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AQA GCSE Maths

Revision Notes

Home / GCSE / Maths / AQA / Revision Notes / 1. Number / 1.1 Arithmetic / 1.1.4 Negative Numbers


1.1.4 Negative Numbers


Negative Numbers

  • Negative numbers are any number less than zero
  • They appear in lots of places from numerical calculations to algebra 
  • You might come across them in real-life problems such as temperature or debt

What are the rules for working with Negative Numbers?

  • For all four operations, you can remember some rules to help
    • Same sign makes a positive
    • Different sign makes a negative

  • When adding and subtracting with negative numbers
    • Two negatives make a positive e.g. 5 space minus space minus space 3 space equals space 5 space plus space 3 space equals space 8
    • A negative and a positive make a negative e.g. 7 space plus space minus 3 space equals space 7 space minus space 3 space equals space 4
  • When multiplying or dividing with negative numbers
    • Two negatives make a positive e.g.
      • negative 12 space divided by negative 4 space equals space 3
      • negative 6 space cross times space minus 4 space equals space 24
    • A negative and a positive make a negative e.g. 
      • negative 12 space divided by space 4 space equals space minus 3
      • 6 space cross times space minus 4 space equals space minus 24
  • For multiplication and division, it's often easier to calculate ignoring any signs, then making a decision about whether the answer should be positive or negative

Exam Tip

  • It can help to think of negative numbers as temperature or hot and cold air
  • Be super careful to remember the rules when adding, subtracting, multiplying and dividing with negatives
  • Your calculator isn't always as clever as you think... you might need to use brackets to make sure it knows a number is negative e.g. negative 3 squared not equal to negative 9



