.
Coverage promise. Every one of the sixteen active Core subtopics and all eighteen Extended subtopics has its own teaching section below, with its own definition panel, method, fully checked worked example and quick check. Nothing in Topic 1 is covered by a heading alone, and no Core explanation depends on anything inside an Extended depth panel.
The pattern behind eight of the ten. Each one is a moment where a quantity was silently converted into something it is not: an interval into a value, an exact number into a decimal, a base into a different base, a rate into a mean. Before you write a final answer, name what kind of object it is.
The pattern. Fourteen of these twenty are the same underlying slip: an operation was applied to the wrong base or the wrong object — a percentage of the new value instead of the old, a cancel across a sum instead of a product, an index across a sum, a bound treated as a value. Before every line of working, name what you are operating on.
One habit above all. Whatever else you drop from this plan, keep this: work exactly for as long as you can, and round only once, at the end. It costs nothing, it applies to every chapter of the course, and it is the single largest source of avoidable lost marks in Topic 1.

What you need to be able to do

I can identify and use natural numbers, integers, prime numbers, square numbers, cube numbers, common factors, common multiples, rational and irrational numbers and reciprocals. Core C1.1
I can convert between numbers and words in both directions, reading and writing large numbers by grouping digits in threes. Core C1.1
I can express a number as a product of its prime factors, and use that form to find the HCF and the LCM of two numbers. Core C1.1
I can use set language and the notation $n(A)$, $A'$, $\xi$, $\cup$ and $\cap$ correctly. Core C1.2
I can describe a set by listing its elements and by a rule in set-builder notation. Core C1.2
I can draw, complete and read a two-set Venn diagram. Core C1.2
I can calculate squares, cubes, square roots, cube roots and other powers and roots, and I can recall the squares and square roots from 1 to 15 and the cubes and cube roots of 1, 2, 3, 4, 5 and 10. Core C1.3
I can use the language and notation of proper fractions, improper fractions, mixed numbers, decimals and percentages, and write a fraction in its simplest form. Core C1.4
I can recognise equivalence and convert between fractions, decimals and percentages in every direction. Core C1.4
I can add, subtract, multiply and divide fractions and mixed numbers, cancelling only common factors. Core C1.6
I can order quantities by magnitude and use $=$, $\ne$, $<$, $>$, $\le$ and $\ge$ correctly, including with negative numbers. Core C1.5
I can use the four operations on integers, fractions and decimals with brackets and the correct order of operations, including signed arithmetic and practical situations such as a temperature change. Core C1.6
I can use positive, zero and negative integer indices. Core C1.7
I can apply the rules of indices $a^ma^n=a^{m+n}$, $a^m\div a^n=a^{m-n}$ and $(a^m)^n=a^{mn}$. Core C1.7
I can use the standard form $A\times10^n$ with $1\le A<10$ and $n$ an integer, convert into and out of it, and calculate with values in standard form. Core C1.8
I can round a value to a specified degree of accuracy, in decimal places or significant figures. Core C1.9
I can make an estimate for a calculation by rounding each number to 1 significant figure, and round a final answer to a reasonable degree of accuracy for the context. Core C1.9
I can give the upper and lower bounds of data rounded to a specified accuracy, using $\le$ for the lower endpoint and $<$ for the upper. Core C1.10
I can give a ratio in its simplest form and divide a quantity in a given ratio. Core C1.11
I can use proportional reasoning and ratios in context — adapting a recipe, using a map scale and determining best value. Core C1.11
I can use common measures of rate, apply other measures of rate from a formula given in the question, and solve problems involving average speed. Core C1.12
I can calculate a given percentage of a quantity, express one quantity as a percentage of another, and calculate a percentage increase or decrease. Core C1.13
I can handle deposit, discount, profit and loss, earnings and percentages over $100\%$. Core C1.13
I can calculate with simple interest and with compound interest, from memory — no formula is given. Core C1.13
I can use a calculator efficiently, enter values appropriately, and interpret the display in context without rounding inside the calculation. Core C1.14
I can calculate with time in the 12-hour and 24-hour systems, read clocks and timetables, and solve problems involving time zones, local times and time differences. Core C1.15
I can calculate with money and convert from one currency to another in the correct direction. Core C1.16
I can use Venn diagrams for two or three sets and to represent relationships between sets, and I can use the notation $\in$, $\notin$, $\varnothing$, $\subseteq$ and $\nsubseteq$ as well as the Core notation. Extended E1.2
I can read and write recurring-decimal dot notation, and convert between a recurring decimal and a fraction in both directions, including a decimal with a non-recurring prefix. Extended E1.4
I can use fractional indices, and evaluate expressions such as $81^{1/2}$, $27^{2/3}$ and $8^{-2/3}$ without a calculator. Extended E1.7
I can find the upper and lower bounds of a result calculated from rounded data — a perimeter, an area, or a speed from a rounded distance and time. Extended E1.10
I can calculate using reverse percentages, and with repeated percentage change. Extended E1.13
I can use exponential growth and decay, including depreciation and population change. Knowledge of $e$ is not required. Extended E1.17
I can understand and use surds, simplify surd expressions, and rationalise a denominator. Extended E1.18

