Types of number

1. Overview

Understanding different types of numbers is fundamental to IGCSE Mathematics (0580). This topic covers identifying and working with natural numbers, integers, prime numbers, square and cube numbers, factors, multiples, rational and irrational numbers, and reciprocals. Mastering these concepts is crucial for success in more advanced topics in algebra, geometry, and number theory. This revision note provides definitions, examples, and exam tips to help you achieve mastery.

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Key Definitions

• Natural Numbers ($\mathbb{N}$): Counting numbers starting from 1 (1, 2, 3...).
• Integers ($\mathbb{Z}$): Whole numbers, including positive, negative, and zero (...-2, -1, 0, 1, 2...).
• Prime Number: A number greater than 1 that has exactly two factors: 1 and itself.
• Square Number: The result of multiplying an integer by itself (e.g., $3 \times 3 = 9$).
• Cube Number: The result of multiplying an integer by itself three times (e.g., $2 \times 2 \times 2 = 8$).
• Factor: A number that divides exactly into another number without leaving a remainder.
• Multiple: A number found in a specific number's "times table."
• Rational Number ($\mathbb{Q}$): A number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.
• Irrational Number: A number that cannot be written as a simple fraction (e.g., $\pi$ or $\sqrt{2}$).
• Reciprocal: The result of dividing 1 by the number (e.g., the reciprocal of $x$ is $\frac{1}{x}$).

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Core Content

Natural Numbers and Integers

Natural numbers are used for counting objects. Integers include natural numbers but add zero and negative values.

• Note: Zero is an integer, but it is neither positive nor negative.

Prime Numbers

Primes are the "building blocks" of numbers.

• List of first few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23...
• Important: 1 is not a prime number (it only has one factor). 2 is the only even prime number.

Factors and Multiples

• Factors of 12: 1, 2, 3, 4, 6, 12.
• Multiples of 12: 12, 24, 36, 48...

Worked Example 1 — Finding the Highest Common Factor (HCF)

Find the HCF of 18 and 24.

1. List all factors of 18: 1, 2, 3, 6, 9, 18.
2. List all factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
3. Identify the highest number in both lists: 6.

• Result: HCF = 6.

Worked Example 2 — Finding Factors

List all the factors of 36.

1. Start with 1 and the number itself: 1 and 36.
2. Check if 2 is a factor: 36 ÷ 2 = 18, so 2 and 18 are factors.
3. Check if 3 is a factor: 36 ÷ 3 = 12, so 3 and 12 are factors.
4. Check if 4 is a factor: 36 ÷ 4 = 9, so 4 and 9 are factors.
5. Check if 5 is a factor: 36 ÷ 5 = 7.2, so 5 is not a factor.
6. Check if 6 is a factor: 36 ÷ 6 = 6, so 6 is a factor.
7. We have reached 6, and the next factor would be greater than 6, so we have found all the factors.

• Result: The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Square and Cube Numbers

You should memorize squares up to $15^2$ and cubes up to $5^3$.

• Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...
• Cubes: 1, 8, 27, 64, 125...

Rational and Irrational Numbers

• Rational: $0.5$ (which is $\frac{1}{2}$), $5$ (which is $\frac{5}{1}$), $0.333...$ (which is $\frac{1}{3}$).
• Irrational: $\pi$ ($3.14159...$), $\sqrt{2}$, $\sqrt{3}$. If a square root is not a perfect square, it is irrational.

Reciprocals

To find the reciprocal of a fraction, "flip" it. To find the reciprocal of a decimal, calculate $1 \div \text{decimal}$.

Worked Example 3 — Reciprocal of a decimal

Find the reciprocal of 0.4.

1. Write as a fraction: $1 \div 0.4$.
2. Convert decimal to fraction: $0.4 = \frac{4}{10}$.
3. Divide 1 by the fraction: $1 \div \frac{4}{10}$.
4. Multiply by the reciprocal: $1 \times \frac{10}{4}$.
5. Simplify: $\frac{10}{4} = \frac{5}{2} = 2.5$.