  • 1. Number
    • 1.1 Arithmetic
      • 1.1.1 Multiplication (non-Calc)
        • 1.1.2 Division (non-Calc)
          • 1.1.3 Mathematical Symbols
            • 1.1.4 Negative Numbers
              • 1.1.5 Order By Size
                • 1.1.6 Time
                • 1.2 Fractions
                  • 1.2.1 Mixed Numbers & Top Heavy Fractions
                    • 1.2.2 Adding & Subtracting Fractions
                      • 1.2.3 Multiplying & Dividing Fractions
                      • 1.3 Basic Percentages
                        • 1.3.1 Basic Percentages
                          • 1.3.2 Percentage Increases & Decreases
                          • 1.4 Reverse Percentages
                            • 1.4.1 Reverse Percentages
                            • 1.5 Compound Interest
                              • 1.5.1 Compound Interest
                                • 1.5.2 Depreciation
                                • 1.6 LCM / HCF / Prime Factors
                                  • 1.6.1 Prime Factors
                                    • 1.6.2 HCFs & LCMs
                                    • 1.7 Roots & Indices
                                      • 1.7.1 Roots & Indices - Basics
                                        • 1.7.2 Roots & Indices - Harder
                                        • 1.8 Rounding & Estimation
                                          • 1.8.1 Rounding & Estimation
                                          • 1.9 Standard Form
                                            • 1.9.1 Standard Form - Basics
                                              • 1.9.2 Standard Form - Harder
                                              • 1.10 Bounds & Error Intervals
                                                • 1.10.1 Bounds & Error Intervals - Basics
                                                  • 1.10.2 Calculations using Bounds
                                                  • 1.11 Recurring Decimals
                                                    • 1.11.1 Recurring Decimals
                                                    • 1.12 Surds
                                                      • 1.12.1 Surds - Basics
                                                        • 1.12.2 Surds - Rationalising Denominators
                                                        • 1.13 Using a Calculator
                                                          • 1.13.1 Using a Calculator
                                                          • 1.14 Counting
                                                            • 1.14.1 Counting
                                                            • 1.15 Best Buy
                                                              • 1.15.1 Best Buy
                                                              • 1.16 Exchange Rates
                                                                • 1.16.1 Exchange Rates
                                                              • 2. Algebra Basics
                                                                • 2.1 Expanding One Bracket
                                                                  • 2.1.1 Expanding One Bracket
                                                                  • 2.2 Simple Factorisation
                                                                    • 2.2.1 Simple Factorisation
                                                                    • 2.3 Expanding Quadratics
                                                                      • 2.3.1 Expanding Two Brackets - Basics
                                                                        • 2.3.2 Expanding Two Brackets - Harder
                                                                          • 2.3.3 Expanding Three Brackets
                                                                          • 2.4 Factorising Quadratics
                                                                            • 2.4.1 Factorising Quadratics - Basics
                                                                              • 2.4.2 Factorising Quadratics - Harder
                                                                                • 2.4.3 Difference of Two Squares
                                                                                  • 2.4.4 Factorising Quadratics - General
                                                                                  • 2.5 Rearranging Formulae
                                                                                    • 2.5.1 Rearranging Formulae - Basics
                                                                                      • 2.5.2 Rearranging Formulae - Harder
                                                                                      • 2.6 Solving Linear Equations
                                                                                        • 2.6.1 Collecting Like Terms
                                                                                          • 2.6.2 Solving Linear Equations
                                                                                            • 2.6.3 Forming Expressions & Equations
                                                                                            • 2.7 Substitution
                                                                                              • 2.7.1 Substitution
                                                                                              • 2.8 Proof/Reasoning
                                                                                                • 2.8.1 Proof/Reasoning - Algebraic
                                                                                                • 2.9 Functions
                                                                                                  • 2.9.1 Functions - Basics
                                                                                                    • 2.9.2 Compound Functions
                                                                                                      • 2.9.3 Inverse Functions
                                                                                                      • 2.10 Algebraic Fractions
                                                                                                        • 2.10.1 Algebraic Fractions - Adding & Subtracting
                                                                                                          • 2.10.2 Algebraic Fractions - Simplifying Fractions
                                                                                                            • 2.10.3 Algebraic Fractions - Multiplying & Dividing
                                                                                                          • 3. Solving Equations & Inequalities
                                                                                                            • 3.1 Quadratic Formula
                                                                                                              • 3.1.1 Quadratic Formula
                                                                                                              • 3.2 Completing the Square
                                                                                                                • 3.2.1 Completing the Square
                                                                                                                • 3.3 Linear Simultaneous Equations
                                                                                                                  • 3.3.1 Linear Simultaneous Equations
                                                                                                                  • 3.4 Quadratic Simultaneous Equations
                                                                                                                    • 3.4.1 Quadratic Simultaneous Equations
                                                                                                                    • 3.5 Inequalities
                                                                                                                      • 3.5.1 Solving Inequalities - Linear
                                                                                                                        • 3.5.2 Solving Inequalities - Quadratic
                                                                                                                        • 3.6 Inequalities on a Graph
                                                                                                                          • 3.6.1 Inequalities on Graphs - Drawing
                                                                                                                            • 3.6.2 Inequalities on Graphs - Interpreting
                                                                                                                            • 3.7 Iteration
                                                                                                                              • 3.7.1 Iteration - Using a Calculator
                                                                                                                                • 3.7.2 Iteration - Applications
                                                                                                                                • 3.8 Equations & Problem Solving
                                                                                                                                  • 3.8.1 Equations & Problem Solving
                                                                                                                                • 4. Sequences
                                                                                                                                  • 4.1 Sequences - Linear
                                                                                                                                    • 4.1.1 Sequences - Linear
                                                                                                                                    • 4.2 Sequences - Quadratic
                                                                                                                                      • 4.2.1 Sequences - Quadratic
                                                                                                                                      • 4.3 Fibonacci & Geometric
                                                                                                                                        • 4.3.1 Sequences - Basics
                                                                                                                                          • 4.3.2 Sequences - Identifying
                                                                                                                                            • 4.3.3 Sequences - Others
                                                                                                                                          • 5. Graphs
                                                                                                                                            • 5.1 Coordinates
                                                                                                                                              • 5.1.1 Coordinates
                                                                                                                                              • 5.2 Drawing Graphs
                                                                                                                                                • 5.2.1 Drawing Graphs - Shapes
                                                                                                                                                  • 5.2.2 Drawing Graphs - Using a Table
                                                                                                                                                    • 5.2.3 Drawing Graphs - Trig Graphs
                                                                                                                                                    • 5.3 Equations of a Line / y = mx + c
                                                                                                                                                      • 5.3.1 Straight Lines - Finding Equations
                                                                                                                                                        • 5.3.2 Straight Lines - Drawing Graphs
                                                                                                                                                        • 5.4 Perpendicular Lines
                                                                                                                                                          • 5.4.1 Perpendicular Lines
                                                                                                                                                          • 5.5 Transformations of Graphs
                                                                                                                                                            • 5.5.1 Transformations of Graphs
                                                                                                                                                            • 5.6 D-T / V-T Graphs
                                                                                                                                                              • 5.6.1 Distance-Time Graphs
                                                                                                                                                                • 5.6.2 Speed-Time Graphs
