Understanding rational and irrational numbers is fundamental in mathematics, particularly within the Cambridge IGCSE curriculum for Mathematics - International - 0607 - Advanced. These classifications help in identifying and categorizing the vast array of numbers encountered in various mathematical contexts, from simple calculations to complex problem-solving scenarios. Grasping the distinction between rational and irrational numbers not only reinforces numerical literacy but also lays the groundwork for higher mathematical concepts and applications.

Key Concepts

Definition of Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the numerator is an integer and the denominator is a non-zero integer. In mathematical terms, a number is rational if it can be written in the form:

$$ \frac{p}{q} $$

where $ p $ and $ q $ are integers, and $ q \neq 0 $. This definition encompasses integers, fractions, terminating decimals, and repeating decimals.

Examples of Rational Numbers

Consider the following examples:

Integers such as 5, -3, and 0 are rational because they can be expressed as $ \frac{5}{1} $, $ \frac{-3}{1} $, and $ \frac{0}{1} $.
Fractions like $ \frac{2}{3} $, $ \frac{-7}{4} $, and $ \frac{10}{5} $ (which simplifies to 2) are rational numbers.
Terminating decimals such as 0.75 and 3.5 can be expressed as $ \frac{3}{4} $ and $ \frac{7}{2} $ respectively.
Repeating decimals like 0.\overline{3} and 2.142857\overline{142857} are also rational numbers, as they can be represented as fractions.

Definition of Irrational Numbers

Irrational numbers, on the other hand, are numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-repeating. This means that they go on forever without repeating a pattern, and there is no fraction $ \frac{p}{q} $ that exactly represents them.

Examples of Irrational Numbers

Prominent examples of irrational numbers include:

$\sqrt{2}$: The square root of 2 is perhaps the most famous irrational number, proven by the ancient Greeks.
$\pi$: Pi, the ratio of a circle's circumference to its diameter, is an irrational number often approximated as 3.14159.
Euler's number $ e $: Approximately 2.71828, $ e $ is fundamental in calculus, particularly in the study of exponential growth and complex numbers.
The golden ratio $ \phi $: Approximately 1.61803, the golden ratio appears in various areas of art, architecture, and nature.

Properties of Rational Numbers

Rational numbers possess several key properties:

Closure under Addition and Multiplication: The sum or product of two rational numbers is always a rational number. For example, $ \frac{1}{2} + \frac{1}{3} = \frac{5}{6} $.
Density: Between any two rational numbers, there exists another rational number. This property indicates that rational numbers are densely packed on the number line.
Order: Rational numbers can be ordered on the number line based on their size relative to each other.

Properties of Irrational Numbers

Irrational numbers are characterized by the following properties:

Non-Repeating and Non-Terminating Decimals: As mentioned earlier, their decimal expansions do not terminate or repeat.
Density: Similar to rational numbers, irrational numbers are also dense on the number line, meaning there are infinitely many irrational numbers between any two integers.
Not Closed under Addition or Multiplication: The sum or product of two irrational numbers can be either rational or irrational. For instance, $ \sqrt{2} + (-\sqrt{2}) = 0 $, which is rational, while $ \sqrt{2} \times \sqrt{2} = 2 $, also rational.

Decimal Representations

Understanding the decimal representations of rational and irrational numbers is crucial:

Rational Numbers: Their decimal forms either terminate after a finite number of digits or eventually begin to repeat a sequence of digits infinitely. For example, $ \frac{1}{4} = 0.25 $ (terminating) and $ \frac{1}{3} = 0.\overline{3} $ (repeating).
Irrational Numbers: They have infinite non-repeating decimal expansions. The lack of a repeating pattern indicates the non-rationality of the number. For example, $ \pi = 3.1415926535\ldots $.

Algebraic Representations

Algebraically, rational and irrational numbers can be distinguished through their expressions:

Rational: Any number that can be expressed as a fraction $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $, is rational.
Irrational: Numbers that cannot be expressed as such fractions are irrational. This often includes roots of non-perfect squares, logarithms of certain numbers, and transcendental numbers like $ e $ and $ \pi $.

