Mathematics Formula Book

Number Theory and Computation

Place value (base 10): each digit position represents a power of 10. In base $b$, each position represents a power of $b$ and digits must be less than $b$.

Standard form (scientific notation): $a \times 10^n$ where $1 \leq a < 10$ and $n$ is an integer.

Rounding: round up if the next digit is 5 or more, otherwise round down. Report significant figures from the first non-zero digit.

Surds: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$, $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$, $\sqrt{a} \times \sqrt{a} = a$.

Rationalise the denominator using the conjugate: $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$.

Number types: natural numbers $\mathbb{N}$ (1, 2, 3, …), integers $\mathbb{Z}$, rationals $\mathbb{Q}$ (expressible as $\frac{p}{q}$, $q \neq 0$), irrationals (non-terminating, non-repeating), reals $\mathbb{R}$.

HCF and LCM via prime factorisation:

• HCF: take each prime to its lowest power in either factorisation.
• LCM: take each prime to its highest power in either factorisation.
• Relationship: $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.

Arithmetic sequences:

$T_n = a + (n-1)d \qquad S_n = \frac{n}{2}[2a + (n-1)d]$

where $a$ is the first term and $d$ is the common difference.

Consumer Arithmetic

Profit and loss:

$\text{Profit/Loss} = \text{Selling Price} - \text{Cost Price}$

$\text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100\%$

A markup is a percentage added to the cost price; a markdown/discount is a percentage reduction from the marked price.

Percentage change:

$\% \text{ change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

Simple interest:

$I = \frac{PRT}{100} \qquad A = P + I = P\!\left(1 + \frac{RT}{100}\right)$

where $P$ is the principal, $R$ is the annual rate (%), and $T$ is the time in years.

Compound interest:

$A = P\!\left(1 + \frac{R}{100}\right)^n$

where $n$ is the number of compounding periods. Compound interest $= A - P$.

Appreciation and depreciation use the same compound formula:

$\text{New value} = \text{Original} \times \left(1 \pm \frac{R}{100}\right)^n$

Use $+$ for appreciation, $-$ for depreciation.

Hire purchase: deposit paid upfront, then equal instalments. Total HP cost = deposit + (instalment $\times$ number of instalments). The HP price is usually more than the cash price.

Currency conversion: multiply by the exchange rate to convert from the base currency; divide to reverse. Always label the currency at each step.

Tax: Value Added Tax (VAT) and income tax are percentages of the taxable amount. Net pay = gross pay minus deductions.

Sets

Notation:

Key results:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$

$A \cup A' = U \qquad A \cap A' = \varnothing \qquad (A')' = A$

De Morgan's laws:

$(A \cup B)' = A' \cap B' \qquad (A \cap B)' = A' \cup B'$

A Venn diagram shows sets as overlapping circles inside a rectangle (the universal set). Fill regions starting from the innermost intersection.

Algebra

Expanding brackets: $(a + b)(c + d) = ac + ad + bc + bd$

Special products:

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

$(a \pm b)^2 = a^2 \pm 2ab + b^2$

Laws of indices (same base only):

Solving linear equations: isolate the variable by performing the same inverse operation on both sides.

Simultaneous linear equations (two unknowns):

• Substitution: rearrange one equation, substitute into the other.
• Elimination: multiply equations so one variable cancels when added or subtracted.

Quadratic formula: for $ax^2 + bx + c = 0$,

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Discriminant $\Delta = b^2 - 4ac$:

Vertex (turning point) of $y = ax^2 + bx + c$:

$x_v = -\frac{b}{2a} \qquad y_v = c - \frac{b^2}{4a}$

Parabola opens upward if $a > 0$ (minimum), downward if $a < 0$ (maximum).

Direct variation: $y = kx$, so $\dfrac{y}{x} = k$ (constant). Graph is a straight line through the origin.

Inverse variation: $y = \dfrac{k}{x}$, so $xy = k$ (constant). Graph is a hyperbola.

Changing the subject: apply inverse operations in reverse PEMDAS order to isolate the required variable.

Solving inequalities: use the same method as equations, but reverse the inequality sign when multiplying or dividing both sides by a negative number.

Number line conventions:

• Open circle $\circ$: endpoint not included ($<$ or $>$).
• Filled circle $\bullet$: endpoint included ($\leq$ or $\geq$).

