EduNinja

1.1 Integers

An integer is a whole number: it can be positive, negative or zero. Examples are $-12$ , $0$ and $37$ . Numbers with a fractional part, such as $4.5$ or $\frac{2}{3}$ , are not integers.

Integers extend without end in both directions. On a number line, positive integers lie to the right of zero and negative integers lie to the left. The sign tells which side of zero the number is on; zero is neither positive nor negative.

Relationship	Example
opposite integers are equally far from zero	$-6$ and $6$
absolute value is distance from zero	$\|-6\|=6$
a minus sign is part of a negative number	$-9$ is nine below zero

Do not confuse a negative integer with a subtraction instruction. In $-5$ , the sign describes the number; in $8-5$ , the symbol tells you to subtract.

A digit's place determines its value. Moving one place left multiplies its value by $10$ ; moving one place right divides its value by $10$ .

Digit in $406\,072$	Place	Value
$4$	hundred-thousands	$400\,000$
$6$	thousands	$6\,000$
$7$	tens	$70$
$2$	ones	$2$

Zero can hold an empty place. In $406\,072$ , the zero in the ten-thousands place prevents the $6$ from being read as sixty thousand, and the zero in the hundreds place keeps $72$ in the final two places.

To write a number from words, place each stated value in its column and insert zeros where a place is missing. For example, six thousand and seventy-nine is $6079$ , not $679$ .

The digit and its value are different: the digit $3$ in $11\,377$ has value $300$ , not $3$ .

A directed number combines a size with a direction from a reference point. Positive and negative signs can represent above or below zero, gain or loss, credit or debt, and movement in opposite directions.

Context	Positive direction	Negative direction
temperature	above $0^\circ$ C	below $0^\circ$ C
elevation	above reference level	below reference level
money balance	credit	debt

A change is found by final value minus initial value. If a temperature rises from $-4^\circ$ C to $3^\circ$ C, the change is $3-(-4)=+7^\circ$ C. The positive answer records a rise.

A difference is a non-negative distance between values. The difference between $-6^\circ$ C and $5^\circ$ C is $11^\circ$ C because the interval crosses zero: $6+5=11$ .

Always define what the sign means in the context. A negative temperature is below the chosen zero; it does not mean that the size of the temperature is invalid.

Integers increase from left to right on a number line. Therefore a number farther right is greater, and a number farther left is smaller.

Among negative integers, the number closer to zero is greater. For example, $-3>-8$ because $-3$ lies to the right of $-8$ .

To order several integers from smallest to largest: place the most negative values first, then zero, then positive values. Within the negatives, larger absolute value means smaller number.

For $-7, 3, -5, -9, 0, 1$ , the ascending order is $-9,-7,-5,0,1,3$ . The reverse list gives descending order.

$-9$ is smaller than $-5$ even though $9>5$ . Comparing only the digits ignores the negative signs.

Addition combines amounts, subtraction finds a change or difference, multiplication combines equal groups, and division shares or finds how many groups fit. With integers, the operation and the signs both affect the result.

Operation	Reliable sign rule	Example
add	same signs: add sizes and keep the sign; different signs: subtract sizes and keep the sign of the larger size	$-9+15=6$
subtract	add the opposite	$-9-(-15)=-9+15=6$
multiply/divide	same signs give positive; different signs give negative	$6\times(-8)=-48$ , $(-64)\div(-4)=16$

First identify the operation, then handle the signs, calculate the unsigned values, and check whether the answer's sign is sensible. Use the inverse operation to check: $-48\div6=-8$ checks $6\times(-8)=-48$ .

Keep units in contextual calculations. If one tunnel is $15\,516$ m long and another is $8\,820$ m long, the difference is $15\,516-8\,820=6\,696$ m.

Division by zero is undefined. Also, two negative signs do not always make a positive: that rule applies to multiplication, division, or subtracting a negative—not to every pair of minus signs.

The hierarchy of operations makes one expression have one agreed value. Brackets can change that value by making a chosen part happen first.

Priority	Operations
1	brackets, working from inner to outer
2	powers
3	multiplication and division, left to right
4	addition and subtraction, left to right

For $6^2+2^3\times5$ , calculate powers first: $36+8\times5$ . Then multiply: $36+40$ . Finally add, giving $76$ .

Brackets can make an intended equality true. In $25+3\times(7-2)$ , the bracket gives $25+3\times5=40$ . Without it, multiplication occurs before subtraction and the value is different.

Multiplication is not automatically before division, and addition is not automatically before subtraction. Operations at the same priority are completed from left to right.

These terms classify integers by divisibility. A factor divides a number exactly; a multiple is produced by multiplying a number by an integer.

Term	Test	Example
even	divisible by $2$	$18$
odd	not divisible by $2$	$19$
prime	positive integer greater than $1$ with exactly two positive factors: $1$ and itself	$19$
factor of $n$	divides $n$ with no remainder	$24$ is a factor of $72$
multiple of $n$	equals $n\times$ an integer	$42$ is a multiple of $7$

$2$ is the only even prime. Every prime greater than $2$ is odd, but not every odd number is prime: $9$ is odd and has factors $1,3,9$ .

Factor and multiple statements reverse: if $6$ is a factor of $42$ , then $42$ is a multiple of $6$ .

$1$ is neither prime nor composite because it has only one positive factor. Also, a number has finitely many positive factors but infinitely many positive multiples.

A prime factor is a factor that is also prime. The positive factors of $18$ are $1,2,3,6,9,18$ , so its prime factors are $2$ and $3$ .

Job	Method	Example for $12$ and $18$
common factors	list factors of each number and take the overlap	$1,2,3,6$
common multiples	list multiples of each number and take the overlap	$36,72,108,\ldots$

A proposed common factor must divide every given number exactly. A proposed common multiple must be divisible by every given number. For example, $6$ divides both $12$ and $18$ , while $36$ is divisible by both.

Prime factors, common factors and common multiples are different sets. This objective asks you to identify them; finding the highest common factor or lowest common multiple is a later step.

