# The How and Why of Mathematics

## Understand everything from counting to complex numbers

### By J. L. Gorham

T

he How and Why of Mathematics started out as a book, then became a blog, and now I guess it's a youtube channel. You can now follow me on twitter.

#### Changelog

2021-09-17 Video: To the families struggling with homeschooling/remote learning during lockdowns
2020-12-29 Video: a geometry construction problem I could actually do!
2020-09-10 Video: Senior AMC 2019 Q30: function recursion
2020-04-13 Video: How to do maths tutoring online
2020-03-29 Video: Long division maths puzzle
2020-03-14 Video: Solutions for Q16 of the 2019 NSW HSC Maths Ext2 exam
2020-01-26 Video: Maths Olympiad Question: IMO 2018 C1
2019-12-25 Video: Bloopers and auto-captions gone wrong! Also merry Christmas!
2019-12-03 Video: Hardest Canadian maths questions - Mathematics 30-1 Diploma Exams
2019-11-17 Video: Akshat's questions, and what I've been up to
2019-08-30 Video: Hardest maths questions (junior) - candles
2019-08-05 Video: The problem with PEMDAS: why calculators disagree
2019-07-19 Video: Maths Olympiad Question - 2019 INMO Q1 (circle geometry)
2019-07-05 Video: How computers can help with maths (solving a geometry problem with sagemath and metapost)
2019-05-21 Video: 2018 JEE Advanced Maths questions I struggled with (Q4,5,6 from paper 1 and Q7 from paper 2)
2019-05-07 Video: 2018 JEE Advanced Paper 1 Maths Q17 and 18 (combinatorics questions)
2019-04-23 Video: Hardest maths questions (olympiad) - 2017 IMO C1
2019-04-09 Video: Hardest maths questions - minimisation
2019-04-02 Video: Making of refactorisation, solids of integration, and computer graphics in future videos
2019-03-19 Video: General solutions to trig equations
2019-03-12 Video: Hardest maths questions - express $8\times9\times10\times11\times12\times13\times14$ as another product of consecutive whole numbers
2019-03-05 Video: Hardest maths questions - simultaneous diophantine equations
2019-02-26 Video: How to solve trig equations with a calculator (and the unit circle)
2019-02-19 Video: How to solve trig equations with exact solutions using the unit circle
2019-02-12 Video: Upcoming videos, thank you to my first 100 subscribers, and happy birthday to me
2019-02-05 Video: Sine, cos, and tan of special angles with help from the unit circle
2019-01-29 Video: Trig identities and the unit circle
2019-01-22 Video: Introduction to the unit circle
2019-01-18 Video: Hardest maths questions - tyres
2019-01-15 Video: Hardest maths questions - find $a^4 + b^4 + c^4$
2019-01-11 Video: Hardest maths questions - inheritance
2019-01-08 Video: Hardest maths questions - solve $\sqrt[3]{n+\sqrt{n^2+8}} + \sqrt[3]{n-\sqrt{n^2+8}} = 8$
2018-12-22 Video: Hardest maths questions - find $f(2003)$ if $f(11) = 11$ and $f(n+3) = \frac{f(n)-1}{f(n)+1}$
2018-12-18 Video: 2018 NSW HSC Mathematics Extension 2 exam Q16
2018-12-14 Video: 2018 NSW HSC Mathematics Extension 2 exam Q15
2018-12-11 Video: 2018 NSW HSC Mathematics Extension 2 exam Q14
2018-12-07 Video: 2018 NSW HSC Mathematics Extension 2 exam Q13
2018-12-05 Video: 2018 NSW HSC Mathematics Extension 2 exam Q12
2018-12-01 Video: 2018 NSW HSC Mathematics Extension 2 exam Q11
2018-11-27 Video: 2018 NSW HSC Mathematics Extension 2 exam Q1-10
2018-11-20 Video: Perpendicular slope equation - a very simple explanation
2018-11-16 Video: Polynomial long division with complex coefficients examples
2018-11-04 Video: Hardest questions from the AMC junior division - the talent quest
2018-10-11 Video: Partial fractions - lots of examples
2018-09-25 Video: Hardest questions from the 2018 AMC senior division
2018-09-24 Video: Hardest questions from the AMC senior division - |x| + |y| + |x+y| ≤ 2
2018-09-17 Video: How to find the y-intercept and equation of a line from coordinates
2018-09-12 Video: How to find gradient or slope given two points on a straight line
2018-07-27 Video: NSW HSC Maths 2017 Extension 2 Exam Solutions Q16 (part 7/7)
2018-07-23 Video: NSW HSC Maths 2017 Extension 2 Exam Solutions Q15 (part 6/7)
2018-07-18 Video: NSW HSC Maths 2017 Extension 2 Exam Solutions Q14 (part 5/7)
2018-07-08 Video: NSW HSC Maths 2017 Extension 2 Exam Solutions Q13 (part 4/7)
2018-07-04 Video: NSW HSC Maths 2017 Extension 2 Exam Solutions Q12 (part 3/7)
2018-07-03 Video: NSW HSC Maths 2017 Extension 2 Exam Solutions Q11 (part 2/7)
2018-07-02 Video: NSW HSC Maths 2017 Extension 2 Exam Solutions Q1-10 (part 1/7)
2018-06-25 Video: Some of my favourite tricky maths problems from past Australian Mathematics Competitions
2018-06-18 Video: The problem with PEMDAS
2018-06-11 Exam walkthrough video for the 2016 HSC Mathematics Extension 2, part 7/7
2018-06-04 Video: Cosine rule proof