Why Number matters

Why it matters: the classification is not decoration. It decides what you are allowed to do. You may only write $\frac pq$ for a rational; you may only build a prime factorisation from a positive integer; and you may only leave $\sqrt2$ as $\sqrt2$ because it is irrational and no decimal is exact. Getting the name right first prevents the method being wrong later.

Common mistakes to avoid

1. “1 is a prime number.” Core and Extended Why wrongA prime has exactly two positive factors. The only positive factor of 1 is 1 itself, so it has one. Correct1 is neither prime nor composite. Prime factorisations therefore never contain a 1, and 2 is the smallest prime. Say this“A prime number has exactly two positive factors, so 1 is not prime.” CheckList the primes below 12. (2, 3, 5, 7, 11.)
2. “The HCF takes the smaller power of every prime you can see.” Core and Extended Why wrongFor $72=2^3\times3^2$ and $90=2\times3^2\times5$, that rule would include the 5 and give 90 — which does not divide 72 at all. CorrectOnly primes appearing in both factorisations go into the HCF. The LCM, by contrast, takes every prime present. Say this“The common primes are 2 and 3 at their lower powers, so the HCF is $2\times3^2=18$.” CheckHCF of $2^4\times5$ and $2^2\times3\times5^2$. ($2^2\times5=20$.)
3. “$n(A)=22$ means 22 students study only Art.” Core and Extended Why wrong$n(A)$ counts the whole circle, including the overlap. Putting 22 in the Art-only region double-counts everyone in the intersection. CorrectEnter the intersection first, then subtract it from each set total to obtain the “only” regions. Say this“Art only is $n(A)-n(A\cap B)$.” Check$n(A)=31$, $n(A\cap B)=15$. How many are in $A$ only? (16.)
4. “$\sqrt{64}=4$, because $4^3=64$.” Core and Extended Why wrong$64$ is both $8^2$ and $4^3$. The index on the root sign decides which fact is needed, and a plain $\sqrt{\phantom{x}}$ means a square root. Correct$\sqrt{64}=8$ and $\sqrt[3]{64}=4$. Say this“A bare root sign is a square root.” CheckEvaluate $\sqrt{729}$ and $\sqrt[3]{729}$. (27 and 9.)
5. “$\frac12+\frac13=\frac25$.” Core and Extended Why wrongAdding numerators and denominators is not addition of fractions. $\frac25=0.4$, but the true sum is $0.8\overline3$ — the “answer” is smaller than one of the things being added. CorrectRewrite over a common denominator: $\frac36+\frac26=\frac56$. Say this“Fractions are added over a common denominator; only the numerators combine.” Check$\frac34+\frac25$. ($\frac{15}{20}+\frac{8}{20}=\frac{23}{20}$.)
6. “In $\dfrac{3+x}{3}$ the threes cancel.” Core and Extended Why wrongThe 3 on top is a term, not a factor of the whole numerator. Test with $x=6$: the true value is $3$, not $6$. CorrectCancel only a factor of the entire numerator against a factor of the entire denominator. $\frac{3(1+x)}{3}$ does cancel, to $1+x$. Say this“Cancel factors, never terms joined by addition.” CheckSimplify $\dfrac{5x+10}{5}$. ($x+2$, after factorising the numerator as $5(x+2)$.)
7. “$-7$ is bigger than $-3$ because 7 is bigger than 3.” Core and Extended Why wrongSize and position are different things. On the number line $-7$ is further left, and further left always means smaller. Correct$-7<-3$. Among negatives, the larger the digits the smaller the number. Say this“The smaller number is the one further to the left on the number line.” CheckOrder $-2,-9,0,-5$ smallest first. ($-9,-5,-2,0$.)
8. “$24\div6\times2=2$, because multiplication comes before division.” Core and Extended Why wrongMultiplication and division share one level of precedence. They are performed in the order they appear, left to right. Correct$24\div6=4$, then $4\times2=8$. The same applies to addition and subtraction: $10-4+3=9$. Say this“Multiplication and division are equal in rank and taken left to right.” Check$36\div9\times3$. (12, not 1.)
9. “$-3^2=9$.” Core and Extended Why wrongWithout a bracket, the index binds only to the 3, so the expression means $-(3^2)$. Correct$-3^2=-9$, while $(-3)^2=9$. Write the bracket whenever the negative is part of the base. Say this“The bracket decides whether the minus sign is part of the base.” CheckEvaluate $-2^4$ and $(-2)^4$. ($-16$ and $16$.)
10. “$(a+b)^2=a^2+b^2$.” Core and Extended Why wrongIndex laws describe products, quotients and powers — not sums. Test it: $(3+4)^2=49$ but $9+16=25$. CorrectExpand the bracket properly. The same warning applies to $\sqrt{a+b}\ne\sqrt a+\sqrt b$. Say this“The index laws apply to products and powers, never across a sum.” CheckIs $\sqrt{25+144}=5+12$? (No: $\sqrt{169}=13$, not 17.)