1.2 Fractions

Equivalent fractions name the same proportion. Multiplying or dividing the numerator and denominator by the same non-zero number changes the parts used to name the proportion, but not its value.

Move	Example	Why it is equivalent
multiply top and bottom by $3$	$\frac{2}{5}=\frac{6}{15}$	each fifth is split into three equal parts
divide top and bottom by $4$	$\frac{8}{60}=\frac{2}{15}$	groups of four parts are combined

A fraction is in simplest form, or lowest terms, when numerator and denominator have no common factor greater than $1$ . Cancel common factors until none remain: $\frac{18}{24}=\frac{3}{4}$ after dividing both by $6$ .

Cross-products can check equivalence: $\frac{a}{b}=\frac{c}{d}$ when $ad=bc$ , provided $b$ and $d$ are non-zero. For $\frac{2}{5}$ and $\frac{6}{15}$ , both cross-products are $30$ .

Never cancel across addition or subtraction, and never change only one part of a fraction. For example, $\frac{8}{60}\ne\frac{8}{15}$ .

A common (vulgar) fraction has the form $\frac{a}{b}$ with $b\ne0$ . A proper fraction has numerator smaller than denominator; an improper fraction has numerator at least as large. A mixed number combines a whole number and a proper fraction.

Conversion	Method	Example
improper to mixed	divide numerator by denominator; quotient is the whole part and remainder is the new numerator	$\frac{11}{4}=2\frac{3}{4}$
mixed to improper	whole $\times$ denominator $+$ numerator; keep the denominator	$3\frac{2}{5}=\frac{17}{5}$

Both forms represent the same value. Since $\frac{11}{4}$ contains two complete groups of four quarters with three quarters left, it equals $2\frac34$ .

Simplify the fractional part when needed. For example, $2\frac{6}{8}=2\frac34$ .

A mixed number means addition: $2\frac34=2+\frac34$ . It does not mean $2\times\frac34$ .

A common denominator is a number that can be used as the denominator of two or more equivalent fractions. It must be a common multiple of the original denominators.

Step	Example for $\frac23$ and $\frac57$
list or recognise a common multiple of $3$ and $7$	$21$
scale each fraction to that denominator	$\frac23=\frac{14}{21}$ ; $\frac57=\frac{15}{21}$
check each numerator was multiplied by the same factor as its denominator	$2\times7=14$ ; $5\times3=15$

Any common multiple gives a valid common denominator, so $42$ would also work. The smallest convenient choice usually keeps the arithmetic shorter.

Common denominators express fractions in equal-sized parts. Once the parts have the same size, their numerators can be compared, added or subtracted meaningfully.

Do not add or multiply the denominators to each other without scaling the numerators. Changing a denominator alone changes the fraction's value.

A fraction describes a proportion, so the same fraction reasoning can compare proportions and scale a quantity. First identify whether the job is to compare values or to take a stated part of an amount.

Job	Reliable method	Worked example
order fractions	rewrite them with a common denominator or common decimal form	$\frac12=\frac{20}{40}$ , $\frac7{10}=\frac{28}{40}$ , $\frac{29}{40}$ , $\frac45=\frac{32}{40}$ , so this is ascending order
find $\frac{a}{b}$ of a quantity	divide by $b$ , then multiply by $a$	$\frac38$ of $240$ kg: $240\div8\times3=90$ kg

When fractions have the same positive denominator, compare numerators. When they have the same positive numerator, the fraction with the smaller denominator is larger because each part is larger.

A proper fraction of a positive quantity must be smaller than the original quantity. An ordering can be checked by estimating against useful benchmarks such as $0$ , $\frac12$ and $1$ .

Do not compare only denominators: $\frac45>\frac7{10}$ even though $5<10$ . The numerator and denominator work together to determine value.

To express a quantity $A$ as a fraction of quantity $B$ , write $\frac{A}{B}$ and simplify. The phrase order matters: the quantity after “of” becomes the denominator.

Step	Example: express $30$ as a fraction of $48$
put the first quantity over the second	$\frac{30}{48}$
make units the same if necessary	both are already counts
cancel common factors	$\frac{30\div6}{48\div6}=\frac58$

For a part of a group, denominator is the total group. If $19$ of $403$ people are crew, the fraction who are crew is $\frac{19}{403}$ .

The result need not be proper. Expressing $12$ as a fraction of $8$ gives $\frac{12}{8}=\frac32$ , because the first quantity is larger than the second.

Never compare quantities with different units until they are converted to the same unit. Also, reversing the order answers a different question: $\frac{30}{48}\ne\frac{48}{30}$ .

Fractions can be added or subtracted only when they name equal-sized parts. Create a common denominator, combine the numerators, keep the common denominator, then simplify.

For $\frac23+\frac57$ , use denominator $21$ : $\frac{14}{21}+\frac{15}{21}=\frac{29}{21}=1\frac{8}{21}$ .

Mixed-number method	Example $3\frac15-2\frac23$
convert to improper fractions	$\frac{16}{5}-\frac83$
use a common denominator	$\frac{48}{15}-\frac{40}{15}$
subtract and simplify	$\frac8{15}$

You may work with whole and fractional parts separately, but regroup one whole when the first fractional part is too small to subtract. Converting to improper fractions avoids that hidden borrowing step.

Do not add or subtract denominators: $\frac13+\frac14$ is not $\frac27$ . Thirds and quarters must first be renamed as equal-sized parts.

A fraction, decimal and percentage can name the same proportion. To convert $\frac{a}{b}$ to a decimal, calculate $a\div b$ . To convert it to a percentage, multiply the decimal by $100\%$ .

Fraction	Decimal	Percentage
$\frac35$	$3\div5=0.6$	$60\%$
$\frac4{9}$	$0.4444\ldots$	$44.4444\ldots\%$
$\frac6{20}$	$0.3$	$30\%$

If an equivalent fraction has denominator $100$ , its numerator is the percentage: $\frac{18}{100}=18\%=0.18$ . Otherwise division always works.

Some decimals terminate; others repeat forever. Keep an ellipsis or recurring notation until the required rounding stage so the fraction's value is not silently changed.

