IB Mathematics: Applications and Interpretation Study Notes (SL/HL)

Topic 1: Number and Algebra

Number systems
- Natural numbers: \mathbb{N} = {0, 1, 2, …}
- Integers: \mathbb{Z} = {…, -2, -1, 0, 1, 2, …}
- Rationals: \mathbb{Q} = {a/b : a, b \in \mathbb{Z}, b \neq 0}
- Irrationals: real numbers that cannot be written in the form a/b with integers a, b and b \neq 0 (examples: \sqrt{2}, \pi)
- Real numbers: all numbers that can be placed on the number line, including rationals and irrationals
- Note: every recurring decimal is rational; every terminating decimal is rational.
SI units (base and derived)
- Base units: Time - second (s); Length - metre (m); Mass - kilogram (kg); Electric current; Thermodynamic temperature; Amount of substance; Luminous intensity
- Derived units (examples):
- Speed/velocity - \text{m s}^{-1}
- Acceleration - \text{m s}^{-2}
- Area - \text{m}^2
- Volume - \text{m}^3
- Density - \text{kg m}^{-3}
- Force - newton (N)
Rounding, decimal approximations & significant figures
- Rounding options: to decimal places (d.p.), to significant figures (s.f.), to the nearest integer, to a power of 10
- Rules: if the digit after the place being rounded is <5, round down; if ≥5, round up
- First s.f.: first non-zero digit
- Examples & notes on maintaining requested decimal places when rounding
- Errors in measurement: concept of rounding errors; sources include estimation and instrument limitations
Absolute value (modulus)
- Definition: |x| is the size of x, ignoring its sign; |x| \ge 0
- Geometric: distance of x from 0 on the number line
- Algebraic: |x| = x if x \ge 0; |x| = -x if x < 0
- Key properties: |xy| = |x| |y|; |x/y| = |x|/|y| for y \neq 0
- Example: for numbers a, b, |a + b| does not generally equal |a| + |b|
- The modulus acts as a grouping symbol: compute inside first
Operations with integers, decimals, fractions; order of operations
- Order of operations: brackets, indices/exponents, multiplication and division (from left to right), addition and subtraction (from left to right)
- Fractions: numerator (top), denominator (bottom)
- Proper fraction: numerator < denominator; Improper fraction: numerator >= denominator
- Mixed numbers, improper fractions, and conversion between forms
- Example identities and simplifications
Prime numbers, factors (divisors) & multiples
- Prime number: a number with exactly two distinct positive divisors: 1 and itself (note: 1 is not prime)
- Prime numbers up to 100 listed (example: 2, 3, 5, 7, 11, 13, …, 97)
- Factors/divisors: numbers that divide the given number exactly
- Multiples: numbers that can be written as a product of the number with integers
- Example: factors of 12: 1, 2, 3, 4, 6, 12; multiples: 12, 24, 36, 48, …
Greatest common factor (GCF) / Highest common factor (HCF) and least common multiple (LCM) (HL only)
- GCF/HCF obtained from prime factorisation (corner method)
- LCM computed from prime factors as the product of highest powers of all primes involved
- Example: GCF(60, 72) = 12; LCM(60, 72) = 360
Straightforward applications of ratio, percentage and proportion
- Direct problems: express one quantity as a percentage of another; percentage increases/decreases; reverse percentages
- Percentage to fraction/decimal conversions and vice versa
- Understand unitary method and ratio method for direct proportion problems
- Examples include fractions like \frac{7}{8}, \frac{11}{15}, etc.; and converting 3/16 to a percentage via decimal first
Manipulation of algebraic expressions; factorisation & expansion