11. “A negative index means a negative answer.” Core and Extended Why wrongThe negative sign is instruction, not value: it says “take the reciprocal”. $2^{-3}$ is $\frac18$, which is positive. Correct$a^{-n}=\frac{1}{a^n}$. For a positive base the answer is always positive. Say this“A negative index means a reciprocal.” CheckEvaluate $5^{-2}$. ($\frac{1}{25}$.)
12. “$18\times10^{3}$ is in standard form.” Core and Extended Why wrongStandard form requires $1\le A<10$, and $18$ fails that. The number is right but the form is not, so the final accuracy mark is lost. Correct$18\times10^3=1.8\times10^1\times10^3=1.8\times10^4$. Say this“The coefficient must be at least 1 and less than 10, so I renormalise.” CheckWrite $0.42\times10^{6}$ in standard form. ($4.2\times10^{5}$.)
13. “$0.004786$ to 2 significant figures is $0.00$.” Core and Extended Why wrongThat is the answer to 2 decimal places. Significant figures start at the first non-zero digit, which here is the 4. Correct$0.0048$. The leading zeros stay because they hold the place value. Say this“Significant figures start counting at the first non-zero digit.” Check$0.0906$ to 2 s.f. ($0.091$.)
14. “A rounded measurement is an exact value.” Core and Extended Why wrong“$8.4$ cm to the nearest $0.1$ cm” describes any length from $8.35$ up to (but not including) $8.45$. Treating it as exactly $8.4$ makes every bound in the question wrong. CorrectWrite the interval first: $8.35\le L<8.45$. Note the asymmetry of $\le$ and $<$. Say this“The lower bound is included because it rounds to the stated value; the upper bound rounds to the next one.” CheckBounds of $240$ to the nearest 10. ($235\le x<245$.)
15. “The largest quotient uses both upper bounds.” Extended only Why wrongDividing by a larger number makes the result smaller. Upper over upper is a perfectly possible value, but it is not the maximum. CorrectLargest quotient $=\dfrac{\text{upper}}{\text{lower}}$; smallest quotient $=\dfrac{\text{lower}}{\text{upper}}$. The same asymmetry applies to differences. Say this“To make it as large as possible, divide as much as possible by as little as possible.” Check$d=150$ m (nearest 10 m), $t=20$ s (nearest 1 s). Upper bound of speed? ($155\div19.5=7.95$ m/s to 3 s.f.)
16. “$45\ \text{min}:2\ \text{h}$ simplifies to $45:2$.” Core and Extended Why wrongA ratio compares quantities in the same unit. Leaving mixed units compares minutes with hours and scales one side by 60. CorrectConvert first: $45:120=3:8$. Say this“Convert to a common unit before simplifying a ratio.” CheckSimplify $1.5\ \mathrm{kg}:400\ \mathrm{g}$. ($15:4$.)
17. “Average speed is the mean of the speeds.” Core and Extended Why wrongThe two legs usually take different times, so they carry different weights. $60$ km at $40$ km/h then $60$ km at $60$ km/h averages $48$ km/h, not $50$. CorrectAverage speed $=\dfrac{\text{total distance}}{\text{total time}}$. Find each leg's time first. Say this“Average speed is total distance over total time.” Check$30$ km at $60$ km/h then $30$ km at $90$ km/h. (72 km/h.)
18. “To reverse a $30\%$ discount, add $30\%$ back on.” Extended only Why wrongThe $30\%$ was a percentage of the original price, and the $30\%$ you would add back is a percentage of the sale price. Different bases, so the operations are not inverses. CorrectDivide by the multiplier: $84\div0.70=\$120$. Adding $30\%$ would give $\$109.20$. Say this“A reverse percentage divides by the multiplier.” CheckA price is $\$156$ after a $20\%$ rise. Original? ($156\div1.20=\$130$.)
19. “Three years of $15\%$ depreciation is $45\%$.” Extended only Why wrongEach year's $15\%$ is taken from a smaller amount than the year before, so the reductions cannot be added. CorrectMultiply: $0.85^3=0.614125$, a total fall of about $38.6\%$, not $45\%$. Say this“Repeated percentage changes multiply their multipliers.” CheckA $10\%$ rise then a $10\%$ fall. ($1.1\times0.9=0.99$: a $1\%$ net decrease.)
20. “$\sqrt{288}=16.97$ is a complete answer.” Extended only Why wrongWhen a question asks for an exact value, a decimal is an approximation of it, not equal to it. Correct$\sqrt{288}=\sqrt{144\times2}=12\sqrt2$. Keep fractions, surds and multiples of $\pi$ exact whenever exactness is asked for or the data is exact. Say this“The largest square factor of 288 is 144, so $\sqrt{288}=12\sqrt2$.” CheckSimplify $\sqrt{98}$. ($7\sqrt2$.)