Historical Context

The discovery of irrational numbers is attributed to the ancient Greek mathematicians, particularly the Pythagoreans. The realization that not all lengths could be expressed as ratios of integers (e.g., the diagonal of a unit square, $ \sqrt{2} $) was a significant milestone in the development of mathematics, challenging previously held beliefs about numbers and their properties.

Applications of Rational and Irrational Numbers

Both rational and irrational numbers have extensive applications across various fields:

Engineering: Precise calculations often require rational numbers, while constants like $ \pi $ are essential for designing structures involving circles and curves.
Computer Science: Understanding number types aids in algorithm design, particularly in areas like numerical analysis and cryptography.
Finance: Rational numbers are used in financial calculations, such as interest rates and loan amortizations.
Physics: Irrational numbers appear in formulas describing fundamental constants and phenomena, such as quantum mechanics and relativity.

Graphical Representation

On the number line, rational and irrational numbers are interspersed densely. However, irrational numbers cannot be pinpointed exactly due to their non-repeating, infinite decimal nature. Rational numbers, being expressible as fractions, can be precisely located based on their numerator and denominator.

Identifying Rational and Irrational Numbers

To determine whether a number is rational or irrational, consider the following steps:

Attempt to express the number as a fraction $ \frac{p}{q} $. If possible, the number is rational.
If the number's decimal expansion terminates or repeats, it is rational.
If neither of the above conditions holds, the number is irrational.

For example, $ 0.\overline{6} = \frac{2}{3} $ (rational), while $ \sqrt{3} $ cannot be expressed as a simple fraction (irrational).

Formal Proofs

Proof that $ \sqrt{2} $ is Irrational

Assume, for contradiction, that $ \sqrt{2} $ is rational. Then it can be written as $ \frac{p}{q} $, where $ p $ and $ q $ are coprime integers (i.e., their greatest common divisor is 1).

Squaring both sides: $ 2 = \frac{p^2}{q^2} $
Multiplying both sides by $ q^2 $: $ 2q^2 = p^2 $
This implies that $ p^2 $ is even, so $ p $ must be even. Let $ p = 2k $ for some integer $ k $.
Substituting back: $ 2q^2 = (2k)^2 = 4k^2 $, simplifying to $ q^2 = 2k^2 $
This implies that $ q^2 $ is even, so $ q $ must also be even.
However, this contradicts the initial assumption that $ p $ and $ q $ are coprime, as both are even and share a factor of 2.

Therefore, $ \sqrt{2} $ cannot be expressed as a ratio of two integers, and thus, it is irrational.

Advanced Concepts

Algebraic Irrational Numbers

Algebraic irrational numbers are roots of non-linear polynomial equations with integer coefficients but cannot be expressed as simple fractions. The classification of numbers into algebraic and transcendental further refines the nature of irrationals.

For example, $ \sqrt[3]{2} $ is the real root of the polynomial equation $ x^3 - 2 = 0 $, making it algebraic. However, it is not a solution to any linear polynomial equation with rational coefficients, confirming its irrationality.

Transcendental Numbers

Transcendental numbers are a subset of irrational numbers that are not roots of any non-zero polynomial equation with integer coefficients. Famous examples include $ \pi $ and $ e $. These numbers hold significant importance in various mathematical theories and applications.

Proof of $ e $ Being Transcendental

The transcendence of $ e $ was established by Charles Hermite in 1873. The proof involves demonstrating that no non-zero polynomial with integer coefficients can have $ e $ as a root. This places $ e $ firmly within the realm of transcendental numbers, separating it from algebraic irrationals like $ \sqrt{2} $.

Measure of Irrationality

The measure of irrationality quantifies how closely an irrational number can be approximated by rational numbers. It is defined as the infimum of the set of exponents $ \mu $ such that the inequality \( \left| x - \frac{p}{q} \right|

For example, Liouville numbers are a class of numbers with an infinite measure of irrationality, making them exceptionally difficult to approximate by rational numbers. This concept plays a role in number theory and Diophantine approximation.

Continued Fractions

Continued fractions provide a way to represent real numbers through an ongoing sequence of integer divisions, offering insights into their properties. For rational numbers, continued fractions terminate, while for irrational numbers, they continue indefinitely without repeating patterns.

For instance, the continued fraction representation of $ \sqrt{2} $ is:

$$ 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}} $$

This infinite continued fraction reflects the irrationality of $ \sqrt{2} $.

