Welcome to the World of Numbers!

Hello IGCSE student! In this chapter, we are diving into the fundamental building blocks of Mathematics: the different types of numbers. Understanding these concepts is essential, as they form the foundation for everything else you will study in algebra, geometry, and beyond.

Don't worry if some of these terms seem familiar but slightly fuzzy—we are going to clearly define each type and give you easy ways to remember them! Let’s get started!

1. The Basic Building Blocks: Natural Numbers and Integers

1.1 Natural Numbers (C1.1/E1.1)

These are the numbers we use for counting.

  • Definition: The set of counting numbers, starting with zero.
  • Notation: \(\{0, 1, 2, 3, 4, ...\}\)
  • Analogy: Think of the natural numbers as the number of apples you could have in a basket (you can have zero apples, one apple, two apples, etc.).

1.2 Integers (C1.1/E1.1)

Integers expand on natural numbers by including their negative counterparts.

  • Definition: The set of all whole numbers, including positive numbers, negative numbers, and zero.
  • Notation: \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\)
  • Key Breakdown:
    • Positive Integers: 1, 2, 3, ...
    • Zero: 0
    • Negative Integers: -1, -2, -3, ...
  • Analogy: Integers are like tracking temperature. It can be above zero (positive), exactly zero, or below zero (negative).

Quick Review: Every natural number is also an integer, but not every integer is a natural number (e.g., -5 is an integer but not natural).

2. Special Types of Whole Numbers

These types of numbers have unique properties based on how they are formed or how they can be divided.

2.1 Prime Numbers (C1.1/E1.1)

Prime numbers are the 'atoms' of multiplication—they can't be broken down further.

  • Definition: A natural number greater than 1 that has exactly two distinct positive factors (divisors): 1 and itself.
  • Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...
  • Important facts:
    • The number 1 is NOT a prime number because it only has one factor (itself).
    • The number 2 is the only even prime number. All other even numbers can be divided by 2.
Did you know?

You can express any whole number greater than 1 as a product of its prime factors. This is called Prime Factor Decomposition (a key skill for finding HCF and LCM, covered later).
Example: \(72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2\)

2.2 Square Numbers (C1.1/E1.1)

These numbers are formed by multiplying an integer by itself.

  • Definition: The result of squaring an integer (\(n \times n\)).
  • Visualisation: Think of arranging dots into a perfect square shape.
  • Examples:
    • \(0^2 = 0\)
    • \(1^2 = 1\)
    • \(2^2 = 4\)
    • \(10^2 = 100\)
  • Key Skill: You should be able to recall squares and square roots from 1 to 15 (e.g., \(\sqrt{169} = 13\)).

2.3 Cube Numbers (C1.1/E1.1)

These numbers are formed by multiplying an integer by itself three times.

  • Definition: The result of cubing an integer (\(n \times n \times n\)).
  • Visualisation: Think of the volume of a perfect cube shape.
  • Examples:
    • \(1^3 = 1\)
    • \(3^3 = 27\)
    • \(10^3 = 1000\)
  • Key Skill: You should be able to recall cubes and cube roots of 1, 2, 3, 4, 5, and 10 (e.g., \(\sqrt[3]{8} = 2\)).

2.4 Triangle Numbers (C1.1/E1.1)

Triangle numbers are related to sequences and patterns.

  • Definition: The total number of items needed to form a triangle pattern, found by summing consecutive natural numbers starting from 1.
  • Formula (for the nth triangle number): \(T_n = \frac{n(n+1)}{2}\) (Though you only need to identify and use them, not necessarily know the formula).
  • Examples:
    • 1st: 1
    • 2nd: \(1 + 2 = 3\)
    • 3rd: \(1 + 2 + 3 = 6\)
    • 4th: \(1 + 2 + 3 + 4 = 10\)

Key Takeaway for Special Numbers: Prime numbers are about division (only 1 and itself). Square, cube, and triangle numbers are about patterns and geometric arrangements.

3. Relationships Between Numbers: Factors and Multiples

When we compare two or more numbers, we often look at their common factors and multiples.

3.1 Common Factors and HCF (C1.1/E1.1)

A factor of a number is a whole number that divides into it exactly, without a remainder.

  • Common Factors: Factors shared by two or more numbers.
  • HCF (Highest Common Factor): The largest number that is a factor of all the given numbers.
Step-by-Step: Finding the HCF

Let's find the HCF of 12 and 18.