- Expand brackets; use identities: (a + b)^2 = a^2 + 2ab + b^2; (a - b)^2 = a^2 - 2ab + b^2; difference of squares: a^2 - b^2 = (a - b)(a + b)
- Factorising polynomials: find p and q such that ax^2 + bx + c = a(x + p)(x + q) or use grouping when a ≠ 1
- Examples: 8x^2 - 50 = 2(4x^2 - 25) = 2(2x - 5)(2x + 5); 36x^2 - 12x + 1 = (6x - 1)^2
- For ax^2 + bx + c with a ≠ 0, factorisation via p + q = b, pq = ac
- When a = 1: x^2 + bx + c = (x + p)(x + q) with p + q = b, pq = c
Rearranging formulae
- Make a given variable the subject of the formula
- Examples include rearranging C = 2πr to solve for r, or r = C/(2π), etc.
- Process often uses inverse operations and algebraic manipulation
Substitution; evaluating expressions by substitution
- Replace variables with given values and simplify
- Example: given a, b, c, evaluate expressions like a^2 + 3b - c, using the provided numbers
Evaluating exponential expressions; laws of indices; HL: rational exponents
- Index laws: am × an = am+n; (am)n = amn; a^m / a^n = a^{m-n}; a^0 = 1
- For rational exponents (HL): a^{m/n} = \sqrt[n]{a^m}
- Examples and simplifications using surds as needed
Use of inequalities;
- Inequalities represented on a number line; include examples of how to show solutions graphically
Simplification of simple expressions involving roots (surds or radicals)
- Surds: irrational radicals; simplify by factoring inside the radical, pull out perfect squares
- Rules for multiplying/dividing surds and for combining like surds
- Rationalising the denominator (HL): if denominator is a + b√p, multiply numerator/denominator by the radical conjugate a - b√p to remove the surd from the denominator
Express numbers in the form a × 10^k (scientific notation)
- Definition: a × 10^k with 1 ≤ a < 10, k ∈ ℤ
- Examples: 64000 = 6.4 × 10^4; 0.0054 = 5.4 × 10^-3
- Currency context: familiarity with world currencies and exchange rates
Currency conversions
- Example: €1 ≈ A$1.50; given a certain amount, compute the converted currency using the rate
- In 2017 example: €400 → A$ = 1.50 × 400; A$940 → € = 940 ÷ 1.5, etc.
Solution of linear equations and inequalities
- Linear equations: solve for x when the equation balances LHS and RHS
- Balance principle: perform the same operation on both sides to isolate the unknown
- Examples: 2x + 3 = 8 → x = 2.5; 4x + 3/5 = -2 → solve stepwise
- Inequalities: solve similarly but reverse the inequality sign when multiplying/dividing by a negative number
Quadratic equations (HL only)
- General form: a x^2 + b x + c = 0, a ≠ 0
- Solutions can be 0, 1, or 2 real roots depending on the discriminant D = b^2 - 4ac
- Methods: factorisation, completing the square, quadratic formula x = [-b ± √D] / (2a)
- Examples: x^2 - 4x + 4 = 0 → (x - 2)^2 = 0 → x = 2; 2x^2 + x - 24 = 0 → (3x - 6)(2x + 4) = 0, etc.
Solving systems of linear equations in two variables
- Methods: substitution or elimination
- Substitution: solve one equation for one variable and substitute into the other
- Elimination: multiply equations to align coefficients and subtract to eliminate a variable
- Example systems show x and y values; e.g., {3x - 2y = 5; y = 3 - 4x} → x = 1, y = -1
Sets and operations on sets
- Definitions: set, element, subset, universal set, empty set, complement
- Operations: union (A ∪ B), intersection (A ∩ B), disjointness, subset relations
- Venn diagrams: visual representation of sets, complements, unions, intersections
HL: Addition and subtraction of algebraic fractions
- Use LCD (lowest common denominator); convert fractions and combine numerators accordingly
- Examples show combining algebraic fractions with variables in denominators