Density and Cardinality

Both rational and irrational numbers are dense on the real number line, meaning between any two real numbers, there exists both a rational and an irrational number. However, their cardinalities differ significantly:

Rational Numbers: Countably infinite, meaning they can be put into a one-to-one correspondence with the natural numbers.
Irrational Numbers: Uncountably infinite, indicating a higher level of infinity than that of rational numbers.

This distinction highlights the vastness of irrational numbers compared to rational numbers, emphasizing their prevalence in the real number system.

Topological Properties

In topology, the real number line is uncountable and complete, but the subsets of rational and irrational numbers have unique properties:

Rational Numbers: They form a dense, countable set with no interior points, making them a type of "nowhere dense" set in the real number line.
Irrational Numbers: As an uncountable dense set, they constitute a "comeager" subset in the real numbers, meaning they are prevalent in the sense of Baire category.

Applications in Calculus

Rational and irrational numbers play pivotal roles in calculus, particularly in the concepts of limits, continuity, and differentiation:

Limits: Understanding the behavior of functions as they approach rational or irrational points is essential in limit calculations.
Continuity: Functions may behave differently at rational and irrational points, influencing their continuity and differentiability.
Differentiation: The properties of numbers ensure the smoothness and predictability of function derivatives, a cornerstone of calculus.

Interdisciplinary Connections

The distinction between rational and irrational numbers intersects with various disciplines:

Physics: Constants like $ \pi $ and $ e $ are fundamental in physical laws and equations governing the universe.
Engineering: Precise measurements and calculations rely on both rational approximations and the inherent irrationality of certain constants.
Computer Science: Algorithms that involve numerical methods must account for the representation and approximation of irrational numbers.
Art and Architecture: The golden ratio $ \phi $, an irrational number, is employed to achieve aesthetically pleasing proportions in design.

Complex Problem-Solving

Advanced problems involving rational and irrational numbers often require multi-step reasoning and the integration of various mathematical concepts. For example:

Problem: Prove that the sum of a rational number and an irrational number is irrational.

Solution:

Assume, for contradiction, that the sum is rational. Let $ r $ be a rational number and $ i $ be an irrational number. Suppose $ r + i = s $, where $ s $ is rational.
Then, $ i = s - r $.
Since $ s $ and $ r $ are both rational, their difference $ s - r $ must also be rational.
This implies that $ i $ is rational, which contradicts the initial assumption that $ i $ is irrational.
Therefore, the sum $ r + i $ must be irrational.

This problem demonstrates the interplay between rational and irrational numbers, showcasing logical reasoning and proof techniques.

Comparison Table

Aspect	Rational Numbers	Irrational Numbers
Definition	Numbers that can be expressed as a ratio of two integers ($ \frac{p}{q} $, where $ q \neq 0 $)	Numbers that cannot be expressed as a ratio of two integers; their decimal expansions are non-terminating and non-repeating
Decimal Representation	Either terminating or repeating decimals	Non-terminating and non-repeating decimals
Examples	0.5, $ \frac{3}{4} $, 7, -2	$ \pi $, $ \sqrt{2} $, $ e $
Density on Number Line	Dense; between any two rationals, there is another rational	Dense; between any two irrationals, there is another irrational
Cardinality	Countably infinite	Uncountably infinite
Closure Properties	Closed under addition, subtraction, multiplication, and division (except by zero)	Not necessarily closed under these operations; results can be rational or irrational

Summary and Key Takeaways

Rational numbers can be expressed as fractions of integers, featuring terminating or repeating decimals.
Irrational numbers cannot be expressed as simple fractions, with infinite non-repeating decimal expansions.
Both types are dense on the number line, but they differ in cardinality and closure properties.
Understanding these distinctions is essential for advanced mathematical concepts and interdisciplinary applications.

Examiner Tip

Tips

- **Mnemonic for Identifying Irrationals:** Remember “**D**ecimal **N**ot **R**epeating or **T**erminating” for irrationals.
- **Proof Techniques:** Practice proof by contradiction to solidify understanding of why certain numbers are irrational.
- **Use Visual Aids:** Utilize number lines to visualize the density and distribution of rational and irrational numbers.
- **Regular Revision:** Frequently revisit definitions and properties to ensure retention and application during exams.