Multiplying a decimal by $100$ moves from decimal form to percentage form, so include the percent sign. $0.6$ and $60\%$ are equal; $0.6\%$ is much smaller.

For any non-zero number $n$ , the unit fraction $\frac1n$ is the multiplicative inverse of $n$ because $n\times\frac1n=1$ . Multiplying by $\frac1n$ undoes multiplication by $n$ .

Statement	Equivalent form	Meaning
divide by $5$	multiply by $\frac15$	take one fifth
$3\div5$	$3\times\frac15=\frac35$	split $3$ into five equal shares
$20\times\frac14$	$20\div4=5$	take one quarter of $20$

Division by $n$ asks how much remains in one of $n$ equal groups. Multiplication by $\frac1n$ takes exactly that one equal share, so the operations have the same effect.

The inverse pair restores the starting value: $x\times n\times\frac1n=x$ for $n\ne0$ .

Zero has no multiplicative inverse because no number multiplied by $0$ gives $1$ . Therefore $\frac10$ and division by zero are undefined.

To multiply fractions, multiply numerators and multiply denominators. Cancel common factors before or after multiplying: $\frac23\times\frac57=\frac{10}{21}$ .

Step	Example $3\frac15\div2\frac23$
convert mixed numbers to improper fractions	$\frac{16}{5}\div\frac83$
multiply by the reciprocal of the divisor	$\frac{16}{5}\times\frac38$
cancel and multiply	$\frac65=1\frac15$

Multiplying by the reciprocal works because $\frac83\times\frac38=1$ ; the reciprocal undoes multiplication by the divisor.

Estimate before calculating. Since $3\frac15$ is a little larger than $2\frac23$ , their quotient should be a little larger than $1$ , which agrees with $1\frac15$ .

Only the divisor is inverted, and it must be non-zero. Do not invert both fractions, and do not multiply mixed-number whole and fractional parts separately.

1.3 Decimals

Decimal notation uses a decimal point to separate whole-number places from fractional places. In $23.47$ , the $23$ is the whole part and $.47$ is forty-seven hundredths.

Position from the point	Value of one unit
first place right	one tenth, $0.1$
second place right	one hundredth, $0.01$
third place right	one thousandth, $0.001$

Decimals locate values between integers on a number line. Between $2.2$ and $2.3$ , ten equal intervals represent hundredths, so $2.28$ is eight hundredths after $2.2$ .

Trailing zeros do not change value: $5.2=5.20$ . A zero between non-zero digits can hold a place, so $5.02$ is not equal to $5.2$ .

The decimal point fixes every place. Do not read $3.05$ as thirty-five hundredths; it is three and five hundredths.

Each move one place left multiplies a digit's value by $10$ ; each move one place right divides its value by $10$ . This rule continues across the decimal point.

Digit in $4.7634$	Place	Value
$7$	tenths	$0.7$
$6$	hundredths	$0.06$
$3$	thousandths	$0.003$
$4$	ten-thousandths	$0.0004$

Expanded form makes the values visible: $4.7634=4+0.7+0.06+0.003+0.0004$ .

Zero holds an empty place. In $0.407$ , the zero in the hundredths place keeps the $7$ in the thousandths place.

Name the value, not just the digit: the $3$ in $4.7634$ has value $0.003$ , not $3$ and not $0.03$ .

To compare decimals, align their decimal points and compare digits from left to right. The first place where the digits differ determines the order.

Original	Equal-length form
$0.078$	$0.078$
$0.7$	$0.700$
$0.87$	$0.870$
$0.08$	$0.080$
$0.707$	$0.707$

Comparing thousandths columns after alignment gives $0.078<0.080<0.700<0.707<0.870$ , so the original numbers are ordered $0.078,0.08,0.7,0.707,0.87$ .

For negative decimals, values farther left on the number line are smaller: $-0.8<-0.35$ . Compare their positive sizes, then reverse the order because both are negative.

More decimal digits do not automatically mean a larger value. $0.707$ has three decimal places but is smaller than $0.87$ .

A terminating decimal has finitely many decimal digits. It can be converted directly to a fraction using a power of $10$ , or to a percentage by multiplying by $100\%$ .

Target	Method	Example
fraction	write the digits over $10$ , $100$ , $1000$ , … according to decimal places, then simplify	$0.72=\frac{72}{100}=\frac{18}{25}$
percentage	multiply the decimal by $100$ and attach $\%$	$0.08=8\%$

For $0.017$ , there are three decimal places, so $0.017=\frac{17}{1000}$ . The numerator and denominator have no common factor greater than $1$ .

A decimal below $1$ becomes a proper fraction. A decimal such as $0.6$ becomes $60\%$ , which is also below $100\%$ .

This conversion method covers terminating decimals here. Do not place a recurring decimal over a guessed power of $10$ ; recurring decimals require the later algebraic method.

A terminating decimal is a fraction because every decimal place is a fractional power-of-ten place. Finite decimal digits therefore form a finite sum of tenths, hundredths, thousandths and so on.

For example, $0.407=\frac4{10}+\frac0{100}+\frac7{1000}=\frac{407}{1000}$ . This is an exact equality, not an approximation.

Decimal places	Power-of-ten denominator	Example
1	$10$	$0.6=\frac6{10}=\frac35$
2	$100$	$0.65=\frac{65}{100}=\frac{13}{20}$
3	$1000$	$0.125=\frac{125}{1000}=\frac18$

Simplifying changes the name of the fraction, not the value. Thus $0.65$ , $\frac{65}{100}$ and $\frac{13}{20}$ are the same number.

A displayed rounded decimal may only approximate a value. The claim is exact when the decimal truly terminates at the final shown digit, not when further digits have merely been hidden by rounding.

A recurring decimal repeats the same digit or block forever. Algebra converts it to a fraction by shifting identical recurring tails into alignment and subtracting them away.

