Types of Number and BODMAS Revision Notes for AQA GCSE Maths

Types of number and BODMAS

What are integers?

An integer is simply another word for a whole number. This includes all positive whole numbers, all negative whole numbers, and zero. Think of integers as the numbers you would use when counting objects or measuring temperature on a basic thermometer.

Examples of integers: -365, 0, 1, 17, 989, 1234567890

Examples of numbers that are NOT integers: 0.5, 2/3, √7, 13¾, -1000.1, 66.66, π

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The key thing to remember is that integers cannot have decimal parts or be written as fractions - they must be complete whole numbers.

Understanding rational and irrational numbers

Every number you encounter in mathematics falls into one of two categories: rational or irrational. This classification is crucial for understanding how different numbers behave in calculations.

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This classification system helps you understand the fundamental nature of numbers and choose appropriate methods for solving problems.

Rational numbers

Rational numbers are the "well-behaved" numbers in mathematics. They can always be expressed as fractions, which makes them easier to work with in many situations. Rational numbers appear in three main forms:

Integers - These are rational because any integer can be written as a fraction (for example, $4 = \frac{4}{1}$ , $-5 = \frac{-5}{1}$ )
Fractions - Where both the numerator and denominator are integers (and the denominator isn't zero), such as $\frac{1}{4}$ , $\frac{3}{7}$ , or $\frac{7}{2}$
Terminating or recurring decimals - These include decimals that either end (like $0.125 = \frac{1}{8}$ ) or repeat in a pattern (like $0.333333... = \frac{1}{3}$ , or $0.143143143... = \frac{143}{999}$ )

Irrational numbers

Irrational numbers are more complex and cannot be written as simple fractions. They appear as never-ending, non-repeating decimals. This means their decimal representation goes on forever without forming a repeating pattern.

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Common examples of irrational numbers include:

Square roots of numbers that aren't perfect squares (like $\sqrt{2}$ and $\sqrt{3}$ )
Surds (expressions involving square roots that cannot be simplified to whole numbers)
$\pi$ (pi), which equals approximately 3.14159... but never terminates or repeats

Understanding the difference between rational and irrational numbers helps you choose the most appropriate methods for solving problems and expressing your answers.

BODMAS: The order of operations

BODMAS is your guide for tackling complex mathematical expressions. It tells you exactly which operations to perform first, ensuring you always get the correct answer. The acronym stands for:

Brackets
Other (operations like squaring, cube roots, etc.)
Division
Multiplication
Addition
Subtraction

When you encounter an expression with multiple operations, work through them in this exact order.

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Division and multiplication have equal priority (work from left to right), as do addition and subtraction.

BODMAS becomes particularly important when dealing with complex expressions involving square roots, powers, and multiple operations. If you're unsure what to do next in a calculation, BODMAS will always guide you to the correct sequence.

Working with reciprocals and square roots

Let's explore how BODMAS applies to more advanced problems involving square roots and reciprocals. A reciprocal of a number is simply 1 divided by that number.

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Worked Example: Finding the reciprocal of $\sqrt{4 + 6 \times (12 - 2)}$

Step 1: Resolve the brackets first $12 - 2 = 10$

Step 2: Perform multiplication $6 \times 10 = 60$

Step 3: Perform addition $4 + 60 = 64$

Step 4: Calculate the square root $\sqrt{64} = 8$

Step 5: Find the reciprocal $\frac{1}{8} = 0.125$

Therefore, the reciprocal of $\sqrt{4 + 6 \times (12 - 2)}$ is $\frac{1}{8}$ or 0.125.

This example demonstrates the step-by-step process for finding the reciprocal. Notice how BODMAS guides each step: first the brackets are resolved, then multiplication, then addition, followed by the square root, and finally the reciprocal calculation.

Checking your work

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Mathematics requires precision, and even small mistakes in applying BODMAS can lead to incorrect answers. It's essential to double-check your working, especially in exam conditions. Take your time with each step and verify that you've followed the correct order of operations.

When practising BODMAS problems, try working through them slowly at first, clearly identifying each operation and its priority. As you become more confident, you'll naturally speed up while maintaining accuracy.

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Key Points to Remember:

Integers are whole numbers (positive, negative, or zero) - no decimal parts allowed
Rational numbers can be written as fractions, including integers and terminating/recurring decimals
Irrational numbers cannot be written as fractions and have never-ending, non-repeating decimal representations
BODMAS gives you the correct order: Brackets, Other operations, Division/Multiplication, Addition/Subtraction
Always work through complex expressions step by step, following BODMAS carefully to avoid mistakes

Types of Number and BODMAS (AQA GCSE Maths): Revision Notes