Basic Mathematics

Place value in base ten

Base ten uses the digits 0–9. Reading from the right, the place values are ones, tens, hundreds, thousands and so on. For example, in 3 426 the digit 4 stands for 4 hundreds.

Types of numbers

• Natural (counting) numbers: 1, 2, 3, …
• Whole numbers: 0, 1, 2, 3, …
• Integers: …, −2, −1, 0, 1, 2, … (include negative numbers).

Operations on whole numbers

• Addition and subtraction are inverse operations.
• Multiplication is repeated addition; division is repeated subtraction.
• When an expression has several operations, follow the order BODMAS.

Fractions

A fraction a/b has a numerator a and a denominator b. Equivalent fractions have the same value (1/2 = 2/4). To add or subtract, use a common denominator; to multiply, multiply numerators and denominators; to divide, multiply by the reciprocal.

Decimals

A decimal is a fraction whose denominator is a power of ten. The decimal point separates the whole-number part from the fractional part. When adding or subtracting, align the decimal points.

Percentage

Percentage means 'out of a hundred'. To change a fraction to a percentage, multiply by 100%. Percentages are used for discounts, interest, and profit or loss.

Common metric units

• Length: millimetre, centimetre, metre, kilometre.
• Mass: gram, kilogram, tonne.
• Capacity: millilitre, litre.
• Time: second, minute, hour.

Converting units

To convert from a larger unit to a smaller one, multiply; from a smaller unit to a larger one, divide. For example, 1 m = 100 cm, so 3 m = 300 cm.

Rounding off

To round to a given place, look at the next digit: if it is 5 or more, round up; otherwise round down. For example, 3.78 ≈ 3.8 to one decimal place.

Significant figures

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.

Basic terms

• A point shows position; a line has length only.
• An angle is formed when two lines meet at a point.
• Types of angles: acute (<90°), right (90°), obtuse (between 90° and 180°), reflex (>180°).

Angle properties

• Angles on a straight line add up to 180°.
• Angles at a point add up to 360°.
• Vertically opposite angles are equal.

Algebraic expressions

An expression combines variables and numbers using operations, e.g. 2x + 3. Like terms (such as 2x and 5x) can be added or subtracted; unlike terms cannot.

Linear equations

An equation states that two expressions are equal. To solve it, perform the same operation on both sides until the variable stands alone, e.g. x + 5 = 12 gives x = 7.

Ratio and proportion

A ratio a : b compares quantities. To share an amount in a ratio, add the parts and find the value of one part. A proportion states that two ratios are equal.

Profit and loss

• Profit = Selling Price − Buying Price (when SP > BP).
• Loss = Buying Price − Selling Price (when BP > SP).
• Percentage profit = (Profit ÷ Buying Price) × 100%.

Coordinates

A point is written as (x, y), where x is the horizontal distance and y the vertical distance from the origin. For example, (3, 2) is 3 units right and 2 units up.

Quadrants

The two axes divide the plane into four regions called quadrants, numbered anticlockwise starting from the top right.

Perimeter

• Rectangle: P = 2(l + w).
• Triangle: the sum of the three sides.
• Circle (circumference): C = 2πr.

Area

• Rectangle: A = l × w.
• Triangle: A = ½ × base × height.
• Circle: A = πr².

Laws of exponents

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1 and a⁻ⁿ = 1/aⁿ

Radicals (surds)

A radical such as √a is a number whose square is a. Surds are simplified using √(ab) = √a × √b, and a denominator is rationalised by multiplying by a suitable surd.

Expansion and factorisation

To expand, remove brackets using the distributive law, e.g. a(b + c) = ab + ac. Factorisation is the reverse — writing an expression as a product of its factors.

Simultaneous equations

Two equations with two unknowns are solved together by substitution or elimination, giving the values that satisfy both equations at once.

Solving by factorisation

Write the equation as a product of two factors equal to zero, then set each factor to zero, e.g. x² − 5x + 6 = 0 gives (x − 2)(x − 3) = 0, so x = 2 or x = 3.

The quadratic formula

When factorisation is difficult, use x = [−b ± √(b² − 4ac)] / 2a. The expression b² − 4ac (the discriminant) tells the number of real roots.

Laws of logarithms

• log(MN) = log M + log N
• log(M/N) = log M − log N
• log(Mⁿ) = n log M

Common logarithms

Common logarithms use base 10 and were traditionally read from tables (characteristic and mantissa) to simplify long calculations.

Conditions for congruent triangles

• SSS — three sides equal.
• SAS — two sides and the included angle equal.
• ASA — two angles and the included side equal.
• RHS — right angle, hypotenuse and one side equal.