Step	Example $x=0.3222\ldots$
identify one non-recurring digit and one recurring digit	$x=0.3222\ldots$
multiply to place matching recurring tails after the decimal point	$10x=3.2222\ldots$ , $100x=32.2222\ldots$
subtract the aligned equations	$100x-10x=32.2222\ldots-3.2222\ldots$
solve and simplify	$90x=29$ , so $x=\frac{29}{90}$

If the repeating block begins immediately, use $x$ and a power-of-ten multiple. For $x=0.4545\ldots$ , $100x-x=45$ , so $99x=45$ and $x=\frac5{11}$ .

The multiplier must shift by the length of the repeating block, while a second multiplier may be needed to pass any non-recurring digits. The two decimals subtracted must have identical infinite tails.

Do not truncate the recurring decimal before subtracting: $0.3222$ is a terminating approximation, whereas $0.3222\ldots$ is the exact recurring value.

1.4 Powers and roots

A square number is the product of an integer with itself, $n^2=n\times n$ . A cube number is the product of three equal integer factors, $n^3=n\times n\times n$ .

Type	Sequence from non-negative integers	Recognition
squares	$0,1,4,9,16,25,36,\ldots$	$25=5^2$
cubes	$0,1,8,27,64,125,\ldots$	$27=3^3$

Some numbers are both: $64=8^2=4^3$ . A negative integer can be a cube, such as $-27=(-3)^3$ , but an integer square cannot be negative.

To test a positive integer, find whether its square root or cube root is an integer.

A number containing a square digit is not necessarily square. The whole number must equal $n^2$ for an integer $n$ .

Squaring and taking the principal square root are inverse operations for non-negative numbers; cubing and taking the cube root are inverse operations for all real numbers.

Power	Inverse statement
$14^2=196$	$\sqrt{196}=14$
$19^3=6859$	$\sqrt[3]{6859}=19$
$(-4)^3=-64$	$\sqrt[3]{-64}=-4$

The symbol $\sqrt{a}$ means the non-negative principal root. Thus $\sqrt{49}=7$ . But the equation $x^2=49$ has two solutions, $x=7$ and $x=-7$ .

Brackets matter: $(-3)^2=9$ , while $-3^2=-(3^2)=-9$ under the operation hierarchy.

There is no real square root of a negative number, but negative numbers do have real cube roots.

In $a^n$ , $a$ is the base and $n$ is the index. Index laws compress repeated multiplication and extend consistently to zero and negative integer powers.

Law, where defined	Result
$a^m\times a^n$	$a^{m+n}$
$a^m\div a^n$	$a^{m-n}$
$(a^m)^n$	$a^{mn}$
$a^0$ , $a\ne0$	$1$
$a^{-n}$ , $a\ne0$	$\frac1{a^n}$

$2^{-3}\times2^9=2^{-3+9}=2^6$ . Also, $5^{-2}=\frac1{25}$ ; the negative index creates a reciprocal, not a negative value.

Multiplication and division laws require the same base. For division and negative powers the base must be non-zero.

Do not multiply indices when multiplying powers: $a^m a^n=a^{m+n}$ . Indices multiply only in a power of a power.

A prime factor decomposition writes an integer greater than $1$ as a product containing only prime factors. Repeated primes are collected as powers.

Step	Example for $720$
divide repeatedly by prime numbers	$720=2\times360=2^2\times180=2^3\times90=2^4\times45$
continue until every remaining factor is prime	$45=3\times3\times5$
collect repeated factors	$720=2^4\times3^2\times5$

Multiply the prime powers to check: $16\times9\times5=720$ . A different valid factor tree must finish with the same prime factors.

The number $1$ is neither prime nor included as a prime factor; including it changes nothing but is not part of the decomposition.

Stopping at composite factors is incomplete. For example, $720=16\times45$ is a factorisation, but not yet a product of prime factors.

The highest common factor divides every given number and is as large as possible. The lowest common multiple is divisible by every given number and is as small as possible.

Number	Prime decomposition
$72$	$2^3\times3^2$
$108$	$2^2\times3^3$

For the HCF, keep only primes present in both and choose the smaller exponent: $2^2\times3^2=36$ .

For the LCM, include every prime present and choose the larger exponent: $2^3\times3^3=216$ . Check that $216$ divides exactly by both $72$ and $108$ .

Do not swap the exponent rules: HCF uses shared minimum powers; LCM uses all maximum powers.

A surd is an exact irrational root, such as $\sqrt2$ or $\sqrt[3]{5}$ , left in root form rather than replaced by a rounded decimal.

Root	Type	Exact simplified value
$\sqrt{49}$	rational	$7$
$\sqrt8$	surd	$2\sqrt2$
$\sqrt[3]{27}$	rational	$3$

Extract perfect-power factors: $\sqrt{ab}=\sqrt a\sqrt b$ for non-negative $a,b$ , so $\sqrt{72}=\sqrt{36\times2}=6\sqrt2$ .

Surd form preserves exact value. A calculator decimal such as $1.414\ldots$ approximates $\sqrt2$ and should not replace it when an exact answer is required.

Roots do not distribute over addition: $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ .

Simplify surds first, then combine only like surds: $3\sqrt8+\sqrt{32}=6\sqrt2+4\sqrt2=10\sqrt2$ .

Job	Exact move	Example
multiply surds	multiply coefficients and radicands	$\sqrt3\times\sqrt{12}=\sqrt{36}=6$
remove a single surd denominator	multiply top and bottom by that surd	$\frac1{\sqrt2}=\frac{\sqrt2}{2}$
remove a binomial surd denominator	multiply by its conjugate	$\frac2{2-\sqrt3}\times\frac{2+\sqrt3}{2+\sqrt3}=4+2\sqrt3$

Conjugates use difference of squares: $(a-b\sqrt c)(a+b\sqrt c)=a^2-b^2c$ , which is rational.

Expand brackets carefully and simplify every square root before collecting terms.

Unlike surds cannot be added: $\sqrt2+\sqrt3$ does not become $\sqrt5$ .

Fractional indices represent roots, while negative indices represent reciprocals. Together they extend the same index laws used for integer powers.

