Welcome to the World of Numbers!
Hello future Mathematician! This chapter, "Types of number," is the foundation of all the exciting mathematics we will be doing. Understanding these basic categories is like sorting your toolkit—you need to know exactly which tool (number type) to use for which job.
Don't worry if some terms sound unfamiliar; we will break them down into simple, easy-to-remember concepts. Let's start building a strong number sense!
Section 1: The Whole Numbers Family
1.1 Natural Numbers (or Counting Numbers)
These are the numbers we use for counting objects.
- Definition: The set of positive whole numbers and zero.
- Notation: \(\{0, 1, 2, 3, 4, ...\}\) (Note: The syllabus specifies 0 is included, though sometimes in pure math, natural numbers start at 1.)
Analogy: If you were asked to count the apples in a basket, you would start at 0 (if empty) or 1. These are your natural numbers.
1.2 Integers
The integers expand our counting system to include negative numbers.
- Definition: All whole numbers, including positive numbers, negative numbers, and zero.
- Examples: \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\)
Key Takeaway: All natural numbers are also integers, but not all integers are natural numbers (because negative numbers aren't natural).
Real-World Example: Integers are crucial for describing temperature (below freezing is negative) or bank balances (debt is negative).
Quick Review Box: Whole numbers are either Natural Numbers (counting from 0) or Integers (including the negatives).
Section 2: Numbers Based on Patterns and Shapes
2.1 Square Numbers (Perfect Squares)
A square number is the result of multiplying an integer by itself.
- Definition: A number \(n\) where \(n = x^2\), where \(x\) is an integer.
- Examples: \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), ...
Did you know? You can physically arrange the quantity of items represented by a square number into a perfect square grid!
2.2 Cube Numbers (Perfect Cubes)
A cube number is the result of multiplying an integer by itself three times.
- Definition: A number \(n\) where \(n = x^3\), where \(x\) is an integer.
- Examples: \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), \(4^3 = 64\), ...
Tip for the Exam: The syllabus expects you to recall squares and their corresponding roots from 1 to 15, and cubes and their corresponding roots of 1, 2, 3, 4, 5 and 10. Start memorizing!
Example task: Write down the value of \(\sqrt{169}\). (Answer: 13, since \(13^2 = 169\)).
2.3 Triangle Numbers
These are numbers that can be arranged to form an equilateral triangle.
- Definition: The sum of consecutive natural numbers starting from 1.
- Examples:
- 1 (1)
- 3 (1 + 2)
- 6 (1 + 2 + 3)
- 10 (1 + 2 + 3 + 4)
- 15, 21, 28, ...
The formula for the \(n\)th triangle number, \(T_n\), is \(T_n = \frac{n(n+1)}{2}\). This can be a useful tool!
Key Takeaway: Square, cube, and triangle numbers follow specific, recognizable growth patterns.
Section 3: Numbers Defined by Divisibility
3.1 Prime Numbers
These numbers are the fundamental building blocks for all other integers (when multiplying).
- Definition: A natural number greater than 1 that has exactly two distinct positive factors: 1 and itself.
- Examples: 2, 3, 5, 7, 11, 13, 17, ...
- Important Rule: 1 is NOT a prime number. 2 is the only even prime number.
3.2 Factors and Highest Common Factor (HCF)
A factor is a number that divides exactly into another number without leaving a remainder.
- Common Factors: Factors shared by two or more numbers.
- Highest Common Factor (HCF): The largest number that divides into two or more numbers exactly.
Finding the HCF (The Prime Factor Method)
The syllabus expects you to express numbers as a product of its prime factors (using index notation often makes this easier).
Example: Find the HCF of 72 and 108.
- Prime Factorise:
\(72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2\)
\(108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3\) - Identify the "Common" Primes: Take the lowest power of the primes they share.
They share \(2\)s (lowest power is \(2^2\)).
They share \(3\)s (lowest power is \(3^2\)).
- Calculate HCF: Multiply the common parts.
\(HCF = 2^2 \times 3^2 = 4 \times 9 = 36\)
3.3 Multiples and Lowest Common Multiple (LCM)
A multiple is the result of multiplying a number by an integer. Multiples are essentially the numbers in the multiplication table of that number.