Examiner tips

One rule that applies to every section below. Do the mathematics exactly for as long as you can, and round only on the final line. An intermediate value rounded to 3 significant figures and then reused will often push the final answer outside the accepted range, and the mark is lost even though the method was right.
Paper numbers are route labels, not difficulty labels. “Paper 1” means the Core non-calculator paper and “Paper 2” means the Extended non-calculator paper. When a past paper, a textbook or a teacher says “Paper 2”, check which qualification is meant before you assume it is yours.
Learn 64 twice. $64$ is both $8^2$ and $4^3$, so $\sqrt{64}=8$ while $\sqrt[3]{64}=4$. Reading the root sign carelessly here is a very cheap mark to lose.
Work in improper fractions. Multiplying or dividing mixed numbers directly does not work: $1\frac12\times2\frac12$ is $\frac32\times\frac52=\frac{15}{4}$, not $2\frac14$. Convert to improper form first, operate, then convert back only if the question asks for a mixed number.
Read the dots before you start. $0.\dot{1}2\dot{3}$ and $0.1\dot{2}\dot{3}$ differ only in where the first dot sits, and they are different numbers — $\frac{41}{333}$ and $\frac{61}{495}$. Write out three or four decimal places in full before choosing the power of 10 to multiply by. This chapter uses the overbar as well, purely because it is easier to read on screen; the two notations mean exactly the same thing.
The distinction between $<$ and $\le$ carries marks in 1.10. A bounds answer written $8.35<L<8.45$ loses the lower endpoint, which is attainable. The correct statement is $8.35\le L<8.45$.
Difference and quotient are the two that catch people out, because the “largest” answer mixes an upper with a lower. Say it as a sentence rather than memorising a formula: to make a difference as large as possible, start with as much as possible and take away as little as possible.
Minutes are not decimals of an hour. $3$ h $45$ min is $3.75$ h, not $3.45$ h. Minutes are sixtieths, not hundredths, so every mixed time has to be converted before it is substituted — and the same conversion reappears in 1.14 when reading a calculator display.
Cambridge writes 24-hour times without a colon — $03\,15$ and $15\,15$. Follow the notation used in the question. Whichever you use, always write four digits: “$3\,15$” is ambiguous.
Service D changes date. It leaves Greenhill at $22\,55$ and reaches Marston at $00\,04$ — the next day. Times in a timetable column always run forwards, so a time that appears to go backwards is telling you that midnight has been crossed.
Before you look. Give yourself 25 minutes and write full working, not just answers. Several of these carry method marks that survive an arithmetic slip — but only if the method is on the page.
Reading your Extended result. A missed question here points at exactly one Extended depth panel: question 1 at the three-set panel in 1.2, 2 at the recurring-decimal panel in 1.4, 3 at the fractional-index panel in 1.7, 4 at the calculated-bounds panel in 1.10, 5 and 6 at the reverse and repeated-change panel in 1.13, 7 at 1.17 and 8 at 1.18.
Carry unrounded values between parts. Several of these questions use an earlier answer in a later part. If you round part (a) and reuse the rounded value, part (b) will drift outside the accepted range.

Verified Curriculum Content

Every chapter note, MCQ explanation, and structured mark scheme is rigorously vetted by Cambridge curriculum specialists.

Chapter overview

A summary of this Mathematics chapter — open a section to read it. The full notes, worked examples and practice questions are in the study modules above.

What is Number about?

