Mathematics

1. Mathematics: The Versatile and Beautiful Foundation

Mathematics is probably the most versatile subject there is.

Ubiquitous applications. Mathematics underpins countless aspects of modern life and technology, from the laws of physics enabling daily technology to modeling stock markets and traffic flow. It is essential in diverse fields like science, geography, art, psychology, medicine, architecture, and engineering. Understanding basic mathematics is invaluable for navigating the technological age.

Intrinsic beauty. Beyond its practical uses, mathematics possesses inherent beauty, appreciated by artists like Kandinsky and Escher who incorporated geometry and tessellations into their work. Natural phenomena, such as the symmetry in butterfly wings or the Fibonacci sequence in sunflowers and shells, also reveal mathematical patterns, demonstrating its presence in the natural world.

Everyday relevance. Many people underestimate their daily use of mathematics, from calculating value for money and understanding probability in betting to managing finances with percentages and interpreting statistics in newspapers. Developing numeracy skills is as crucial as literacy for informed decision-making and career development, boosting both employability and self-esteem.

2. Numbers: The Essential Building Blocks

Our system of writing numbers is called the decimal system, because it is based on ten, the number of fingers and thumbs we have.

Place value system. The decimal system uses digits 0-9, where the position of a digit determines its value (units, tens, hundreds, etc.). Zero is crucial for indicating empty columns, allowing representation of numbers like 5023 (five thousand and twenty-three) where hundreds are absent. Larger numbers are grouped in threes for readability.

Arithmetic operations. The four fundamental operations—addition, subtraction, multiplication, and division—are the bedrock of mathematics. Each has a specific symbol and name for its result: sum (+), difference (-), product (×), and quotient (÷). Multiplication and division are inverse processes, as are addition and subtraction.

Special number types. Numbers can be categorized into various types, each with unique properties.

• Whole numbers: 0, 1, 2, 3...
• Natural numbers: 1, 2, 3... (also counting or positive whole numbers)
• Integers: ..., -3, -2, -1, 0, 1, 2, 3... (positive, negative, and zero)
• Even/Odd numbers: Divisible by 2 or not.
• Square/Triangle/Rectangle/Cube numbers: Based on geometric arrangements or repeated multiplication.

3. Angles: Measuring Turn and Direction

An angle is formed when two straight lines meet...

Defining angles. Angles measure the amount of turn from one line to another, with the meeting point called the vertex. A complete turn is 360 degrees (°), a half turn is 180° (a straight angle), and a quarter turn is 90° (a right angle). Lines meeting at 90° are perpendicular.

Types of angles. Angles are classified based on their size relative to a right angle and a straight angle.

• Acute angle: Less than 90°.
• Obtuse angle: Between 90° and 180°.
• Reflex angle: Greater than 180°.
  Angles can be named using a letter inside, the vertex letter, or three letters with the vertex in the middle (e.g., ∠ABC).

Angle facts and properties. Understanding relationships between angles is key to solving geometric problems.

• Angles on a straight line sum to 180° (supplementary angles).
• Angles at a point sum to 360°.
• Vertically opposite angles (formed by intersecting lines) are equal.
• Corresponding angles (F-shape) and alternate angles (Z-shape) formed by a transversal crossing parallel lines are equal.

4. Fractions, Decimals, and Percentages: Parts of the Whole

A fraction is one or more equal parts of a whole.

Fractions basics. Fractions represent parts of a whole, written with a numerator (top number) and a denominator (bottom number). Equivalent fractions represent the same value (e.g., 1/2 = 2/4). Simplifying fractions involves dividing the numerator and denominator by their highest common factor to reach the lowest terms.

Types and operations. Fractions can be proper (numerator < denominator), improper (numerator > denominator), or mixed numbers (whole number + proper fraction). Operations follow specific rules:

• Adding/Subtracting: Requires a common denominator.
• Multiplying: Multiply numerators and denominators.
• Dividing: Multiply by the reciprocal (inverted second fraction).
  Fractions are also used to express one quantity as a part of another, requiring consistent units.

Decimals and percentages. Decimals extend the place value system to represent fractions with denominators of 10, 100, 1000, etc. Percentages are fractions out of 100 (per cent). Conversions between these forms are fundamental:

• Decimal to fraction: Use place value (0.7 = 7/10).
• Fraction to decimal: Divide numerator by denominator (3/4 = 0.75).
• Percentage to decimal: Divide by 100 (25% = 0.25).
• Decimal to percentage: Multiply by 100 (0.5 = 50%).
  Decimals can be terminating (end) or recurring (repeat).