  1. List the factors:
    • Factors of 12: 1, 2, 3, 4, 6, 12
    • Factors of 18: 1, 2, 3, 6, 9, 18
  2. Identify Common Factors: 1, 2, 3, 6
  3. Identify the Highest: The HCF is 6.

Memory Aid: HCF is about dividing, so the answer must be smaller than or equal to the original numbers.

3.2 Common Multiples and LCM (C1.1/E1.1)

A multiple of a number is the result of multiplying that number by an integer (it’s basically its times table).

  • Common Multiples: Multiples shared by two or more numbers.
  • LCM (Lowest Common Multiple): The smallest positive number that is a multiple of all the given numbers.
Step-by-Step: Finding the LCM

Let's find the LCM of 4 and 6.

  1. List the multiples:
    • Multiples of 4: 4, 8, 12, 16, 20, 24, ...
    • Multiples of 6: 6, 12, 18, 24, 30, ...
  2. Identify the Lowest Common Multiple: The LCM is 12.

Memory Aid: LCM is about multiplication, so the answer must be larger than or equal to the original numbers.

Common Mistake to Avoid:

Mixing up HCF and LCM! If you get an LCM that is smaller than one of the original numbers, you have confused it with the HCF. If you get an HCF that is larger than the original numbers, you have confused it with the LCM.

4. Extending the Number System: Rational and Irrational

So far, all the numbers we've looked at (integers, squares, primes) are relatively straightforward. Now we look at numbers that involve fractions and decimals, which introduces the key division between rational and irrational numbers.

4.1 Rational Numbers (C1.1/E1.1)

Rational numbers are numbers that make sense! They can be written as a clean fraction.

  • Definition: Any number that can be expressed as a fraction \(\frac{p}{q}\), where p and q are integers, and \(q \neq 0\).
  • Examples:
    • All integers (e.g., \(5 = \frac{5}{1}\)).
    • All terminating decimals (e.g., \(0.75 = \frac{3}{4}\)).
    • All recurring decimals (e.g., \(0.333... = \frac{1}{3}\) or \(0.181818... = \frac{2}{11}\)).

4.2 Irrational Numbers (C1.1/E1.1)

Irrational numbers are the opposite of rational—you cannot write them as a simple fraction.

  • Definition: A number that cannot be expressed as a fraction \(\frac{p}{q}\). When written as a decimal, it is non-terminating (never ends) and non-recurring (never repeats).
  • Most Common Examples You Need to Know:
    • \(\pi\) (Pi, approximately 3.14159...)
    • The square root of non-square integers (e.g., \(\sqrt{2}, \sqrt{3}, \sqrt{10}\)). These are known as surds.

Analogy: Imagine trying to measure the side of a square that has an area of exactly 2. The side length would be \(\sqrt{2}\). No matter how small a fraction you use, you can never describe \(\sqrt{2}\) perfectly—it's irrational!

4.3 Reciprocals (C1.1/E1.1)

The reciprocal of a number is what you multiply the number by to get 1.

  • Definition: The reciprocal of a number x is \(\frac{1}{x}\).
  • Rule: To find the reciprocal, simply flip the number (if it is a fraction).
  • Examples:
    • The reciprocal of 5 is \(\frac{1}{5}\).
    • The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
    • The reciprocal of \(0.25\) (\(\frac{1}{4}\)) is 4.

Key Takeaway: The entire IGCSE Number curriculum lives within the set of Real Numbers, which is simply all rational and all irrational numbers combined.

Chapter Summary: Key Number Terminology

Make sure you can define and provide examples for every term below—these are your core vocabulary for the Number section!

Number Classifications Checklist
  • Natural Numbers: \(\{0, 1, 2, ...\}\)
  • Integers: Whole numbers (positive, negative, and zero).
  • Prime Numbers: Only factors are 1 and itself (must be \(>1\)).
  • Square Numbers: \(n^2\)
  • Cube Numbers: \(n^3\)
  • Triangle Numbers: Sum of consecutive numbers.
  • Rational Numbers: Can be written as \(\frac{p}{q}\) (terminating or recurring decimals).
  • Irrational Numbers: Cannot be written as a fraction (\(\pi\), surds).
  • Common Factors: Shared divisors.
  • HCF: The largest shared factor.
  • Common Multiples: Shared products in the times table.
  • LCM: The smallest shared multiple.
  • Reciprocals: \(1/x\).