Similar triangles

If two triangles are similar, the ratios of their corresponding sides are equal. This is used to find unknown lengths and to make scale drawings and maps.

Types of transformation

• Reflection — a flip across a mirror line.
• Rotation — a turn about a fixed point through an angle.
• Translation — a slide in a given direction.
• Enlargement — a change of size by a scale factor about a centre.

The theorem

In a right-angled triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides: a² + b² = c². It is used to find an unknown side when the other two are known.

The three ratios

• sin θ = opposite ÷ hypotenuse
• cos θ = adjacent ÷ hypotenuse
• tan θ = opposite ÷ adjacent

Applications

These ratios are used to find unknown sides and angles, and to solve problems involving angles of elevation and depression.

Set operations

• Union (A ∪ B): elements in A or B or both.
• Intersection (A ∩ B): elements in both A and B.
• Complement (A′): elements not in A.

Venn diagrams

Venn diagrams use circles within a rectangle (the universal set) to show the relationships between sets and to solve problems on numbers of elements.

Presenting data

• Frequency table — shows how often each value occurs.
• Bar chart — bars whose heights show frequency.
• Pie chart — a circle divided into sectors proportional to frequency.

Representing relations

A relation can be shown by a set of ordered pairs, an arrow diagram, a table or a graph. The domain is the set of inputs and the range is the set of outputs.

Function notation

A function may be written as f(x). To evaluate f(x) for a value, substitute the value for x, e.g. if f(x) = 2x + 1 then f(3) = 7.

Graphs of functions

A linear function gives a straight-line graph; a quadratic function gives a curve called a parabola.

Mean, median and mode

• Mean — the sum of values divided by their number.
• Median — the middle value when data is arranged in order.
• Mode — the value that occurs most often.

Rates

A rate is a ratio of two different quantities, such as speed = distance ÷ time, or density = mass ÷ volume.

Direct and inverse variation

In direct variation, as one quantity increases the other increases in the same ratio. In inverse variation, as one increases the other decreases.

Arithmetic progression (AP)

An AP has a common difference d. The nth term is aₙ = a + (n − 1)d, where a is the first term.

Geometric progression (GP)

A GP has a common ratio r. The nth term is aₙ = arⁿ⁻¹. GPs model growth such as compound interest.

Parts of a circle

• Radius, diameter and chord.
• Arc, sector and segment.
• Tangent — a line touching the circle at one point.

Circle theorems

• The angle at the centre is twice the angle at the circumference on the same arc.
• Angles in the same segment are equal.
• The angle in a semicircle is 90°.

Latitude and longitude

Latitudes are circles parallel to the Equator (0°–90° N or S); longitudes (meridians) run from pole to pole (0°–180° E or W) from the Greenwich meridian.

Distances

Distances along a great circle can be found from the angle subtended at the centre, since 1° along a great circle ≈ 60 nautical miles.

Double-entry principle

Every transaction is recorded twice — as a debit in one account and a credit in another — so that total debits equal total credits.

Trial balance

A trial balance lists the balances of all accounts to check that total debits equal total credits before preparing final accounts.

Key formulae

• Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]
• Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Gradient m = (y₂ − y₁)/(x₂ − x₁)

Equation of a line

A straight line can be written as y = mx + c, where m is the gradient and c is the y-intercept.

Sectors and arcs

For a sector of angle θ in a circle of radius r: arc length = (θ/360) × 2πr and area = (θ/360) × πr².

Volume

• Prism/cylinder: V = base area × height.
• Cone: V = ⅓πr²h.
• Sphere: V = 4⁄3 πr³.

Single events

Probability of an event = (number of favourable outcomes) ÷ (total number of possible outcomes).

Combined events

For independent events, P(A and B) = P(A) × P(B); for mutually exclusive events, P(A or B) = P(A) + P(B).

Sine and cosine rules

• Sine rule: a/sin A = b/sin B = c/sin C.
• Cosine rule: a² = b² + c² − 2bc·cos A.

Vector operations

Vectors are added by joining them head to tail; the resultant runs from the start of the first to the end of the last. The magnitude of a vector (x, y) is √(x² + y²).

Operations

Matrices of the same order are added or subtracted element by element. Multiplication combines rows of the first matrix with columns of the second.

Transformations

A 2×2 matrix can transform points on the plane, representing reflections, rotations and enlargements.

Steps

• Form inequalities from the constraints.
• Draw the inequalities and identify the feasible region.
• Test the corner points in the objective function to find the optimum.