5. Two-Dimensional Shapes: Exploring Plane Geometry

Two-dimensional shapes are flat shapes, such as squares and circles.

Basic shapes. Plane geometry studies flat shapes like triangles and quadrilaterals. Triangles are three-sided shapes with an angle sum of 180°. Quadrilaterals are four-sided shapes with an angle sum of 360°. These angle sums can be proven by dividing the shapes into triangles.

Types and properties. Shapes are classified by side lengths, angles, and symmetries.

• Triangles: Equilateral (3 equal sides/angles), Isosceles (2 equal sides/angles), Scalene (no equal sides/angles). Can be acute, obtuse, or right-angled.
• Quadrilaterals: Square, rectangle, parallelogram, rhombus, trapezium, kite. Each has specific properties regarding sides, angles, and diagonals.
• Polygons: Shapes with three or more straight sides. Regular polygons have equal sides and angles.

Symmetry and congruence. Shapes can have lines of symmetry (mirror lines) and rotational symmetry (fitting onto itself when rotated). Congruent shapes are identical in size and shape. Tessellations occur when shapes fit together without gaps or overlaps, often related to their interior angles summing to 360° at a point.

6. Statistics: Collecting, Organizing, and Analyzing Data

Statistics is the branch of mathematics concerned with the collection, organization and analysis of data...

Data collection. Data can be gathered from existing sources, experiments, or surveys. Surveys often use samples to represent a larger population, emphasizing the need for representative samples to avoid bias. Data collection sheets or tally charts help organize raw data into frequency tables.

Data presentation. Visual displays make data easier to understand.

• Pictograms: Use symbols to represent frequencies.
• Bar charts: Use bar heights (or lengths) to show frequencies.
• Pie charts: Use sectors (slices) to show proportions of a whole.
• Line graphs: Show trends over time, joining points when values change gradually.
• Scatter graphs: Show relationships between two sets of data points.

Data types and grouping. Numerical data is either discrete (exact values, often counted) or continuous (measured, not exact). Non-numerical data is categorical. For large datasets, data can be grouped into class intervals, presented in grouped frequency tables and visualized with bar charts or frequency polygons (joining mid-points of bars).

7. Directed Numbers: Working with Positives and Negatives

Numbers greater than zero are called positive... Numbers less than zero are called negative numbers.

Understanding directed numbers. Directed numbers include positive numbers (greater than zero, usually written without a sign) and negative numbers (less than zero, written with a minus sign). They represent values with direction, often visualized on a number line or thermometer scale. Ordering directed numbers involves comparing their positions on the number line.

Operations with directed numbers. Adding and subtracting directed numbers follows specific rules, which can be visualized on a number line or understood through patterns.

• Adding a negative is like subtracting a positive (e.g., 3 + (-2) = 3 - 2 = 1).
• Subtracting a negative is like adding a positive (e.g., 3 - (-2) = 3 + 2 = 5).
  Thinking of debt can also help: adding debt decreases your balance, taking away debt increases it.

Multiplication and division. The rules for multiplying and dividing directed numbers depend on the signs of the numbers involved.

• Same signs (positive × positive, negative × negative) result in a positive product/quotient.
• Different signs (positive × negative, negative × positive) result in a negative product/quotient.
  Calculators can handle directed number operations, often requiring a specific key for negative input.

8. Graphs: Visualizing Relationships and Data

Coordinates are used in mathematics to describe the position of a point.

Using coordinates. Cartesian coordinates use a horizontal x-axis and a vertical y-axis intersecting at the origin (0,0) to specify a point's position as an ordered pair (x, y). The x-coordinate is the distance along the x-axis, and the y-coordinate is the distance along the y-axis. Negative coordinates extend the system left of the y-axis and below the x-axis.

Straight-line graphs. Linear equations (not involving powers of x like x² or x³) graph as straight lines. Plotting points from a table of values and joining them reveals the line. The equation y = mx + c represents a straight line with gradient 'm' and a y-intercept at (0, c). Lines parallel to axes have simple equations: x = constant (vertical) or y = constant (horizontal).

Interpreting graphs. Graphs visually represent relationships between quantities.

• Conversion graphs: Show equivalence between different units (e.g., kg and lb).
• Distance-time graphs: Show journeys, with slope indicating speed (steeper = faster, horizontal = stationary).
• Regions: Inequalities can describe areas on a graph, often bounded by lines parallel to the axes (e.g., x ≥ 1).