Did You Know

1. The number $ \pi $ is not only irrational but also transcendental, meaning it cannot be the solution of any polynomial equation with integer coefficients.
2. The golden ratio $ \phi $ appears in famous structures like the Parthenon and is believed to contribute to their aesthetically pleasing proportions.
3. Despite being irrational, $ e $ is essential in describing growth processes, such as population growth and compound interest, in real-world scenarios.

Common Mistakes

1. **Confusing Terminating with Repeating Decimals:** Students often mistake terminating decimals as the only rational decimals, forgetting that repeating decimals are also rational.
Incorrect: 0.25 is irrational because it doesn’t repeat.
Correct: 0.25 is rational because it terminates.
2. **Assuming All Roots are Irrational:** Not all square roots are irrational. For example, $ \sqrt{4} = 2 $ is rational.
3. **Errors in Simplifying Fractions:** When simplifying fractions to identify rational numbers, students may incorrectly reduce them, leading to wrong conclusions about their nature.

FAQ

What defines a rational number?

A rational number can be expressed as the ratio of two integers, where the denominator is not zero.

Can a repeating decimal be irrational?

No, a repeating decimal is always a rational number because it can be expressed as a fraction.

Is $ \sqrt{9} $ irrational?

No, $ \sqrt{9} = 3 $, which is an integer and hence a rational number.

Why is $ \pi $ considered irrational?

Because $ \pi $ cannot be expressed as a fraction of two integers, and its decimal expansion is non-terminating and non-repeating.

How can you determine if a number is irrational?

Check if the number cannot be expressed as a fraction of two integers and has a non-terminating, non-repeating decimal expansion.

Are all square roots irrational numbers?

No, only square roots of non-perfect squares are irrational. For example, $ \sqrt{4} = 2 $ is rational, whereas $ \sqrt{2} $ is irrational.