                                                                                                                                                                • 5.7 Solving Equations using Graphs
                                                                                                                                                                  • 5.7.1 Solving Equations using Graphs
                                                                                                                                                                  • 5.8 Equation of a Circle
                                                                                                                                                                    • 5.8.1 Circles - Equation & Graphs
                                                                                                                                                                      • 5.8.2 Circles - Finding Tangents
                                                                                                                                                                      • 5.9 Estimating Areas & Gradients of Graphs
                                                                                                                                                                        • 5.9.1 Estimating Areas & Gradients of Graphs
                                                                                                                                                                      • 6. Ratios, Proportion & Rate of Change
                                                                                                                                                                        • 6.1 Ratios
                                                                                                                                                                          • 6.1.1 Ratios
                                                                                                                                                                          • 6.2 Direct & Inverse Proportion
                                                                                                                                                                            • 6.2.1 Direct & Inverse Proportion
                                                                                                                                                                            • 6.3 Speed, Density & Pressure
                                                                                                                                                                              • 6.3.1 Speed, Density & Pressure
                                                                                                                                                                                • 6.3.2 Speed, Density & Pressure - Harder
                                                                                                                                                                              • 7. Geometry & Measures
                                                                                                                                                                                • 7.1 Angles in Parallel Lines
                                                                                                                                                                                  • 7.1.1 Angles in Parallel Lines
                                                                                                                                                                                  • 7.2 Angles in Polygons
                                                                                                                                                                                    • 7.2.1 Angles in Polygons
                                                                                                                                                                                    • 7.3 Similarity - Lengths
                                                                                                                                                                                      • 7.3.1 Similarity - Lengths
                                                                                                                                                                                      • 7.4 Similarity - Areas & Volumes
                                                                                                                                                                                        • 7.4.1 Similarity - Areas & Volumes
                                                                                                                                                                                        • 7.5 Congruent Triangles
                                                                                                                                                                                          • 7.5.1 Congruent Triangles
                                                                                                                                                                                          • 7.6 Transformations
                                                                                                                                                                                            • 7.6.1 Transformations - Rotation
                                                                                                                                                                                              • 7.6.2 Transformations - Reflection
                                                                                                                                                                                                • 7.6.3 Transformations - Translation
                                                                                                                                                                                                  • 7.6.4 Transformations - Enlargement
                                                                                                                                                                                                    • 7.6.5 Combined Transformations
                                                                                                                                                                                                    • 7.7 Transformations - Enlargement (Negative Scale Factor)
                                                                                                                                                                                                      • 7.7.1 Transformations - Enlargement (Negative Scale Factor)
                                                                                                                                                                                                      • 7.8 Area - Triangles & Quadrilaterals
                                                                                                                                                                                                        • 7.8.1 Area - Formulae
                                                                                                                                                                                                          • 7.8.2 Area - Adding & Subtracting
                                                                                                                                                                                                          • 7.9 Circles - Area & Circumference
                                                                                                                                                                                                            • 7.9.1 Circles - Area & Circumference
                                                                                                                                                                                                            • 7.10 Circles - Sector & Arcs
                                                                                                                                                                                                              • 7.10.1 Circles - Sector Areas & Arc Lengths
                                                                                                                                                                                                              • 7.11 Problem Solving with Areas
                                                                                                                                                                                                                • 7.11.1 Problem Solving with Areas
                                                                                                                                                                                                                  • 7.11.2 Converting Lengths & Areas
                                                                                                                                                                                                                  • 7.12 3D Shapes
                                                                                                                                                                                                                    • 7.12.1 3D Shapes - Surface Area
                                                                                                                                                                                                                      • 7.12.2 3D Shapes - Volume
                                                                                                                                                                                                                      • 7.13 Pythagoras
                                                                                                                                                                                                                        • 7.13.1 Pythagoras Theorem
                                                                                                                                                                                                                        • 7.14 SOHCAHTOA
                                                                                                                                                                                                                          • 7.14.1 SOHCAHTOA
                                                                                                                                                                                                                            • 7.14.2 3D Pythagoras & SOHCAHTOA
                                                                                                                                                                                                                            • 7.15 Sine/Cos Rules & Area of a Triangle
                                                                                                                                                                                                                              • 7.15.1 Sine & Cosine Rules, Area of Triangle - Basics
                                                                                                                                                                                                                                • 7.15.2 Sine & Cosine Rules, Area of Triangle - Harder
                                                                                                                                                                                                                                • 7.16 Circle Theorems
                                                                                                                                                                                                                                  • 7.16.1 Circle Theorems - Angles at Centre & Circumference
                                                                                                                                                                                                                                    • 7.16.2 Circle Theorems - Tangents
                                                                                                                                                                                                                                      • 7.16.3 Circle Theorems - Cyclic Quadrilaterals
                                                                                                                                                                                                                                        • 7.16.4 Circle Theorems - Alternative Segment
                                                                                                                                                                                                                                        • 7.17 Loci & Construction
                                                                                                                                                                                                                                          • 7.17.1 Loci & Construction