• Result: 2.5.

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Extended Content (Extended Curriculum Only)

Set Notation with Natural Numbers

Extended students must be comfortable using mathematical symbols to describe sets of numbers:

• $n \in \mathbb{N}$ means "$n$ is an element of the set of Natural Numbers."
• If a question defines $x \in \{ \text{positive integers} \}$, it is the same as saying $x$ is a natural number.

Lowest Common Multiple (LCM) with Variables

Finding the LCM of expressions involving variables requires finding the LCM of the coefficients and then including the highest power of each variable present in the expressions. This ensures the LCM is divisible by both original expressions.

Worked Example 4 — Lowest Common Multiple (LCM) with Variables

Find the LCM of $4x^2$ and $6x$.

1. Find the LCM of the coefficients (4 and 6): The multiples of 4 are 4, 8, 12... The multiples of 6 are 6, 12... LCM of 4 and 6 is 12.
2. Find the LCM of the variables ($x^2$ and $x$): Take the highest power of each variable. Here, it is $x^2$.
3. Combine the LCM of the coefficients and variables: $12x^2$.

• Result: LCM = $12x^2$.

Worked Example 5 — Identifying Prime Numbers

Determine whether 91 is a prime number.

1. Recall that a prime number has only two factors: 1 and itself.
2. Check for divisibility by prime numbers less than $\sqrt{91} \approx 9.5$. The prime numbers to check are 2, 3, 5, and 7.
3. 91 is not divisible by 2 (it's not even).
4. 91 is not divisible by 3 (9 + 1 = 10, which is not divisible by 3).
5. 91 is not divisible by 5 (it doesn't end in 0 or 5).
6. Check divisibility by 7: $91 \div 7 = 13$.
7. Since 91 is divisible by 7, it has factors other than 1 and itself.

• Result: 91 is not a prime number.

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Key Equations

While this topic focuses on definitions, the following notations are essential:

• $\frac{1}{x}$: The formula for the reciprocal of $x$.
• $\sqrt{x}$: Square root (Inverse of $x^2$).
• $\sqrt[3]{x}$: Cube root (Inverse of $x^3$).
• $a \times b$: The product of two numbers.

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Common Mistakes to Avoid

• ❌ Wrong: Writing "sixteen thousand and thirty seven" as $16000037$.
• ✅ Right: Use place value columns. $16,000 + 37 = 16,037$.
• ❌ Wrong: Saying $111$ is a prime number without checking.
• ✅ Right: Check divisibility rules. $1+1+1=3$, so $111$ is divisible by 3 ($3 \times 37 = 111$). It is not prime.
• ❌ Wrong: Adding numbers when asked for the "product."
• ✅ Right: "Product" always means multiply. Note that the product of two negative numbers is positive (e.g., $-5 \times -2 = 10$), which might be larger than a product of small positive numbers.
• ❌ Wrong: Forgetting 1 or the number itself when listing factors.
• ✅ Right: Always list factors in pairs (e.g., for 18: 1 & 18, 2 & 9, 3 & 6) to ensure none are missed.
• ❌ Wrong: Confusing square root and cube root on your calculator.
• ✅ Right: Double-check you're using the cube root (∛) function on your calculator, not the square root (√) or straight multiplication.

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Exam Tips

• Command Words:
  • "List": Write down all possibilities (e.g., factors).
  • "Identify/Write down": Usually a 1-mark question requiring a single number from a given list.
• Calculator Use: Use the $\sqrt{\text{ }}$ and $\sqrt[3]{\text{ }}$ buttons to check if a number is a perfect square or cube. If the result is a whole number, it is a perfect square/cube.
• Check your zeros: When writing large numbers like "nine billion," remember:
  • Million = 6 zeros ($1,000,000$)
  • Billion = 9 zeros ($1,000,000,000$)
• Show Working: Even for HCF/LCM, show your prime factor trees or lists. If you make a small multiplication error but show your method, you can still earn "Method Marks."
• Read Carefully: Pay close attention to place values when writing numbers from words. A small error can lead to a wrong answer.