1. Number

1.1 Types of Numbers

1.1.1 Square numbers

1.1.2 Natural numbers

1.1.3 Cube numbers

1.1.4 Prime numbers

1.1.5 Triangle numbers

1.1.6 Integers (positive, zero, and negative)

1.1.7 Common factors

1.1.8 Common multiples

1.1.9 Rational and irrational numbers

1.1.10 Reciprocals

1.2 Percentages

1.2.1 Percentage increase/decrease

1.2.2 Simple and compound interest

1.2.3 Reverse percentages

1.2.4 Calculating percentages of a quantity

1.2.5 Expressing one quantity as a percentage of another

1.3 Using a Calculator

1.3.1 Efficient calculator use

1.3.2 Entering values correctly

1.3.3 Interpreting calculator displays

1.4 Sets

1.4.1 Set notation and terminology

1.4.2 Venn diagrams (limited to two or three sets)

1.4.3 Universal set, subsets, intersection, and union

1.5 Powers and Roots

1.5.1 Squares and square roots

1.5.2 Cubes and cube roots

1.5.3 Other powers and roots

1.6 Time

1.6.1 Time calculations (seconds, minutes, hours, days, etc.)

1.6.2 12-hour and 24-hour clock conversions

1.6.3 Reading timetables

1.7 Fractions, Decimals, and Percentages

1.7.1 Proper and improper fractions

1.7.2 Mixed numbers

1.7.3 Decimal and percentage conversions

1.8 Money

1.8.1 Calculations with money

1.8.2 Currency conversions

1.9 Exponential Growth and Decay

1.9.1 Understanding exponential growth and decay

1.9.2 Applications in depreciation and population changes

1.10 Surds

1.10.1 Understanding and simplifying surds

1.10.2 Rationalizing the denominator

1.11 Ordering

1.11.1 Comparing and ordering numbers using =, ≠, >, <, ≥, ≤

1.12 The Four Operations

1.12.1 Operations with integers, fractions and decimals

1.12.2 Order of operations (including brackets)

1.13 Indices

1.13.1 Understanding and using indices (positive, zero, negative, and fractional)

1.13.2 Applying rules of indices

1.14 Standard Form

1.14.1 Expressing numbers in standard form (A × 10ⁿ)

1.14.2 Converting to and from standard form

1.14.3 Calculating with standard form

1.15 Estimation

1.15.1 Rounding values (decimal places and significant figures)

1.15.2 Estimating calculations

1.15.3 Rounding answers appropriately

1.16 Ratio and Proportion

1.16.1 Simplifying ratios

1.16.2 Dividing a quantity in a given ratio

1.16.3 Using proportional reasoning

1.17 Rates

1.17.1 Common rates (e.g. hourly wages, exchange rates, flow rates)

1.17.2 Solving problems involving average speed

2. Statistics

2.1 Scatter Diagrams

2.1.1 Understanding positive, negative, and zero correlation

2.1.2 Drawing and using a straight line of best fit

2.1.3 Using a graphic display calculator to find and apply the equation of linear regression

2.1.4 Drawing and interpreting scatter diagrams

2.2 Cumulative Frequency Diagrams

2.2.1 Drawing and interpreting cumulative frequency tables and diagrams

2.2.2 Estimating and interpreting the median, percentiles, quartiles and interquartile range from cumulati

2.3 Classifying Statistical Data

2.3.1 Classifying and tabulating statistical data

2.4 Interpreting Statistical Data

2.4.1 Reading, interpreting, and drawing inferences from tables and statistical diagrams

2.4.2 Comparing sets of data using tables, graphs and statistical measures

2.4.3 Understanding restrictions on drawing conclusions from data

2.5 Discrete and Continuous Data

2.5.1 Distinguishing between discrete and continuous data

2.6 Averages and Measures of Spread

2.6.1 Calculating mean, median, mode, quartiles, range, and interquartile range for individual data

2.6.2 Calculating an estimate of the mean for grouped discrete or continuous data

2.6.3 Identifying the modal class from a grouped frequency distribution

2.7 Averages on a Calculator

2.7.1 Using a graphic display calculator to calculate mean, median, and quartiles for discrete data

2.7.2 Using a graphic display calculator to calculate mean for grouped data

2.8 Statistical Charts and Diagrams

2.8.1 Drawing and interpreting bar charts, pie charts, pictograms, stem-and-leaf diagrams and simple frequ

3. Algebra

3.1 Inequalities

3.1.1 Constructing, solving, and interpreting linear inequalities

3.1.2 Solving inequalities using a graphic display calculator

3.1.3 Representing inequalities graphically

3.1.4 Listing inequalities that define a given region

3.1.5 Representing and interpreting inequalities on a number line

3.2 Sequences

3.2.1 Continuing number sequences and patterns

3.2.2 Recognizing patterns and term-to-term rules

3.2.3 Finding and using the nth term for sequences (linear, quadratic, cubic, exponential)

3.2.4 Using subscript notation for sequences

3.3 Proportion

3.3.1 Expressing direct and inverse proportion in algebraic terms

3.3.2 Using proportion equations to solve problems