Key ideas to remember

Every other subtopic in Topic 1 is identical in both routes. Subtopics 1.1, 1.3, 1.5, 1.6, 1.8, 1.9, 1.11, 1.12, 1.14, 1.15 and 1.16 have Core and Extended statements that ask for the same mathematics, so eleven of the eighteen sections below carry both tags and need no route decision from you at all.
Coverage promise. Every one of the sixteen active Core subtopics and all eighteen Extended subtopics has its own teaching section below, with its own definition panel, method, fully checked worked example and quick check. Nothing in Topic 1 is covered by a heading alone, and no Core explanation depends on anything inside an Extended depth panel.
The pattern behind eight of the ten. Each one is a moment where a quantity was silently converted into something it is not: an interval into a value, an exact number into a decimal, a base into a different base, a rate into a mean. Before you write a final answer, name what kind of object it is.
The pattern. Fourteen of these twenty are the same underlying slip: an operation was applied to the wrong base or the wrong object — a percentage of the new value instead of the old, a cancel across a sum instead of a product, an index across a sum, a bound treated as a value. Before every line of working, name what you are operating on.
One habit above all. Whatever else you drop from this plan, keep this: work exactly for as long as you can, and round only once, at the end. It costs nothing, it applies to every chapter of the course, and it is the single largest source of avoidable lost marks in Topic 1.

What you need to be able to do

I can identify and use natural numbers, integers, prime numbers, square numbers, cube numbers, common factors, common multiples, rational and irrational numbers and reciprocals. Core C1.1
I can convert between numbers and words in both directions, reading and writing large numbers by grouping digits in threes. Core C1.1
I can express a number as a product of its prime factors, and use that form to find the HCF and the LCM of two numbers. Core C1.1
I can use set language and the notation $n(A)$, $A'$, $\xi$, $\cup$ and $\cap$ correctly. Core C1.2
I can describe a set by listing its elements and by a rule in set-builder notation. Core C1.2
I can draw, complete and read a two-set Venn diagram. Core C1.2
I can calculate squares, cubes, square roots, cube roots and other powers and roots, and I can recall the squares and square roots from 1 to 15 and the cubes and cube roots of 1, 2, 3, 4, 5 and 10. Core C1.3
I can use the language and notation of proper fractions, improper fractions, mixed numbers, decimals and percentages, and write a fraction in its simplest form. Core C1.4
I can recognise equivalence and convert between fractions, decimals and percentages in every direction. Core C1.4
I can add, subtract, multiply and divide fractions and mixed numbers, cancelling only common factors. Core C1.6
I can order quantities by magnitude and use $=$, $\ne$, $<$, $>$, $\le$ and $\ge$ correctly, including with negative numbers. Core C1.5
I can use the four operations on integers, fractions and decimals with brackets and the correct order of operations, including signed arithmetic and practical situations such as a temperature change. Core C1.6
I can use positive, zero and negative integer indices. Core C1.7
I can apply the rules of indices $a^ma^n=a^{m+n}$, $a^m\div a^n=a^{m-n}$ and $(a^m)^n=a^{mn}$. Core C1.7
I can use the standard form $A\times10^n$ with $1\le A<10$ and $n$ an integer, convert into and out of it, and calculate with values in standard form. Core C1.8
I can round a value to a specified degree of accuracy, in decimal places or significant figures. Core C1.9
I can make an estimate for a calculation by rounding each number to 1 significant figure, and round a final answer to a reasonable degree of accuracy for the context. Core C1.9
I can give the upper and lower bounds of data rounded to a specified accuracy, using $\le$ for the lower endpoint and $<$ for the upper. Core C1.10
I can give a ratio in its simplest form and divide a quantity in a given ratio. Core C1.11
I can use proportional reasoning and ratios in context — adapting a recipe, using a map scale and determining best value. Core C1.11
I can use common measures of rate, apply other measures of rate from a formula given in the question, and solve problems involving average speed. Core C1.12
I can calculate a given percentage of a quantity, express one quantity as a percentage of another, and calculate a percentage increase or decrease. Core C1.13
I can handle deposit, discount, profit and loss, earnings and percentages over $100\%$. Core C1.13
I can calculate with simple interest and with compound interest, from memory — no formula is given. Core C1.13
I can use a calculator efficiently, enter values appropriately, and interpret the display in context without rounding inside the calculation. Core C1.14
I can calculate with time in the 12-hour and 24-hour systems, read clocks and timetables, and solve problems involving time zones, local times and time differences. Core C1.15
I can calculate with money and convert from one currency to another in the correct direction. Core C1.16
I can use Venn diagrams for two or three sets and to represent relationships between sets, and I can use the notation $\in$, $\notin$, $\varnothing$, $\subseteq$ and $\nsubseteq$ as well as the Core notation. Extended E1.2
I can read and write recurring-decimal dot notation, and convert between a recurring decimal and a fraction in both directions, including a decimal with a non-recurring prefix. Extended E1.4
I can use fractional indices, and evaluate expressions such as $81^{1/2}$, $27^{2/3}$ and $8^{-2/3}$ without a calculator. Extended E1.7
I can find the upper and lower bounds of a result calculated from rounded data — a perimeter, an area, or a speed from a rounded distance and time. Extended E1.10
I can calculate using reverse percentages, and with repeated percentage change. Extended E1.13
I can use exponential growth and decay, including depreciation and population change. Knowledge of $e$ is not required. Extended E1.17
I can understand and use surds, simplify surd expressions, and rationalise a denominator. Extended E1.18