Topic 2: Functions

Mappings and notation
- A function maps elements of the domain (input) to elements of the range (output)
- Representations: sets of ordered pairs, tables, diagrams, and graphs
Domain and range concepts
- Domain: set of all possible input values for which the function is defined
- Range: set of all possible output values produced by the function

Topic 3: Geometry and Trigonometry

Pythagoras’ theorem and its converse
- Theorem: for a right-angled triangle, c^2 = a^2 + b^2 where c is the hypotenuse
- Converse: if a^2 + b^2 = c^2, then the triangle is right-angled
- Hypotenuse is the longest side (opposite the right angle)
- Distinguish between a, b as legs and c as hypotenuse
Midpoint of a line segment; distance between two points
- Given A(x1, y1) and B(x2, y2):
- Midpoint M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right)
- Distance d between A and B: \sqrt{(x1 - x2)^2 + (y1 - y2)^2}
- 3D extension: midpoint and distance formula in 3D
Geometric concepts: point, line, plane
- Point: location with no size; Line: infinite sequence of points; Plane: flat infinite surface
Bearings and angle measurement
- Three-figure bearings: measure from North/South towards East/West; true bearings measured clockwise from true north (0°–360°) and written in three digits
- Compass bearings use only acute angles; true bearings use full 0–360° range
The triangle sum theorem
- In any triangle, sum of interior angles = 180°
- Exterior angle theorem: exterior angle equals sum of the two opposite interior angles
Right-angle trigonometry; SOH CAH TOA
- Ratios for a given angle A in a right triangle:
- sin A = opposite/hypotenuse
- cos A = adjacent/hypotenuse
- tan A = opposite/adjacent
- SOH CAH TOA mnemonic for remembering the ratios
Simple geometric transformations
- Translation: slide the figure by a vector (a, b); right/left by a, up/down by b
- Rotation: turn about a centre by a given angle and direction; invariant point: the centre stays fixed
- Reflection: flip about a mirror line (perpendicular bisector of the image pair)
- Enlargement (scale factor): centre of enlargement and a scale factor k; k > 1 enlarges, 0 < k < 1 reduces; negative k implies opposite-side enlargement
The circle and related terms
- Circle: set of all points at a fixed distance (radius r) from a fixed point (centre)
- Radius: distance from centre to any point on the circle; diameter: a chord through the centre; chord: line joining two circle points
- Arc: part of the circle between two points; minor and major arcs; semicircle is half a circle
- Sector: region between two radii and the circle
- Tangent: a line touching the circle at exactly one point; tangent is perpendicular to the radius at that point
- Segment: region between a chord and the circle
Perimeter and area of plane figures; polygons
- Polygon: straight-sided closed figure on a plane
- Triangles: classification by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse)
- Exterior angles: sum to 360° around a point; interior angle sum: for an n-sided polygon, sum = 180°(n − 2)
- Quadrilaterals: special types with properties (parallelograms, rectangles, squares, rhombuses, kites, trapezoids); area and perimeter formulas depend on shape
Three-dimensional shapes and their volumes/surface areas
- Prisms, pyramids, spheres, cylinders, cones
- Volume formulas for common solids (e.g., cuboid V = l w h; cylinder V = π r^2 h; cone V = (1/3)π r^2 h; pyramid V = (1/3) base area × height)
- Surface area concepts via nets; curved surfaces like cylinders have lateral surface area, etc.
Spheres, cylinders; surface area and volume in practice
- Hollow cylinder outer surface area: A = 2π r h (no ends); with ends: A = 2π r h + π r^2
- Solid cylinder: A = 2π r h + 2π r^2
- Volume of cuboids and prisms, pyramids and cones; Archimedes’ principles around spheres
Cumulative notes on transformation packages and practical examples