3.3.3 Identifying the best variation model for given data

3.4 Introduction to Algebra

3.4.1 Understanding variables and expressions

3.4.2 Substituting values into expressions and formulas

3.5 Algebraic Manipulation

3.5.1 Simplifying expressions by collecting like terms

3.5.2 Expanding products of algebraic expressions

3.5.3 Factorizing expressions (ax + bx + kay + kby, a²x² - b²y², a² + 2ab + b², ax² + bx + c, ax³ + bx² +

3.6 Algebraic Fractions

3.6.1 Manipulating algebraic fractions

3.6.2 Factorizing and simplifying rational expressions

3.7 Indices

3.7.1 Understanding and using indices (positive, zero, negative, and fractional)

3.7.2 Applying rules of indices

3.8 Equations

3.8.1 Constructing expressions, equations, and formulas

3.8.2 Solving linear equations in one unknown

3.8.3 Solving fractional equations with numerical and algebraic denominators

3.8.4 Solving simultaneous linear equations in two variables

3.8.5 Solving quadratic equations (factorization, quadratic formula, using a graphic display calculator)

3.8.6 Changing the subject of formulas (when the subject appears twice or involves powers/roots)

4. Transformations and Vectors

4.1 Transformations

4.1.1 Rotation of a shape about a center through multiples of 90°

4.1.2 Enlargement of a shape from a center using a positive, describing, and performing transformations

4.1.3 Translation of a shape using a vector

4.1.4 Performing and describing combinations of transformations

4.1.5 Determining the reverse of a transformation

4.1.6 Recognizing, describing and performing transformations

4.1.7 Reflection of a shape in a straight line

4.2 Vectors in Two Dimensions

4.2.1 Describing a translation using a vector

4.2.2 Adding and subtracting vectors

4.2.3 Multiplying a vector by a scalar

4.3 Magnitude of a Vector

4.3.1 Calculating the magnitude of a vector using √(x² + y²)

5. Geometry

5.1 Circle Theorems I

5.1.1 Angle properties in circles (segment, cyclic quadrilateral, alternate segment)

5.1.2 Angle properties in circles (semicircle, tangent, center)

5.2 Circle Theorems II

5.2.1 Symmetry properties in circles (equal chords, perpendicular bisector)

5.2.2 Symmetry properties in circles (tangents from an external point)

5.3 Geometrical Terms

5.3.1 Understanding and using basic geometric terms (point, vertex, line, plane, parallel, perpendicular,

5.3.2 Angle properties (right, acute, obtuse, reflex, interior, exterior)

5.3.3 Shape properties (similarity, congruence, scale factor)

5.3.4 Recognizing and interpreting the vocabulary of triangles, quadrilaterals, polygons, and solids

5.4 Angle Measurement in Degrees

5.4.1 Measuring and drawing lines and angles

5.4.2 Using and interpreting three-figure bearings

5.5.2 Using relationships between lengths, areas, and volumes of similar solids

5.5.3 Simplifying and solving problems involving similarity

5.6 Symmetry

5.6.1 Recognizing line symmetry and order of rotational symmetry in two-dimensional shapes

5.6.2 Identifying symmetry properties of prisms, cylinders, pyramids, and cones

5.7 Angles

5.7.1 Calculating unknown angles using basic properties (angles at a point, angles on a straight line, ver

5.7.2 Calculating unknown angles in shapes (sum of angles in a triangle and quadrilateral)

5.7.3 Using angle properties of parallel lines (corresponding, alternate and co-interior angles)

5.7.4 Identifying and using angle properties of regular and irregular polygons

6. Functions

6.1 The Logarithmic Function

6.1.1 Understanding the logarithmic function as the inverse of the exponential function

6.1.2 Converting between exponential and logarithmic form y = a^x as x = logₐ(y)

6.1.3 Solving exponential equations using logarithms

6.2 Graphs of Functions

6.2.1 Recognizing function types from their graphs (linear, quadratic, cubic, reciprocal, exponential, tri

6.2.2 Determining one or two coefficients (a, b, c, or d) for given function graphs

6.2.3 Finding values in a function from its graph

6.3 Sketching Graphs on a Calculator

6.3.1 Sketching the graph of a function using a graphic display calculator

6.3.2 Producing a table of values for a function

6.3.3 Plotting points on a graph

6.3.4 Finding zeros, local maxima, or local minima

6.3.5 Finding the intersection of function graphs

6.3.6 Identifying the vertex of a quadratic function

6.4 Functions and Notation

6.4.1 Understanding functions, domain, and range

6.4.2 Using function notation

6.4.3 Finding inverse functions f⁻¹(x)

6.4.4 Forming composite functions gf(x) = g(f(x))

6.5 Finding a Quadratic Function

6.5.1 Finding a quadratic function given vertex and another point

6.5.2 Finding a quadratic function given x-intercepts and a point

6.5.3 Determining a quadratic function when a = 1 with given vertex or x-intercepts