Form	Meaning	Example
$a^{1/n}$	$\sqrt[n]{a}$	$8^{1/3}=2$
$a^{m/n}$	$(\sqrt[n]{a})^m$	$8^{2/3}=2^2=4$
$a^{-p}$	$\frac1{a^p}$	$625^{-1/2}=\frac1{25}$

Apply the root and reciprocal meanings in either safe order: $16^{-3/4}=1/(16^{1/4})^3=1/8$ .

For real-number work, even roots require a non-negative radicand. Negative powers require a non-zero base.

A negative index does not make the value negative, and $a^{m/n}$ does not mean $a^m\div a^n$ .

1.5 Set language and notation

A set is a well-defined collection of distinct objects, called elements or members. Curly brackets list the elements, for example $P=\{2,3,5,7\}$ .

Description	Set
vowels in the word MATHEMATICS	$\{A,E,I\}$
positive factors of $12$	$\{1,2,3,4,6,12\}$

A rule must decide unambiguously whether an object belongs. ‘Prime numbers below 10’ defines a set; ‘nice numbers’ does not unless ‘nice’ is precisely defined.

Order and repetition do not change a set: $\{1,2,3\}=\{3,2,1,1\}$ . Each distinct element is listed once in standard form.

The braces name the collection, not a calculation. $\{2,4\}$ is a set with two elements, whereas $2+4$ is a numerical expression.

Set symbols state membership and combine collections precisely.

Notation	Meaning
$x\in A$ / $x\notin A$	$x$ is / is not an element of $A$
$A\cap B$	elements in both $A$ and $B$
$A\cup B$	elements in $A$ or $B$ , including both
$E$	the universal set for the problem
$\varnothing$	the empty set, with no elements

If $A=\{1,2,3,4\}$ and $B=\{3,4,5\}$ , then $A\cap B=\{3,4\}$ and $A\cup B=\{1,2,3,4,5\}$ .

Membership relates an object to a set: $3\in A$ . Intersection and union relate two sets: $A\cap B$ and $A\cup B$ .

In set language, ‘or’ is inclusive: an element in both sets belongs to $A\cup B$ . Do not omit the overlap.

The universal set $E$ contains every element under consideration in a particular problem. Its contents depend on the stated context.

The empty set $\varnothing$ contains no elements, so $n(\varnothing)=0$ . It can arise when two sets share no members: $A\cap B=\varnothing$ .

Situation	Result
$E=\{1,2,3,4,5,6\}$ , $A=$ even numbers	$A=\{2,4,6\}$
$A=\{2,4,6\}$ , $B=\{1,3,5\}$	$A\cap B=\varnothing$

Every set used in the problem is interpreted inside $E$ . Changing $E$ can change what lies outside a named set.

$\varnothing$ is not the same as $\{0\}$ . The first has zero elements; the second has one element, namely zero.

The complement $A'$ is the set of all elements in the universal set $E$ that are not in $A$ .

Given	Complement
$E=\{1,2,3,4,5,6,7,8\}$
$A=\{2,4,6,8\}$	$A'=\{1,3,5,7\}$

On a Venn diagram, shade every region inside the universal rectangle but outside the circle for $A$ to represent $A'$ .

Read operations from the inside out. $(A\cup B)'$ means everything outside both circles, while $A'\cap B$ means in $B$ but not in $A$ .

A complement is relative to $E$ , not an unlimited collection of every object that is not in $A$ .

A Venn diagram represents sets as regions inside the universal rectangle. Overlapping circles show elements that satisfy more than one set condition.

Region	Meaning
overlap of $A$ and $B$	$A\cap B$
anywhere in either circle	$A\cup B$
in $A$ but outside $B$	$A\cap B'$
outside both circles	$(A\cup B)'$

Place intersection elements first, then elements belonging to only one set, then elements outside all named sets. Check every element of $E$ appears exactly once.

With three sets, begin at the central triple intersection, then fill pair-only regions, single-set regions and finally the outside region.

An element in an overlap is not copied into each circle’s separate region. Its one position already shows membership of both sets.

An algebraic set is defined by a condition on its elements, for example $A=\{x:x\in\mathbb{Z},\ 1\le x<6\}=\{1,2,3,4,5\}$ .

If every element of $A$ is also an element of $B$ , then $A$ is a subset of $B$ , written $A\subset B$ in this specification.

Sets	Decision
$A=\{2,4,6\}$ , $B=\{1,2,3,4,5,6\}$	$A\subset B$
$C=\{2,7\}$ , same $B$	$C\not\subset B$ because $7\notin B$

Translate the algebraic rule into allowed values, respecting the stated number domain. To disprove a subset claim, one counterexample is enough.

Do not confuse membership with subset notation: $2\in A$ , but $\{2\}\subset A$ .

A number written in a Venn region counts elements in that exact region. Totals for a set are found by adding every region inside its circle.

Information	Region value
$n(A)=28$ , $n(A\cap B)=9$	$A$ only $=28-9=19$
$n(B)=21$ , $n(A\cap B)=9$	$B$ only $=21-9=12$
$n(E)=40$	outside $=40-(19+9+12)=0$

Work from the most overlapped region outward. Subtract known overlap counts from set totals, then subtract all circle regions from the universal total.

For three sets, fill the triple intersection first. A stated pair intersection includes the triple region, so subtract it to obtain the pair-only region.

Adding $n(A)+n(B)$ double-counts the overlap. For two sets, $n(A\cup B)=n(A)+n(B)-n(A\cap B)$ .

$n(A)$ means the number of distinct elements in set $A$ ; it is a number, not a set.

Set expression	Count
$A=\{2,3,5,7\}$	$n(A)=4$
$A\cap B=\{3,5\}$	$n(A\cap B)=2$
$A\cap B=\varnothing$	$n(A\cap B)=0$

On a Venn diagram, add the counts in every region described by the expression inside $n(\ )$ . For $n(A')$ , add all regions outside $A$ but still inside $E$ .

Count distinct members only. Repeated writing does not create extra elements in a set.

$n(A)$ is not the same object as $A$ : if $A=\{4,6,8\}$ , then $A$ is a set and $n(A)=3$ .

