1 Number

Learning objectives

1A Ordinary number operationsThe ordinary processes of number manipulation The ‘four operations’ and combination of them by use of brackets 1B Prime numbers, factors and multiplesPrime numbers, factors, multiples To include finding HCF and LCM in simple cases 1C Indices, powers and rootsIndices, powers and roots Use index notation and index laws for multiplication and division involving integer, fractional and negative powers 1D SurdsSimple manipulation of surds Students should understand what surds represent and their use for exact answers Manipulation will be simple For example: 5√3 + 2√3 = 7√3; √48 = 4√3; 10 × 1/√5 = 2√5 1E Rationalising denominatorsRationalising the denominator For example: 15/(√7 − 2)1F Number setsNatural numbers, integers and rational and irrational numbers Recognitions of these sets Proofs of irrationality will not be required 1G Units, measures and moneyWeights, measures and money Carry out calculations using standard units of mass, length, area, volume and capacity, time and average speed Metric and SI units only Carry out calculations using money, including converting between currencies (where conversion is required, the rate of conversion will always be given)1H Fractions, decimals, ratio and percentageFractions, decimals, ratio, proportion and percentage Students will be expected to interchange any of these methods of fractional representation and to select the most appropriate to given situations Ratios and proportions are required in, at most, three proportions, i.e. a : b or a : b : c Students will be expected to use the four operations with fractions and decimals, and use percentages, ratio and/or proportion in problems 1I AccuracyExpressing numbers to a given degree of accuracy Correction to a given number of decimal places or significant figures 1J Upper and lower boundsSolve problems using upper and lower bounds where values are given to a degree of accuracy 1K Standard formNumbers in standard form a × 10^n, where n is an integer and 1 ≤ a < 10 Solve problems involving standard form Questions may involve the application of any of the techniques listed in 1 to problems of everyday personal, domestic or community life

When several operations appear together, their priority fixes the meaning: evaluate brackets first, then powers or roots, then multiplication and division, then addition and subtraction. Operations at the same priority are completed from left to right.

18-3(4-1)+8\div2=18-9+4=13

Work one priority level at a time and rewrite the whole expression after each step. A minus sign attached to a number is part of that number, so $(-3)^2=9$ but $-3^2=-(3^2)=-9$ .

Do not automatically do multiplication before division, or addition before subtraction: each pair has equal priority, so read from left to right. Enter brackets explicitly on a calculator when the expression contains them.

A prime number has exactly two positive factors, $1$ and itself. Writing a positive integer as a product of primes exposes every factor and makes highest common factor (HCF) and lowest common multiple (LCM) systematic.

Task	Prime-factor choice
HCF	take each common prime with the smaller exponent
LCM	take every prime present with the larger exponent

28=2^2 imes7,\quad120=2^3 imes3 imes5[2pt]\operatorname{HCF}=2^2=4,\quad\operatorname{LCM}=2^3 imes3 imes5 imes7=840

HCF must divide both numbers; LCM must be a multiple of both. The number $1$ is neither prime nor composite.

Form	Result	Condition
$a^m a^n$	$a^{m+n}$	same base
$a^m\div a^n$	$a^{m-n}$	same non-zero base
$(a^m)^n$	$a^{mn}$	multiply the indices
$a^0$	$1$	$a
e0$
$a^{-n}$	$1/a^n$	$a
e0$
$a^{p/q}$	$\sqrt[q]{a^p}$	use a real root in the stated domain

16^{3/4}=(\sqrt[4]{16})^3=2^3=8,\qquad rac{x^5}{x^{-2}}=x^7;(x
e0)

Multiplication combines repeated factors, so indices add; division cancels factors, so indices subtract. A negative index means reciprocal, not a negative value.

Index laws combine powers only when the bases match. In particular, $a^m+a^n$ cannot usually be replaced by one power.

A surd is an irrational root kept in exact form. Simplify it by extracting the largest square factor; only like surds can then be added or subtracted.

\sqrt{48}=\sqrt{16 imes3}=4\sqrt3,\qquad5\sqrt3+2\sqrt3=7\sqrt3

Products use $\sqrt a\sqrt b=\sqrt{ab}$ when the roots are defined. For example, $(2\sqrt3)(3\sqrt6)=6\sqrt{18}=18\sqrt2$ .

Do not split addition inside a root: $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ . Decimal approximations lose the exact value, so keep the surd until an approximation is requested.

Rationalising removes a surd from the denominator without changing the value. For a binomial denominator, multiply numerator and denominator by its conjugate: keep the terms but reverse the sign between them.

(a-b)(a+b)=a^2-b^2

rac{15}{\sqrt7-2} imesrac{\sqrt7+2}{\sqrt7+2}=rac{15(\sqrt7+2)}{7-4}=5(\sqrt7+2)

The conjugate is used because the middle surd terms cancel. Multiply the entire numerator and denominator, and simplify only after checking that the original denominator is non-zero.