- Common Multiples: Multiples shared by two or more numbers.
- Lowest Common Multiple (LCM): The smallest positive number that is a multiple of two or more numbers.
Finding the LCM
Example: Find the LCM of 72 and 108. (Using the prime factors from above: \(72 = 2^3 \times 3^2\) and \(108 = 2^2 \times 3^3\)).
- Identify "All" Primes: Take the highest power of all prime factors found in either number.
Highest power of 2 is \(2^3\).
Highest power of 3 is \(3^3\).
- Calculate LCM: Multiply these highest powers together.
\(LCM = 2^3 \times 3^3 = 8 \times 27 = 216\)
Key Takeaway: HCF involves taking the smallest powers of common prime factors; LCM involves taking the largest powers of *all* prime factors.
Section 4: Rational vs. Irrational Numbers
This section is about classifying numbers based on whether they can be perfectly expressed as a simple fraction.
4.1 Rational Numbers
These are the "tame" numbers whose decimals are predictable.
- Definition: Any number that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\).
- Decimal Form: When written as a decimal, a rational number either terminates (stops) or repeats (forms a pattern).
Examples:
- \(\frac{1}{2} = 0.5\) (Terminates)
- \(5\) (Can be written as \(\frac{5}{1}\))
- \(-0.333...\) (Can be written as \(-\frac{1}{3}\)) (Repeats)
4.2 Irrational Numbers
These are the "wild" numbers—they cannot be written as a perfect fraction.
- Definition: Any real number that cannot be written as a simple fraction \(\frac{a}{b}\).
- Decimal Form: The decimal representation is non-terminating (never ends) and non-repeating (no predictable pattern).
Examples:
- The famous constant Pi (\(\pi\)): 3.14159265...
- The Golden Ratio (\(\phi\)): 1.61803398...
- The Square Root of a non-square number: \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{7}\). These are known as surds.
Common Mistake to Avoid: Just because a decimal fills up your calculator screen, that doesn't make it irrational! If it was originally a fraction (like \(\frac{2}{7}\)), it is still rational, even though its display might be truncated.
Key Takeaway: Rational = fraction or repeating/terminating decimal. Irrational = cannot be written as a fraction, and the decimal goes on forever without repeating.
Section 5: Reciprocals
5.1 Understanding Reciprocals
The reciprocal is a simple, useful concept related to multiplication.
- Definition: The reciprocal of a number \(x\) is \(\frac{1}{x}\). When you multiply a number by its reciprocal, the result is 1.
Step-by-Step for Finding Reciprocals
For a fraction, you just flip it! For a whole number, imagine it as a fraction over 1.
- Number: \(5\)
Step 1: Write as fraction: \(\frac{5}{1}\)
Step 2: Flip: \(\frac{1}{5}\)
Check: \(5 \times \frac{1}{5} = 1\)
- Number: \(\frac{3}{4}\)
Reciprocal: \(\frac{4}{3}\)
- Number: \(1.25\)
Step 1: Write as fraction: \(\frac{5}{4}\) (Since \(1.25 = 1 \frac{1}{4}\))
Step 2: Flip: \(\frac{4}{5}\) or \(0.8\)
Important Note: The number 0 has no reciprocal, because division by zero is undefined (\(\frac{1}{0}\) is impossible).
Key Takeaway: Reciprocal means "one divided by the number." It's the inverse operation for multiplication.
Summary of Key Terms (C1.1/E1.1)
| Type of Number | Description | Example |
| Natural Numbers | Positive whole numbers, including 0. | 0, 1, 2, 3, 4, ... |
| Integers | All whole numbers (positive, negative, and zero). | ..., -2, -1, 0, 1, 2, ... |
| Prime Numbers | Only divisible by 1 and itself. | 2, 3, 5, 7, 11 |
| Rational Numbers | Can be written as a fraction \(\frac{a}{b}\). Terminating or repeating decimals. | 0.75, \(\frac{2}{3}\), -4 |
| Irrational Numbers | Cannot be written as a fraction. Non-terminating, non-repeating decimals. | \(\pi\), \(\sqrt{2}\) |
Keep practicing identifying these types! Being fluent in the language of numbers will make the rest of the syllabus much smoother. You've got this!