In a practical problem, each set represents a precisely defined property, such as students studying French or customers buying a product.

Words	Set expression
both football and tennis	$F\cap T$
football or tennis or both	$F\cup T$
tennis but not football	$T\cap F'$
neither activity	$(F\cup T)'$

Define the universal group and each set before calculating. Translate one phrase at a time, place or count the overlap first, and check that all regions sum to the stated total.

Return to the context when stating an answer: a region count should be described as people, items or outcomes, with any stated units.

Everyday ‘or’ can sound exclusive, but set union is inclusive unless the problem explicitly says ‘but not both’ or ‘exactly one’.

1.6 Percentages

A percentage tells how many equal parts out of 100 are being considered. The symbol $\%$ means ‘per 100’, so $37\%=37/100$ .

Percentage	Per-100 meaning
$8\%$	8 parts in every 100
$100\%$	the whole amount
$125\%$	one whole and 25 extra parts per 100

The actual whole need not contain 100 objects. If 25% of 60 students travel by bus, the same proportion is $25/100=1/4$ , so 15 students travel by bus.

Percentages put different-sized groups on the same per-100 scale, which makes proportions comparable.

A percentage can exceed 100% and can be below 1%. It is a proportion, not automatically an amount.

To express an amount as a percentage of a reference amount, divide by the reference amount and multiply by 100%.

Question	Calculation
31 500 as a percentage of 42 000	$\frac{31500}{42000}\times100\%=75\%$
18 as a percentage of 24	$\frac{18}{24}\times100\%=75\%$

The words ‘of another number’ identify the denominator. Ask: percentage of which whole or reference value?

If the first number is smaller than the reference, the answer is below 100%; if it is larger, the answer is above 100%.

Reversing the fraction changes the comparison. ‘ $a$ as a percentage of $b$ ’ uses $a/b$ , not $b/a$ .

Because percent means per 100, divide the percentage number by 100. This gives both a fraction and, after division, a decimal.

Percentage	Fraction	Decimal
$23\%$	$23/100$	$0.23$
$45\%$	$45/100=9/20$	$0.45$
$2.5\%$	$2.5/100=1/40$	$0.025$
$125\%$	$125/100=5/4$	$1.25$

Dividing by 100 moves the decimal point two places left; multiplying a decimal by 100 converts it back to a percentage.

Write the percentage over 100, remove any decimal in the numerator if needed, then simplify the fraction fully.

$0.6\%=0.006$ , not $0.6$ . The percent sign already includes division by 100.

A percentage of an amount is multiplication by its decimal or fractional equivalent: $p\%$ of $Q$ is $(p/100)Q$ .

Task	Operator	Result
$45\%$ of 800	$0.45\times800$	360
$12.5\%$ of 64	$0.125\times64$	8
$150\%$ of 40	$1.5\times40$	60

Use whichever equivalent operator is easiest: $25\%=0.25=1/4$ and $10\%=0.1$ .

Successive percentage operators multiply. This multiplicative idea underpins percentage change, interest and depreciation.

‘15% of 120’ means $0.15\times120$ , not $120-15$ and not $120\div15$ .

Increase by $p\%$ using multiplier $1+p/100$ ; decrease by $p\%$ using multiplier $1-p/100$ .

Change	Multiplier	Example from 240
increase 15%	$1.15$	$240\times1.15=276$
decrease 15%	$0.85$	$240\times0.85=204$

A percentage change may also be found from $\frac{\text{new}-\text{original}}{\text{original}}\times100\%$ . Use the original value as the denominator.

In a word problem, calculate each required percentage amount, keep units, and round only when the context or question requires it.

Adding 15% means adding 15% of the original amount, not adding the number 15. An increase and an equal percentage decrease do not cancel.

A final amount after a percentage change equals the original amount multiplied by a change multiplier. Reverse the change by dividing by that multiplier.

Information	Equation	Original
sale price £17.50 after 30% off	$0.70x=17.50$	$x=17.50/0.70=£25$
price 9.45 after 8% rise	$1.08x=9.45$	$x=9.45/1.08=8.75$

Identify what percentage the final value represents: after 17% off it is 83%; after a 12% rise it is 112%.

Apply the stated change to the recovered original to check that it returns the given final value.

Do not undo a 30% decrease by increasing the final value by 30%. Divide by $0.70$ because the final value has a different base.

Compound change applies each period to the current value, so the multiplier is applied repeatedly.

Situation	Value after $n$ periods
compound interest at $r\%$	$P(1+r/100)^n$
depreciation at $r\%$	$P(1-r/100)^n$

6000 dirham at 1.5% compound interest for four years becomes $6000(1.015)^4=6368.18\ldots$ ; the interest earned is $6368.18\ldots-6000=368.18\ldots$ .

Keep full calculator precision through the powers, then round the final money value as instructed.

Compound interest is not $P+nrP/100$ ; that adds the same simple-interest amount every period and ignores growth on earlier interest.

Represent each percentage change by a multiplier and multiply the multipliers in time order.

Sequence	Combined multiplier	Overall change
increase 30%, then decrease 20%	$1.30\times0.80=1.04$	4% increase
depreciate 15% for two years	$0.85^2=0.7225$	27.75% decrease

After finding the combined multiplier $m$ , the total percentage change is $(m-1)\times100\%$ ; a negative result indicates a decrease.

If the final value is known, divide by the product of all change multipliers to recover the starting value.

Do not add signed percentage changes. A 30% rise followed by a 20% fall acts on different base values.

Model a compound-interest account with $A=P(1+r/100)^n$ , where $P$ is principal, $r$ is the annual percentage rate, $n$ is the number of compounding periods and $A$ is the final amount.

Unknown	Rearrangement
final amount	$A=P(1+r/100)^n$
principal	$P=A/(1+r/100)^n$
rate	$r=100[(A/P)^{1/n}-1]$

If 6000 grows to 6311.16 after two years at 1.5% and a third year at rate $r$ , then $6000(1.015)^2(1+r/100)=6311.16$ , giving $r=2.1\%$ .