6.6 Asymptotes

6.6.1 Understanding the concept of asymptotes

6.6.2 Identifying asymptotes parallel to the axes on a graph

6.7 Transforming Graphs of Functions

6.7.1 Describing and identifying transformations of graphs (translations, reflections)

6.7.2 Transforming functions y = f(x) into y = f(x) + k or y = f(x + k)

7. Mensuration

7.1 Units of Measure

7.1.1 Using metric units of mass, length, area volume, and capacity in practical situations

7.1.2 Converting between different units of measurement (e.g. cm² to m², m³ to liters)

7.2 Area and Perimeter

7.2.1 Calculating the perimeter and area of rectangles, triangles, parallelograms, and trapeziums

7.3 Circles, Arcs, and Sectors

7.3.1 Calculating the circumference and area of a circle

7.3.2 Finding arc length and sector area (as fractions of a circle)

7.3.3 Working with both minor and major sectors

7.4 Surface Area and Volume

7.4.1 Calculating surface area and volume of solids (cuboid, prism, cylinder, sphere, pyramid, cone)

7.4.2 Understanding and applying given formulas for curved and total surface areas and volumes

7.5 Compound Shapes and Parts of Shapes

7.5.1 Calculating perimeters and areas of compound shapes and parts of shapes

7.5.2 Finding surface areas and volumes of compound and partial solids (e.g. frustum of a cone)

8. Coordinate Geometry

8.1 Coordinates

8.1.1 Using and interpreting Cartesian coordinates in two dimensions

8.2 Gradient of Linear Graphs

8.2.1 Finding the gradient of a straight line

8.2.2 Calculating the gradient from two given points

8.3 Length and Midpoint

8.3.1 Calculating the length of a line segment from given coordinates

8.3.2 Finding the coordinates of the midpoint of a line segment

8.4 Equations of Linear Graphs

8.4.1 Interpreting and obtaining the equation of a straight-line graph

8.4.2 Expressing equations in different forms (ax + by = c, y = mx + c, x = k)

8.4.3 Finding the equation of a straight line from its graph

8.4.4 Determining the gradient and y-intercept from an equation

8.5 Parallel Lines

8.5.1 Finding the gradient and equation of a line parallel to a given line

8.6 Perpendicular Lines

8.6.1 Finding the gradient and equation of a line perpendicular to a given line

8.6.2 Finding the equation of a perpendicular bisector

9. Trigonometry

9.1 Pythagoras’ Theorem

9.1.1 Applying Pythagoras’ theorem to find unknown sides in right-angled triangles

9.1.2 Finding the length of a chord in a circle

9.1.3 Calculating the distance of a chord from the center of a circle

9.1.4 Finding the distance between two points on a coordinate grid

9.2 Right-Angled Triangles

9.2.1 Using sine, cosine, and tangent ratios for calculations involving right-angled triangles

9.2.2 Solving problems in two dimensions using trigonometry and Pythagoras’ theorem

9.2.3 Understanding that the perpendicular distance from a point to a line is the shortest distance to the

9.2.4 Solving problems involving angles of elevation and depression

9.3 Exact Trigonometric Values

9.3.1 Knowing the exact values of sine and cosine for 0°, 30°, 45°, 60°, 90°

9.3.2 Knowing the exact values of tangent for 0°, 30°, 45°, 60°

9.4 Trigonometric Functions

9.4.1 Recognizing, sketching and interpreting the graphs of y = sin x, y = cos x, y = tan x for 0° ≤ x ≤ 3

9.4.2 Solving trigonometric equations involving sin x, cos x, tan x for 0° ≤ x ≤ 360°

9.5 Non-Right-Angled Triangles

9.5.1 Using the sine rule and cosine rule for solving triangle problems

9.5.2 Calculating the area of a triangle using the formula 1/2 ab sin C

9.6 Pythagoras’ Theorem and Trigonometry in 3D

9.6.1 Applying Pythagoras’ theorem and trigonometry in three-dimensional problems

9.6.2 Finding the angle between a line and a plane

10. Probability

10.1 Introduction to Probability

10.1.1 Understanding and using the probability scale from 0 to 1

10.1.2 Understanding and using probability notation

10.1.3 Calculating the probability of a single event

10.1.4 Understanding that the probability of an event not occurring is 1 - P(A)

10.2 Relative and Expected Frequencies

10.2.1 Understanding relative frequency as an estimate of probability

10.2.2 Calculating expected frequencies using probability

10.3 Probability of Combined Events

10.3.1 Probability of combined events using sample space and Venn diagrams

10.3.2 Using probability notation for combined events including P(A ∩ B) (intersection) and P(A ∪ B) (union

10.3.3 Understanding and applying probability rules including P(A or B) = P(A) + P(B) for mutually exclusiv

10.3.4 Probability of combined events using tree diagrams (with and without replacement)

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Rational and Irrational Numbers

Introduction