Set	Meaning	Examples
$\mathbb N$	natural counting numbers	$1,2,3,\ldots$ ; follow the question's convention about $0$
$\mathbb Z$	integers	$\ldots,-2,-1,0,1,2,\ldots$
$\mathbb Q$	numbers expressible as $p/q$ , $q
e0 $\|$ -3 $,$ 2/5 $,$ 0.\overline7$
irrational	real but not rational	$\sqrt2$ , $\pi$

\mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R,\qquad ext{irrationals}=\mathbb R\setminus\mathbb Q

Terminating and recurring decimals are rational. Simplify an expression before classifying it: for example, $\sqrt{20}/\sqrt5=\sqrt4=2$ is natural, integer and rational.

A root symbol does not automatically make a number irrational: $\sqrt{49}=7$ . Proofs of irrationality are not required here.

Choose units that match the quantity, write the given conversion as an equality, and multiply by a conversion factor equal to $1$ . For area or volume, the linear conversion factor must be squared or cubed.

Quantity	Useful metric/SI link
length	$1 ext{ m}=100 ext{ cm}$
area	$1 ext{ m}^2=10\,000 ext{ cm}^2$
volume/capacity	$1 ext{ m}^3=1000 ext{ L}$ ; $1 ext{ L}=1000 ext{ cm}^3$
mass	$1 ext{ kg}=1000 ext{ g}$
time	$1 ext{ h}=3600 ext{ s}$

ext{average speed}=rac{ ext{total distance}}{ ext{total time}},\qquad2.4 ext{ m}^2=24,000 ext{ cm}^2

For currency, use the rate exactly as stated. If $1 ext{ GBP}=1.18 ext{ EUR}$ , multiply pounds by $1.18$ to obtain euros and divide euros by $1.18$ to obtain pounds. Round money only at the requested stage.

Average speed is based on total distance and total time; it is not normally the mean of separate speeds. Two incomplete zero-mark Question Bank fragments were excluded from the evidence.

Form	Conversion/use
fraction $a/b$	exact part of a whole; use common denominators for addition/subtraction
decimal	divide numerator by denominator; convenient for calculation
percentage	multiply a proportion by $100\%$ ; useful for comparison
ratio	compares parts in the same units; simplify by a common factor

rac ab\divrac cd=rac ab imesrac dc\quad(c,d
e0),\qquad ext{percentage multiplier}=1\pmrac r{100}

To divide $£420$ in the ratio $2:3:5$ , total the shares: $2+3+5=10$ . One share is $£42$ , so the amounts are $£84$ , $£126$ and $£210$ .

A ratio gives relative shares, not the actual total. For repeated percentage change, apply the multiplier each time; adding the percentages ignores the changed starting amount. Ratio and proportion questions use at most three parts in this syllabus.

Accuracy	Where counting starts	Example
decimal places (dp)	first digit after the decimal point	$18.376$ to $2$ dp is $18.38$
significant figures (sf)	first non-zero digit	$0.004786$ to $3$ sf is $0.00479$

Locate the final digit to keep, inspect the next digit, then increase the kept digit by $1$ if the next digit is $5$ or more. Replace omitted whole-number digits with zeros when needed to preserve place value.

Leading zeros are not significant, but zeros between non-zero digits are. Keep extra calculator digits during working and round the final answer unless the question directs otherwise.

x ext{ rounded to step }u ext{ as }r\quad\Longrightarrow\quad r-rac u2\le x<r+rac u2

Choose the combination that makes the required result largest or smallest. For positive quantities: an upper sum uses upper bounds; an upper difference uses the first upper and second lower; an upper product uses upper bounds; an upper quotient uses an upper numerator and lower denominator.

A_{\max}=(610.5)(155.5-75.5)=(610.5)(80)=48,840 ext{ cm}^2

Here $610$ , $155$ and $76$ are each correct to the nearest centimetre. The net height is a subtraction, so its upper bound uses $155.5-75.5$ , not $155.5-76.5$ .

A rounded value is not itself an upper or lower bound. State the interval first, and reconsider the extremum rule if a quantity can be negative.

a imes10^n,\qquad1\le a<10,\quad n\in\mathbb Z

Move the decimal point to make a coefficient between $1$ and $10$ ; the number of places moved gives the power of $10$ . Positive indices represent large place values and negative indices represent small positive place values.

Operation	Method
multiply	multiply coefficients and add indices
divide	divide coefficients and subtract indices
add/subtract	first rewrite with the same power of $10$

(3.6 imes10^7)(2 imes10^{-3})=7.2 imes10^4,\qquad 4 imes10^{100}+3.6 imes10^{101}=4.0 imes10^{101}

Normalise the final coefficient: $36 imes10^5=3.6 imes10^6$ is standard form, but $36 imes10^5$ is not. Preserve units and round only when requested.

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1 Number

Learning objectives

Calculate accurately with the four operations

Use prime factors to find HCF and LCM

Apply index laws to powers and roots

Keep surds exact and simplify them

Rationalise a surd denominator

Recognise natural, integer, rational and irrational numbers

Convert units and money without changing the quantity

Choose between fractions, decimals, ratios and percentages

Round to decimal places or significant figures

Choose upper and lower bounds for a calculation

Calculate with numbers in standard form

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