Match the exponent to the number of compounding periods. When rates change, use a separate multiplier for each rate interval.

Interest earned is $A-P$ , whereas the account balance is $A$ . Read which quantity the question asks for.

1.7 Ratio and proportion

A ratio compares quantities multiplicatively and in a stated order. The ratio $a:b$ means that for every $a$ parts of the first quantity there are $b$ parts of the second.

Job	Example
write in order	32 teeth to 24 teeth is $32:24$
simplify	$32:24=4:3$
express as $1:n$	$5:8=1:1.6$
link to a fraction	in $4:3$ , the first share is $4/(4+3)=4/7$ of the total

Convert quantities to the same units before forming or simplifying a ratio: $2 ext{ m}:50 ext{ cm}=200:50=4:1$ .

Multiplying or dividing every part by the same non-zero factor gives an equivalent ratio.

Order matters: $4:3$ and $3:4$ compare opposite directions. Do not simplify by subtracting the same number from both parts.

To share a total in a ratio, treat the ratio numbers as equal-sized part counts.

Step	Share £416 in $5:3$
add ratio parts	$5+3=8$ parts
find one part	$416\div8=52$
multiply for each share	$5 imes52=260$ , $3 imes52=156$

For a three-part ratio such as $4:3:1$ , add all three parts before finding the value of one part.

Check that the shares total the original quantity and that their simplified ratio matches the given ratio.

Do not divide the total separately by each ratio number. The denominator is the sum of all ratio parts.

Directly proportional quantities change by the same scale factor. If one quantity is multiplied by $k$ , the linked quantity is also multiplied by $k$ .

Information	Proportional step
8 calculators cost £62.80	one costs $62.80\div8=7.85$
cost of 12 calculators	$7.85 imes12=£94.20$
equivalent scale-factor method	$62.80 imes(12/8)=£94.20$

Use a unitary method when one-unit value is useful, or move directly by a scale factor when the relationship is clear.

For direct proportion, the ratio $y/x$ is constant, so the model can be written $y=kx$ .

Adding the same difference is not proportional scaling. The relationship must pass through zero: zero items cost zero at a fixed unit price.

A direct-proportion table contains pairs linked by one constant multiplier. Find that multiplier or reduce to one unit, then complete every missing entry.

Cakes	Flour
15	360 g
1	$360\div15=24$ g
38	$38 imes24=912$ g

If $y$ varies directly as $x$ , write $y=kx$ . Use one known pair to calculate $k$ , then substitute the required value.

Keep corresponding columns in consistent units. For example, convert $0.85 ext{ kg}$ to $850 ext{ g}$ before comparing it with $912 ext{ g}$ .

Do not assume a table is proportional merely because both columns increase. Verify that $y/x$ stays constant.

A word problem must first be translated into the comparison being held constant: ratio parts, a unit rate, a scale factor or a map scale.

Context	Useful comparison
best value	cost per unit or amount per currency unit
recipe	amount per serving, then scale
map	actual distance $=$ map distance $imes$ scale
mixture	total parts, then each ingredient's fraction

A 150 g bag costing £1.80 costs $1.80/150=£0.012$ per gram. A 400 g bag costing £5 costs $5/400=£0.0125$ per gram, so the small bag is better value.

Compare like with like, show the common unit rate or equivalent quantity, and state the conclusion in context.

A larger pack is not automatically better value. Its price and quantity must be compared using the same unit.

1.8 Degree of accuracy

Rounding to a power of 10 means choosing the nearest multiple of that place value: $10^1$ for tens, $10^2$ for hundreds, $10^3$ for thousands, and so on.

Round 6739 to	Look at	Result
nearest 10	units digit 9	6740
nearest 100	tens digit 3	6700
nearest 1000	hundreds digit 7	7000

Locate the rounding digit, inspect the digit immediately to its right, increase the rounding digit if that next digit is 5 or more, then replace later integer digits with zeros.

For negative integers, choose the nearest multiple on the number line; for example, $-6739$ rounds to $-6700$ to the nearest hundred.

Do not count digits from the left without identifying place value. Rounding to $10^2$ means the nearest hundred, not two significant figures.

Decimal places count digits after the decimal point. Significant figures begin at the first non-zero digit and describe the precision of the whole value.

Number	Instruction	Result
45.621	nearest whole number	46
45.621	2 decimal places	45.62
0.004786	2 significant figures	0.0048
58 749	3 significant figures	58 700

Mark the last digit to keep, inspect the next digit, round up for 5–9, and retain placeholder zeros when they communicate magnitude or required decimal places.

Leading zeros are not significant; zeros between non-zero digits are significant. Trailing zeros after a decimal can show stated precision.

Two decimal places and two significant figures usually give different answers. Identify which counting system the question states.

A rounded value represents an interval of possible original values. Half a rounding unit lies below the stated value and half lies above it.

Stated value	Rounding unit	Interval
4.3 kg to nearest 0.1 kg	0.1	$4.25\le w<4.35$
125 cm to nearest cm	1	$124.5\le l<125.5$
2400 to nearest 100	100	$2350\le n<2450$

The lower bound is included because it rounds up to the stated value. The upper bound is excluded because that exact value rounds to the next result.

For a stated number of decimal places or significant figures, first identify the value of the last retained digit; that is the rounding unit.

The bound offset is half a rounding unit, not half the rounded value and not always $0.5$ .

An estimate replaces values with nearby easy numbers so a calculation can be checked mentally and its order of magnitude judged.

Exact expression	Suitable estimate
$68.3 imes42.8\div0.021$	$70 imes40\div0.02=140000$
$19.8 imes4.13$	$20 imes4=80$
$598\div0.31$	$600\div0.3=2000$

Rounding each value to one significant figure is a reliable default, but choose compatible numbers that keep the approximation easy and reasonably close.

Compare the estimate with the calculator result. A large disagreement in size or decimal position signals an input or operation error.

An estimate is not expected to equal the exact result. It must be simple enough to evaluate and close enough to test plausibility.