9. Measurement: Quantifying the Physical World

The metric system is a system of units based on the metre...

The metric system. The metric system is a decimal-based system using prefixes (milli-, centi-, kilo-, etc.) to denote fractions or multiples of base units.

• Length: metre (m), millimetre (mm), centimetre (cm), kilometre (km).
• Mass: gram (g), milligram (mg), kilogram (kg), tonne (t).
• Capacity: litre (l), millilitre (ml), centilitre (cl).
  Conversions within the metric system involve multiplying or dividing by powers of 10 (10, 100, 1000).

Imperial units. Traditional English units, though largely superseded by metric, are still in use for some purposes.

• Length: inch (in), foot (ft), yard (yd), mile.
• Mass: ounce (oz), pound (lb).
• Capacity: pint (pt), gallon.
  Relationships between imperial units are often less straightforward than metric (e.g., 12 inches = 1 foot, 16 ounces = 1 pound).

Conversions and suitability. Converting between metric and imperial units is often necessary, using approximate equivalents (e.g., 1 inch ≈ 2.5 cm, 1 mile ≈ 1.6 km). Choosing appropriate units for measurement depends on the scale of the object or distance being measured (e.g., cm or mm for page width, kg or tonnes for car weight).

10. Perimeter and Area: Defining Boundaries and Surfaces

The perimeter of a shape is the length of its boundary.

Perimeter. Perimeter is the total length of the boundary of a two-dimensional shape. For polygons, it's found by summing the lengths of all sides. Rectangles have a formula: Perimeter = 2 × length + 2 × width. For complex shapes, tracing the boundary and adding lengths works.

Area. Area measures the amount of surface a shape covers, typically quantified in square units (e.g., cm²). It can be found by counting unit squares within a shape, approximating for irregular shapes.

Area formulae. Specific formulae exist for common shapes:

• Rectangle: Area = length × width.
• Square: Area = side × side.
• Parallelogram: Area = base × height (perpendicular height).
• Triangle: Area = ½ × base × height (perpendicular height).
• Trapezium: Area = ½ × (sum of parallel sides) × vertical distance between them.
  Complex shapes can often be divided into simpler shapes (rectangles, triangles, trapeziums) to calculate their total area.

11. Algebraic Expressions: The Language of Symbols

The Arabic word algebra originally meant the study of equations but has come to mean the whole branch of mathematics in which letters and symbols are used to generalize mathematical relations.

Writing expressions. Algebra uses letters (variables) to represent numbers, allowing for generalization of mathematical relationships. Expressions combine variables, numbers, and operations (addition, subtraction, multiplication, division). Multiplication signs are often omitted (e.g., 5y for 5 × y), and numbers usually precede letters (e.g., 3a).

Simplifying expressions. Expressions can be simplified by collecting like terms (terms containing the same variables raised to the same powers). For example, 3a + 4a simplifies to 7a, but 3a + 4b cannot be simplified further as 'a' and 'b' are unlike terms.

Evaluating expressions. If the values of the variables are known, an expression can be evaluated by substituting the numbers for the letters and performing the calculations following the order of operations (BiDMAS). Squaring (x²) and cubing (x³) involve multiplying a variable by itself two or three times, respectively.

Brackets and factorization. Brackets group terms, indicating operations within them should be done first. Expanding brackets involves multiplying the term outside by each term inside (e.g., a(b+c) = ab + ac). Factorizing is the reverse process, rewriting an expression as a product of factors, often involving finding common factors and using brackets (e.g., 7x - 21 = 7(x - 3)).

12. Equations and Inequalities: Finding Unknowns

An equation is a mathematical statement that two expressions are equal...

Understanding equations. Equations state that two expressions are equal, often containing unknown numbers represented by letters (variables). Solving an equation means finding the value(s) of the unknown(s) that make the statement true; these values are called solutions or roots.

Solving linear equations. Linear equations involve variables raised only to the power of one (e.g., 2x + 1 = 9). They can be solved by performing the same operation (adding, subtracting, multiplying, dividing) to both sides to isolate the variable. Thinking of an equation as a balance helps maintain equality.

Equations with brackets. Equations containing brackets are typically solved by first expanding the brackets and then simplifying the resulting expression before isolating the variable.

Solving inequalities. Inequalities compare expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Solving inequalities uses similar techniques to solving equations, but multiplying or dividing both sides by a negative number requires reversing the inequality sign.

Last updated: May 19, 2025