                                                                                                                                                                                                                                          • 7.18 Bearings & Scale
                                                                                                                                                                                                                                            • 7.18.1 Bearings & Scale
                                                                                                                                                                                                                                            • 7.19 Vectors
                                                                                                                                                                                                                                              • 7.19.1 Vectors - Basics
                                                                                                                                                                                                                                                • 7.19.2 Vectors - Finding Paths
                                                                                                                                                                                                                                                  • 7.19.3 Vectors - Proving Things
                                                                                                                                                                                                                                                  • 7.20 Drawing Plans & Elevations
                                                                                                                                                                                                                                                    • 7.20.1 Drawing Plans & Elevations
                                                                                                                                                                                                                                                    • 7.21 Trigonometry - Exact Values
                                                                                                                                                                                                                                                      • 7.21.1 Trigonometry - Exact Values
                                                                                                                                                                                                                                                    • 8. Probability
                                                                                                                                                                                                                                                      • 8.1 Basic Probability
                                                                                                                                                                                                                                                        • 8.1.1 Basic Probability
                                                                                                                                                                                                                                                        • 8.2 Venn Diagrams & Two Way Tables
                                                                                                                                                                                                                                                          • 8.2.1 Probability - Venn Diagrams
                                                                                                                                                                                                                                                            • 8.2.2 Probability - Two Way Tables
                                                                                                                                                                                                                                                              • 8.2.3 Set Notation & Venn Diagrams
                                                                                                                                                                                                                                                              • 8.3 Combined Probability
                                                                                                                                                                                                                                                                • 8.3.1 Combined Probability - Basics
                                                                                                                                                                                                                                                                  • 8.3.2 Combined Probability - Harder
                                                                                                                                                                                                                                                                • 9. Statistics
                                                                                                                                                                                                                                                                  • 9.1 Mean, Median & Mode
                                                                                                                                                                                                                                                                    • 9.1.1 Mean, Median & Mode
                                                                                                                                                                                                                                                                      • 9.1.2 Averages from Tables & Charts
                                                                                                                                                                                                                                                                        • 9.1.3 Averages from Grouped Data
                                                                                                                                                                                                                                                                          • 9.1.4 Calculations with the Mean
                                                                                                                                                                                                                                                                            • 9.1.5 IQR & Range
                                                                                                                                                                                                                                                                            • 9.2 Box Plots
                                                                                                                                                                                                                                                                              • 9.2.1 Box Plots
                                                                                                                                                                                                                                                                              • 9.3 Stem & Leaf Diagrams
                                                                                                                                                                                                                                                                                • 9.3.1 Stem & Leaf Diagrams
                                                                                                                                                                                                                                                                                • 9.4 Frequency Polygons
                                                                                                                                                                                                                                                                                  • 9.4.1 Frequency Polygons
                                                                                                                                                                                                                                                                                  • 9.5 Scatter Graphs (inc. Time Series)
                                                                                                                                                                                                                                                                                    • 9.5.1 Scatter Graphs
                                                                                                                                                                                                                                                                                      • 9.5.2 Time Series Graphs
                                                                                                                                                                                                                                                                                      • 9.6 Histograms
                                                                                                                                                                                                                                                                                        • 9.6.1 Histograms
                                                                                                                                                                                                                                                                                        • 9.7 Cumulative Frequency Graphs
                                                                                                                                                                                                                                                                                          • 9.7.1 Cumulative Frequency
                                                                                                                                                                                                                                                                                          • 9.8 Bar Charts
                                                                                                                                                                                                                                                                                            • 9.8.1 Bar Charts
                                                                                                                                                                                                                                                                                            • 9.9 Population & Samples
                                                                                                                                                                                                                                                                                              • 9.9.1 Comparing Distributions
                                                                                                                                                                                                                                                                                                • 9.9.2 Capture - Recapture


                                                                                                                                                                                                                                                                                                DOWNLOAD PDF

                                                                                                                                                                                                                                                                                              Author: Lucy

                                                                                                                                                                                                                                                                                              Lucy has been a passionate Maths teacher for over 12 years, helping to engage interest and develop confidence in the subject at all levels. As Head of Department and Director of Maths, Lucy has advised schools across both academy trusts and an entire city, where her role was to support and coach teachers to improve Maths teaching for all.


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