Why Number matters

Common mistakes to avoid

1. “1 is a prime number.” Core and Extended Why wrongA prime has exactly two positive factors. The only positive factor of 1 is 1 itself, so it has one. Correct1 is neither prime nor composite. Prime factorisations therefore never contain a 1, and 2 is the smallest prime. Say this“A prime number has exactly two positive factors, so 1 is not prime.” CheckList the primes below 12. (2, 3, 5, 7, 11.)
2. “The HCF takes the smaller power of every prime you can see.” Core and Extended Why wrongFor $72=2^3\times3^2$ and $90=2\times3^2\times5$, that rule would include the 5 and give 90 — which does not divide 72 at all. CorrectOnly primes appearing in both factorisations go into the HCF. The LCM, by contrast, takes every prime present. Say this“The common primes are 2 and 3 at their lower powers, so the HCF is $2\times3^2=18$.” CheckHCF of $2^4\times5$ and $2^2\times3\times5^2$. ($2^2\times5=20$.)
3. “$n(A)=22$ means 22 students study only Art.” Core and Extended Why wrong$n(A)$ counts the whole circle, including the overlap. Putting 22 in the Art-only region double-counts everyone in the intersection. CorrectEnter the intersection first, then subtract it from each set total to obtain the “only” regions. Say this“Art only is $n(A)-n(A\cap B)$.” Check$n(A)=31$, $n(A\cap B)=15$. How many are in $A$ only? (16.)
4. “$\sqrt{64}=4$, because $4^3=64$.” Core and Extended Why wrong$64$ is both $8^2$ and $4^3$. The index on the root sign decides which fact is needed, and a plain $\sqrt{\phantom{x}}$ means a square root. Correct$\sqrt{64}=8$ and $\sqrt[3]{64}=4$. Say this“A bare root sign is a square root.” CheckEvaluate $\sqrt{729}$ and $\sqrt[3]{729}$. (27 and 9.)
5. “$\frac12+\frac13=\frac25$.” Core and Extended Why wrongAdding numerators and denominators is not addition of fractions. $\frac25=0.4$, but the true sum is $0.8\overline3$ — the “answer” is smaller than one of the things being added. CorrectRewrite over a common denominator: $\frac36+\frac26=\frac56$. Say this“Fractions are added over a common denominator; only the numerators combine.” Check$\frac34+\frac25$. ($\frac{15}{20}+\frac{8}{20}=\frac{23}{20}$.)
6. “In $\dfrac{3+x}{3}$ the threes cancel.” Core and Extended Why wrongThe 3 on top is a term, not a factor of the whole numerator. Test with $x=6$: the true value is $3$, not $6$. CorrectCancel only a factor of the entire numerator against a factor of the entire denominator. $\frac{3(1+x)}{3}$ does cancel, to $1+x$. Say this“Cancel factors, never terms joined by addition.” CheckSimplify $\dfrac{5x+10}{5}$. ($x+2$, after factorising the numerator as $5(x+2)$.)
7. “$-7$ is bigger than $-3$ because 7 is bigger than 3.” Core and Extended Why wrongSize and position are different things. On the number line $-7$ is further left, and further left always means smaller. Correct$-7<-3$. Among negatives, the larger the digits the smaller the number. Say this“The smaller number is the one further to the left on the number line.” CheckOrder $-2,-9,0,-5$ smallest first. ($-9,-5,-2,0$.)
8. “$24\div6\times2=2$, because multiplication comes before division.” Core and Extended Why wrongMultiplication and division share one level of precedence. They are performed in the order they appear, left to right. Correct$24\div6=4$, then $4\times2=8$. The same applies to addition and subtraction: $10-4+3=9$. Say this“Multiplication and division are equal in rank and taken left to right.” Check$36\div9\times3$. (12, not 1.)
9. “$-3^2=9$.” Core and Extended Why wrongWithout a bracket, the index binds only to the 3, so the expression means $-(3^2)$. Correct$-3^2=-9$, while $(-3)^2=9$. Write the bracket whenever the negative is part of the base. Say this“The bracket decides whether the minus sign is part of the base.” CheckEvaluate $-2^4$ and $(-2)^4$. ($-16$ and $16$.)
10. “$(a+b)^2=a^2+b^2$.” Core and Extended Why wrongIndex laws describe products, quotients and powers — not sums. Test it: $(3+4)^2=49$ but $9+16=25$. CorrectExpand the bracket properly. The same warning applies to $\sqrt{a+b}\ne\sqrt a+\sqrt b$. Say this“The index laws apply to products and powers, never across a sum.” CheckIs $\sqrt{25+144}=5+12$? (No: $\sqrt{169}=13$, not 17.)