Relations, Functions, and Graphs

Function notation: $f(x)$ means the output of function $f$ for input $x$. Evaluate by substituting the input for every occurrence of the variable.

Domain: the set of permitted inputs. Range (image): the set of actual outputs.

Vertical line test: a graph represents a function if and only if every vertical line crosses it at most once.

Composite functions: $(f \circ g)(x) = f(g(x))$: $g$ is applied first, then $f$. In general, $f \circ g \neq g \circ f$.

Inverse function $f^{-1}$: swap $x$ and $y$, then solve for $y$.

Verification: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Linear function $y = mx + c$:

• Gradient $m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\text{rise}}{\text{run}}$
• $y$-intercept $= c$ (where the line crosses the $y$-axis)
• $x$-intercept: set $y = 0$ and solve

Parallel lines have equal gradients. Perpendicular lines satisfy $m_1 m_2 = -1$.

Equation of a line through $(x_1, y_1)$ with gradient $m$:

$y - y_1 = m(x - x_1)$

Midpoint of $(x_1, y_1)$ and $(x_2, y_2)$:

$M = \left(\frac{x_1 + x_2}{2},\;\frac{y_1 + y_2}{2}\right)$

Distance between two points:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Graphical solution of simultaneous equations: plot both lines; the solution is their intersection point.

Linear inequalities on a graph: use a dashed boundary line for strict inequalities ($<$, $>$), a solid line for $\leq$ or $\geq$. The feasible region is the set of points satisfying all constraints. The optimum of an objective function occurs at a vertex of the feasible region.

Geometry

Angle facts:

Triangle angles: interior angles sum to $180°$; an exterior angle equals the sum of the two non-adjacent interior angles.

Polygon angle sums for an $n$-sided polygon:

• Interior angle sum: $(n - 2) \times 180°$
• Each interior angle (regular): $\dfrac{(n-2) \times 180°}{n}$
• Each exterior angle (regular): $\dfrac{360°}{n}$

Circle theorems:

Congruence conditions (triangles): SSS, SAS, ASA (or AAS), RHS.

Similarity: corresponding angles equal; corresponding sides in the same ratio. If scale factor is $k$:

• Lengths scale by $k$
• Areas scale by $k^2$
• Volumes scale by $k^3$

Transformations:

Rotation matrix for $\theta°$ anticlockwise about the origin:

$\begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix}$

Measurement

Perimeter and area:

Circle arc and sector (angle $\theta$ in degrees):

$\text{Arc length} = \frac{\theta}{360} \times 2\pi r \qquad \text{Sector area} = \frac{\theta}{360} \times \pi r^2$

Surface area and volume:

Speed, distance, time:

$D = ST \qquad S = \frac{D}{T} \qquad T = \frac{D}{S}$

Convert km/h to m/s by multiplying by $\dfrac{5}{18}$; convert m/s to km/h by multiplying by $\dfrac{18}{5}$.

Scale drawings: $\text{actual length} = \text{map length} \times \text{scale factor}$

Density and pressure:

$\rho = \frac{m}{V} \qquad P = \frac{F}{A}$

Unit conversions (area and volume): when converting lengths by a factor of $k$, areas scale by $k^2$ and volumes scale by $k^3$.

Margin of error: when a measurement is rounded to the nearest unit $u$, the true value lies within half a unit above or below: $\text{measured value} \pm \frac{u}{2}$.

Trigonometry

SOHCAHTOA (right-angled triangles only):

$\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}$

Pythagoras' theorem:

$a^2 + b^2 = c^2 \qquad c^2 - a^2 = b^2$

Exact values:

Angles of elevation and depression: both measured from the horizontal. The angle of elevation from $A$ to $B$ equals the angle of depression from $B$ to $A$.

Sine rule (any triangle, labelled with sides $a, b, c$ opposite angles $A, B, C$):

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Use when given: two angles and one side (AAS), or two sides and a non-included angle (SSA, watch for ambiguous case).

Cosine rule:

$a^2 = b^2 + c^2 - 2bc\cos A$

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Use when given: two sides and the included angle (SAS), or all three sides (SSS).

Area of triangle:

$\text{Area} = \frac{1}{2}ab\sin C$

Bearings: measured clockwise from North, always written as three digits (e.g., 045°, 270°). Draw a North line at every point in a bearing problem.