Topic 4: Statistics and Probability

Data collection and representation
- Types of graphs: bar charts, pie charts, pictograms, line graphs
- Bar charts: frequency represented by bar height/length; Pie charts: sectors proportional to frequencies; pictograms require a key
- Line graphs show how a quantity varies with another; independent variable on x-axis; dependent variable on y-axis
Discrete vs. continuous data; measures of central tendency
- Mean (arithmetic average): \bar{x} = (sum of values) / n
- Median: middle value in ordered data; for even n, mean of two middle values
- Mode: most frequently occurring value
- Range: difference between max and min values
Simple statistics: examples and interpretation
- Given a dataset, compute mean, median, mode; interpret in terms of data centre and spread
Probability basics; simple events & frequencies
- Relative frequency approximates theoretical probability; more trials → better approximation
- Expected number of occurrences of event A: E(A) = n × P(A), where n is the number of trials
- Venn diagrams for sorting data; union, intersection, complement; probability of A ∪ B, A ∩ B; conditional probability P(A|B)
- Independent vs dependent events; if independent, P(A ∩ B) = P(A) × P(B)
Tree diagrams and networks for probability
- Tree diagrams help model sequential events; important for conditional probabilities and dependent events
Probability exercises and techniques
- Methods include Venn diagrams, tree diagrams, sample spaces, and counting principles
- Various exercises on: single events, two events, conditional probabilities, mutually exclusive events, and compound events

Topic 5: Calculus (Speed, Distance, Time; related exercises)

Kinematic relationships
- Distance, speed, and time relationships:
- s = d/t, d = s × t, t = d/s
- Basic rearrangements used to solve problems about motion
Exercises and application areas
- Practical problems involving speed, distance and time; translating word problems into equations using s, d, t

Connections to other topics and broader context

Interconnections across topics:
- Algebraic manipulation underpins solving equations, inequalities, and systems (Topic 1 & 5)
- Geometry and trigonometry (Topic 3) rely on algebra and functions (Topic 2) for modelling and solving problems
- Statistics (Topic 4) requires understanding of functions, data representations and basic probability theory (Topics 1 & 4)
- Transformations and geometric reasoning link to coordinate geometry in higher topics
Real-world relevance:
- Currency exchange, unit conversions, scientific notation, and measurement errors reflect practical data handling
- Proportions and ratios appear in everyday contexts (recipes, mixing, scaling, distributions)
- Measurements and geometry inform design, engineering, and navigation (bearings, angles, distances)

Formulas to remember (LaTeX)

Pythagoras: $c^2 = a^2 + b^2$
Distance: $d = \sqrt{(x1 - x2)^2 + (y1 - y2)^2}$
Midpoint: $M = \left(\frac{x1+x2}{2}, \frac{y1+y2}{2}\right)$
Circle relations: circumference $C = 2\pi r = \pi d$ ; area $A = \pi r^2$ ; arc length $\text{arc length} = \dfrac{\phi}{360^\circ} \times C$ ; sector area $A_{\text{sector}} = \dfrac{\phi}{360^\circ} \times \pi r^2$
Exponents: $a^m \cdot a^n = a^{m+n},\quad \left(a^m\right)^n = a^{mn},\quad \dfrac{a^m}{a^n} = a^{m-n}$
Surds/rationalising: If the denominator has the form $a+ b\sqrt{c}$ , multiply by its rationalising conjugate $a - b\sqrt{c}$ to obtain a rational denominator
Scientific notation: a \times 10^k\quad(1 \le a < 10)
Linear equations: balance principle; example solving steps depend on setup
Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant: $D = b^2 - 4ac$
Set operations: union $A \cup B$ , intersection $A \cap B$ , complement $A'$
Probability basics: conditional $P(A|B) = \frac{P(A \cap B)}{P(B)}$ , addition rule $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Speed-distance-time relation: $s = \frac{d}{t}, \quad d = s t, \quad t = \frac{d}{s}$
Currency conversion (example form): if €1 = A$1.50, then €x = \$1.50 x; similarly, $ to € conversions follow the reciprocal rate

Note: This set of notes emphasizes the major and minor points introduced in the transcript, with emphasis on definitions, key formulas (in LaTeX), and representative examples. Use these notes alongside practice problems to solidify understanding across SL and HL topics.