11. “A negative index means a negative answer.” Core and Extended Why wrongThe negative sign is instruction, not value: it says “take the reciprocal”. $2^{-3}$ is $\frac18$, which is positive. Correct$a^{-n}=\frac{1}{a^n}$. For a positive base the answer is always positive. Say this“A negative index means a reciprocal.” CheckEvaluate $5^{-2}$. ($\frac{1}{25}$.)
12. “$18\times10^{3}$ is in standard form.” Core and Extended Why wrongStandard form requires $1\le A<10$, and $18$ fails that. The number is right but the form is not, so the final accuracy mark is lost. Correct$18\times10^3=1.8\times10^1\times10^3=1.8\times10^4$. Say this“The coefficient must be at least 1 and less than 10, so I renormalise.” CheckWrite $0.42\times10^{6}$ in standard form. ($4.2\times10^{5}$.)
13. “$0.004786$ to 2 significant figures is $0.00$.” Core and Extended Why wrongThat is the answer to 2 decimal places. Significant figures start at the first non-zero digit, which here is the 4. Correct$0.0048$. The leading zeros stay because they hold the place value. Say this“Significant figures start counting at the first non-zero digit.” Check$0.0906$ to 2 s.f. ($0.091$.)
14. “A rounded measurement is an exact value.” Core and Extended Why wrong“$8.4$ cm to the nearest $0.1$ cm” describes any length from $8.35$ up to (but not including) $8.45$. Treating it as exactly $8.4$ makes every bound in the question wrong. CorrectWrite the interval first: $8.35\le L<8.45$. Note the asymmetry of $\le$ and $<$. Say this“The lower bound is included because it rounds to the stated value; the upper bound rounds to the next one.” CheckBounds of $240$ to the nearest 10. ($235\le x<245$.)
15. “The largest quotient uses both upper bounds.” Extended only Why wrongDividing by a larger number makes the result smaller. Upper over upper is a perfectly possible value, but it is not the maximum. CorrectLargest quotient $=\dfrac{\text{upper}}{\text{lower}}$; smallest quotient $=\dfrac{\text{lower}}{\text{upper}}$. The same asymmetry applies to differences. Say this“To make it as large as possible, divide as much as possible by as little as possible.” Check$d=150$ m (nearest 10 m), $t=20$ s (nearest 1 s). Upper bound of speed? ($155\div19.5=7.95$ m/s to 3 s.f.)
16. “$45\ \text{min}:2\ \text{h}$ simplifies to $45:2$.” Core and Extended Why wrongA ratio compares quantities in the same unit. Leaving mixed units compares minutes with hours and scales one side by 60. CorrectConvert first: $45:120=3:8$. Say this“Convert to a common unit before simplifying a ratio.” CheckSimplify $1.5\ \mathrm{kg}:400\ \mathrm{g}$. ($15:4$.)
17. “Average speed is the mean of the speeds.” Core and Extended Why wrongThe two legs usually take different times, so they carry different weights. $60$ km at $40$ km/h then $60$ km at $60$ km/h averages $48$ km/h, not $50$. CorrectAverage speed $=\dfrac{\text{total distance}}{\text{total time}}$. Find each leg's time first. Say this“Average speed is total distance over total time.” Check$30$ km at $60$ km/h then $30$ km at $90$ km/h. (72 km/h.)
18. “To reverse a $30\%$ discount, add $30\%$ back on.” Extended only Why wrongThe $30\%$ was a percentage of the original price, and the $30\%$ you would add back is a percentage of the sale price. Different bases, so the operations are not inverses. CorrectDivide by the multiplier: $84\div0.70=\$120$. Adding $30\%$ would give $\$109.20$. Say this“A reverse percentage divides by the multiplier.” CheckA price is $\$156$ after a $20\%$ rise. Original? ($156\div1.20=\$130$.)
19. “Three years of $15\%$ depreciation is $45\%$.” Extended only Why wrongEach year's $15\%$ is taken from a smaller amount than the year before, so the reductions cannot be added. CorrectMultiply: $0.85^3=0.614125$, a total fall of about $38.6\%$, not $45\%$. Say this“Repeated percentage changes multiply their multipliers.” CheckA $10\%$ rise then a $10\%$ fall. ($1.1\times0.9=0.99$: a $1\%$ net decrease.)
20. “$\sqrt{288}=16.97$ is a complete answer.” Extended only Why wrongWhen a question asks for an exact value, a decimal is an approximation of it, not equal to it. Correct$\sqrt{288}=\sqrt{144\times2}=12\sqrt2$. Keep fractions, surds and multiples of $\pi$ exact whenever exactness is asked for or the data is exact. Say this“The largest square factor of 288 is 144, so $\sqrt{288}=12\sqrt2$.” CheckSimplify $\sqrt{98}$. ($7\sqrt2$.)