To maximise or minimise an expression, choose the combination of input bounds that makes the entire expression largest or smallest.

Positive quantities	Upper value uses	Lower value uses
product $ab$	$a_Ub_U$	$a_Lb_L$
quotient $a/b$	$a_U/b_L$	$a_L/b_U$
difference $a-b$	$a_U-b_L$	$a_L-b_U$

For an outer rectangle $8.3 imes7.2$ minus an inner rectangle $6.2 imes5.3$ , all lengths correct to 0.1 cm, the upper shaded area is $8.35 imes7.25-6.15 imes5.25=28.25 ext{ cm}^2$ .

To give a result to a suitable degree of accuracy, calculate both outcome bounds and round only to a precision for which both bounds produce the same stated value.

Using every upper bound does not always maximise a composite expression. A subtracted area or denominator may need its lower bound.

1.9 Standard form

Standard form writes a non-zero number as $a imes10^n$ , where $1\le |a|<10$ and $n$ is an integer. The power of 10 records the place-value shift.

Ordinary number	Standard form
71 800 000	$7.18 imes10^7$
0.00042	$4.2 imes10^{-4}$
$-630000$	$-6.3 imes10^5$

For multiplication, multiply coefficients and add powers; for division, divide coefficients and subtract powers. For addition or subtraction, first express terms with the same power of 10.

Calculation	Normalised result
$(3 imes10^4)(2 imes10^5)$	$6 imes10^9$
$(8 imes10^7)/(4 imes10^3)$	$2 imes10^4$
$4.5 imes10^6+7 imes10^5$	$4.5 imes10^6+0.7 imes10^6=5.2 imes10^6$

A result such as $18 imes10^5$ is not in standard form because the coefficient is not below 10; rewrite it as $1.8 imes10^6$ .

Translate each contextual quantity into a consistent unit, preserve its standard-form structure through the calculation, then interpret and round the final result as requested.

Step	Healthcare spending per person
form a rate	total spending $\div$ population
Austria	$(4.2 imes10^{10})/(8.7 imes10^6)=4.82758\ldots imes10^3$
Luxembourg	$(3.7 imes10^9)/(6.3 imes10^5)=5.87301\ldots imes10^3$
compare	$5873.01\ldots-4827.58\ldots=1045$ dollars nearest whole

Separate coefficient arithmetic from exponent arithmetic, but keep enough calculator precision until the final requested rounding.

Use exponent size to check order of magnitude. Dividing $10^{10}$ by $10^6$ should produce a value around $10^4$ , subject to the coefficient ratio.

Do not subtract exponents when subtracting numbers. Exponent subtraction belongs to division; numerical subtraction requires compatible place values.

1.10 Applying number

Applying number means translating a real situation into quantities, operations and constraints, then interpreting the result in the original context.

Stage	Question to ask
identify	What is known, unknown and constrained?
standardise	Do units, currencies or time formats need conversion?
calculate	Which operations and order model the situation?
interpret	Must the result be rounded up, rounded down or stated with units?

A budget cannot exceed the available money; a required number of journeys, packs or workers is normally rounded up; a number of complete items cut from material may need rounding down.

Estimate first, keep units beside intermediate quantities, and check whether the final value is realistic and satisfies every condition.

A mathematically correct decimal can still be an invalid practical answer. Context determines whether fractional items are possible and how rounding should be handled.

Metric calculations require compatible units. Convert every measurement to a chosen common unit before adding, comparing or using a formula.

Quantity	Key equivalence
length	$1 ext{ m}=100 ext{ cm}=1000 ext{ mm}$
mass	$1 ext{ kg}=1000 ext{ g}$
capacity	$1 ext{ litre}=1000 ext{ ml}$
area	$1 ext{ m}^2=10000 ext{ cm}^2$
volume	$1 ext{ m}^3=1000000 ext{ cm}^3$

Area conversion factors are squared and volume conversion factors are cubed. Since $1 ext{ m}=100 ext{ cm}$ , $1 ext{ m}^3=100^3 ext{ cm}^3$ .

A $10 ext{ m} imes2.4 ext{ m} imes2.4 ext{ m}$ container holds $40 ext{ cm}=0.4 ext{ m}$ cubes in counts $25 imes6 imes6=900$ when aligned with its edges.

Do not use the linear factor 100 for area or volume. Also, a volume ratio alone may not prove that boxes fit; check whole-number edge counts and orientation.

Time and money calculations depend on place-value systems and conversion rates. State the direction of every conversion before multiplying or dividing.

Job	Method
elapsed time	convert to one unit or bridge across clock times
money total	align decimal places and round final currency to the smallest unit
currency conversion	multiply by target units per source unit; divide to reverse
value comparison	convert to one currency and compare cost per common unit

If $1$ dollar $=6.57$ krone, dollars to krone uses $imes6.57$ and krone to dollars uses $\div6.57$ .

For competing oil offers, calculate litres per dollar or cost per litre after converting both prices into the same currency; only then compare value.

Time is not base 100: $2.75$ hours is $2$ hours $45$ minutes, not $2$ hours $75$ minutes. A currency rate must be applied in the stated direction.

1.11 Electronic calculators

A scientific calculator evaluates a mathematical expression according to operation hierarchy. Accurate use begins by translating the printed expression into an unambiguous key sequence.

Stage	Reliable action
plan	identify numerator, denominator, powers, roots and brackets
enter	use bracket keys to preserve the printed structure
inspect	check the display before pressing equals
record	copy all requested display figures before rounding
verify	estimate the size and sign of the result

For $((125.6)/(4.7))^2$ , enter the division in brackets before squaring. For a fraction containing several terms, bracket the entire numerator and denominator.

Store or retain unrounded intermediate values. If the question says ‘write down all the figures on your calculator display’, do not shorten the decimal or give a rounded form.

Check angle mode before trigonometry and use the calculator's scientific-notation, fraction/decimal and previous-answer functions deliberately rather than assuming their state.

A calculator follows the entered syntax, not the intended expression. A plausible-looking display is not proof of correct entry; brackets and an independent estimate are essential.

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