Vectors

Column vector notation:

$\vec{v} = \begin{pmatrix}x\\y\end{pmatrix} \qquad \text{or} \qquad \vec{v} = x\mathbf{i} + y\mathbf{j}$

Magnitude:

$\lvert\vec{v}\rvert = \sqrt{x^2 + y^2}$

Addition and subtraction:

$\begin{pmatrix}a\\b\end{pmatrix} \pm \begin{pmatrix}c\\d\end{pmatrix} = \begin{pmatrix}a \pm c\\b \pm d\end{pmatrix}$

Scalar multiplication: multiply every component by the scalar. Multiplying by $k$ scales the magnitude by $\lvert k \rvert$ and reverses direction if $k < 0$.

Position vector of point $P$: $\overrightarrow{OP} = \mathbf{p}$ (from origin $O$ to $P$).

Displacement vector: $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$

Parallel vectors: $\vec{u}$ and $\vec{v}$ are parallel if $\vec{u} = k\vec{v}$ for some scalar $k$.

Collinear points: $A$, $B$, $C$ are collinear if $\overrightarrow{AC} = k\overrightarrow{AB}$.

Matrices

Matrix dimensions: an $m \times n$ matrix has $m$ rows and $n$ columns.

Addition/subtraction: add or subtract corresponding entries; only defined for matrices of the same dimensions.

Scalar multiplication: multiply every entry by the scalar.

Matrix multiplication: $AB$ is defined only if the number of columns of $A$ equals the number of rows of $B$. The $(i,j)$ entry of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$.

For $2 \times 2$ matrices $A$ and $B$:

$AB = \begin{pmatrix}ae+bg & af+bh \\ ce+dg & cf+dh\end{pmatrix}$

Note: in general, $AB \neq BA$.

Determinant of a $2 \times 2$ matrix:

$\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad - bc$

An inverse exists if and only if $\det \neq 0$; the matrix is then called non-singular.

Inverse of a $2 \times 2$ matrix:

$\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1} = \frac{1}{ad - bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$

Solving simultaneous equations with matrices: for $A\mathbf{x} = \mathbf{b}$, the solution is $\mathbf{x} = A^{-1}\mathbf{b}$ (provided $A^{-1}$ exists).

Common transformation matrices (applied by multiplying from the left):

Reflection in the $x$-axis: $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$

Reflection in the $y$-axis: $\begin{pmatrix}-1&0\\0&1\end{pmatrix}$

Reflection in $y = x$: $\begin{pmatrix}0&1\\1&0\end{pmatrix}$

Rotation 90° anticlockwise about $O$: $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$

Rotation 180° about $O$: $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$

Enlargement factor $k$, centre $O$: $\begin{pmatrix}k&0\\0&k\end{pmatrix}$

Statistics and Probability

Mean (ungrouped data):

$\bar{x} = \frac{\sum x}{n}$

Mean (grouped/frequency data):

$\bar{x} = \frac{\sum fx}{\sum f}$

where $x$ is the midpoint of each class and $f$ is the frequency.

Median: the middle value when data are ordered. For $n$ values, the median is at position $\dfrac{n+1}{2}$.

Mode: the value (or class) with the highest frequency.

Range: $\text{maximum} - \text{minimum}$

Interquartile range (IQR): $Q_3 - Q_1$

Semi-interquartile range: $\dfrac{Q_3 - Q_1}{2}$

Cumulative frequency (ogive): plot cumulative frequency against the upper class boundary. Read $Q_1$ at $\frac{n}{4}$, median at $\frac{n}{2}$, and $Q_3$ at $\frac{3n}{4}$.

Pie chart sector angle:

$\text{Angle} = \frac{\text{frequency}}{\text{total}} \times 360°$

Theoretical probability:

$P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$

Complement: $P(E') = 1 - P(E)$

Mutually exclusive events: $P(A \cup B) = P(A) + P(B)$

Independent events: $P(A \cap B) = P(A) \times P(B)$

Tree diagrams: multiply probabilities along branches; add probabilities across branches for the same outcome. The probabilities on branches from any node must sum to 1.

Experimental probability:

$P(E) \approx \frac{\text{number of times event occurred}}{\text{total number of trials}}$

As the number of trials increases, experimental probability approaches theoretical probability.