Examiner tips

One rule that applies to every section below. Do the mathematics exactly for as long as you can, and round only on the final line. An intermediate value rounded to 3 significant figures and then reused will often push the final answer outside the accepted range, and the mark is lost even though the method was right.
Paper numbers are route labels, not difficulty labels. “Paper 1” means the Core non-calculator paper and “Paper 2” means the Extended non-calculator paper. When a past paper, a textbook or a teacher says “Paper 2”, check which qualification is meant before you assume it is yours.
Learn 64 twice. $64$ is both $8^2$ and $4^3$, so $\sqrt{64}=8$ while $\sqrt[3]{64}=4$. Reading the root sign carelessly here is a very cheap mark to lose.
Work in improper fractions. Multiplying or dividing mixed numbers directly does not work: $1\frac12\times2\frac12$ is $\frac32\times\frac52=\frac{15}{4}$, not $2\frac14$. Convert to improper form first, operate, then convert back only if the question asks for a mixed number.
Read the dots before you start. $0.\dot{1}2\dot{3}$ and $0.1\dot{2}\dot{3}$ differ only in where the first dot sits, and they are different numbers — $\frac{41}{333}$ and $\frac{61}{495}$. Write out three or four decimal places in full before choosing the power of 10 to multiply by. This chapter uses the overbar as well, purely because it is easier to read on screen; the two notations mean exactly the same thing.
The distinction between $<$ and $\le$ carries marks in 1.10. A bounds answer written $8.35<L<8.45$ loses the lower endpoint, which is attainable. The correct statement is $8.35\le L<8.45$.
Difference and quotient are the two that catch people out, because the “largest” answer mixes an upper with a lower. Say it as a sentence rather than memorising a formula: to make a difference as large as possible, start with as much as possible and take away as little as possible.
Minutes are not decimals of an hour. $3$ h $45$ min is $3.75$ h, not $3.45$ h. Minutes are sixtieths, not hundredths, so every mixed time has to be converted before it is substituted — and the same conversion reappears in 1.14 when reading a calculator display.
Cambridge writes 24-hour times without a colon — $03\,15$ and $15\,15$. Follow the notation used in the question. Whichever you use, always write four digits: “$3\,15$” is ambiguous.
Service D changes date. It leaves Greenhill at $22\,55$ and reaches Marston at $00\,04$ — the next day. Times in a timetable column always run forwards, so a time that appears to go backwards is telling you that midnight has been crossed.
Before you look. Give yourself 25 minutes and write full working, not just answers. Several of these carry method marks that survive an arithmetic slip — but only if the method is on the page.
Reading your Extended result. A missed question here points at exactly one Extended depth panel: question 1 at the three-set panel in 1.2, 2 at the recurring-decimal panel in 1.4, 3 at the fractional-index panel in 1.7, 4 at the calculated-bounds panel in 1.10, 5 and 6 at the reverse and repeated-change panel in 1.13, 7 at 1.17 and 8 at 1.18.
Carry unrounded values between parts. Several of these questions use an earlier answer in a later part. If you round part (a) and reuse the rounded value, part (b) will drift outside the accepted range.

Number

Core Revision Module

Revision & Practice Book

Practice & Resources

Chapter overview

What is Number about?

Key ideas to remember

What you need to be able to do

Why Number matters

Common mistakes to avoid

Examiner tips

Academiq Edu Review Team

Number

Core Revision Module

Revision & Practice Book

Practice & Resources

Chapter overview

What is Number about?

Key ideas to remember

What you need to be able to do

Why Number matters

Common mistakes to avoid

Examiner tips

Academiq Edu Review Team