2018-05-28 Exam walkthrough video for the 2016 HSC Mathematics Extension 2, part 6/7
2018-05-07 Exam walkthrough video for the 2016 HSC Mathematics Extension 2, part 5/7
2018-05-03 Exam walkthrough video for the 2016 HSC Mathematics Extension 2, part 4/7
2018-05-01 Exam walkthrough video for the 2016 HSC Mathematics Extension 2, part 3/7
2018-04-27 Exam walkthrough video for the 2016 HSC Mathematics Extension 2, part 2/7
2018-04-26 Exam walkthrough video for the 2016 HSC Mathematics Extension 2, part 1/7
2017-07-15 Derivation of the angle sum formula for sine using Ptolemy's theorem
2017-07-10 Derivation of multiple angle formula for tan
2017-05-07 The practice system can now email you a reminder to revise your cards. Go here to subscribe.
2017-05-04 Practice questions about dimensional analysis and exponents in algebra
2017-04-07 Practice questions about percentages, simple equations, averages, and ratios
2017-03-28 Simple division and simple subtraction practice questions
2017-03-23 A logic puzzle demonstrating the general idea of symmetry
2017-02-23 Practice questions for adding and subtracting negative numbers
2017-02-22 Practice questions for the quadratic formula
2017-01-05 Lots of simultaneous equations from electronics question which makes a good motivation for linear algebra
2016-11-14 Matrix question about a squirrel
2016-11-10 Question about counting outcomes in probability and the sum of two dice
2016-10-15 Question about latitude and longitude
2016-10-09 Practice questions on solving inequalities where you have to graph a parabola, and integrating $\sin^2x$ and $\cos^2x$
2016-10-08 Practice questions on log laws
2016-10-07 Practice questions on graphing the derivative, solving for an exponent, motion terminology: velocity, speed, and acceleration, scientific notation, derivatives of trigonometric functions, integrals of trigonometric functions, and product/quotient/chain rules
2016-10-04 Practice questions on slope
2016-10-03 Practice questions on solving trig equations
2016-10-02 Practice questions on the special angles and their cosines, etc.
2016-09-30 Some practice questions on simplifying algebraic fractions and the difference of two squares
2016-09-04 Completing the square
2016-08-22 Question about the area of a triangle in a rectangle
2016-08-05 A question about the shortest distance between two points, touching a straight line on the way (using Pythagoras), and the solution using calculus
2016-08-02 Adding features to practice page: progress bar, change timezone
2016-07-26 Added practice questions for finding the distance between two points to Pythagoras' theorem
2016-07-25 Introduction
2016-07-22 Making the site display nicely on small screens
2016-07-20 Added alternative solutions to the grid walks question
2016-07-15 Section on Taylor series
2016-07-04 Another question on Pythagoras' theorem
2016-06-15 Added some practice questions to subsection about solving equations with multiple solutions
2016-06-12 Made ‘Extras’ appendix for various ramblings that don't really belong anywhere. Mostly technical software details. SRS source code, How the diagrams are made, and Lowercase numbers in mathematics.
2016-06-02 Solving an equation involving cube roots
2016-05-28 Using multiplication to reverse a division and vice versa
2016-05-26 Golden ratio in the diagonals of pentagons
2016-05-22 A squared rectangle problem
2016-05-17 Added links to MetaPost source code so you can see how the diagrams are created.
2016-05-12 Geometry question about isosceles triangles in pentagon
2016-05-05 Handwriting in algebra
2016-04-16 Distance to Sun and Moon
2016-04-10 Spaced repetition software is more-or-less finished. See for example times tables
2016-04-02 Integration by parts
2016-04-01 Pythagorean triples
2016-02-04 Intro question to integration by parts
2016-01-20 Made gif images as alternative to svg for browsers that don't support svg
2016-01-10 Working on spaced repetition feature...
2016-01-05 Solving for an expression rather than a single variable
2015-12-25 Number of walks through a grid
2015-12-23 Subsection about solving equations with multiple solutions
2015-12-22 Added comma to symbols appendix
2015-12-21 Modified my $\LaTeX$ to html converter so that I can make standalone subsections.
2015-12-14 Newton's method for square roots
2015-11-06 Added ‘assumed knowledge’ feature to questions
2015-11-03 Inverses in modular arithmetic, chinese remainder theorem, checking if a number is prime
2015-10-22 The extended Euclidean algorithm, another example
2015-10-21 Modular exponentiation, the hundred pigs problem
2015-10-12 Number of handshakes
2015-10-10 Number of edges and vertices in a dodecahedron and added diagram to part c answer
2015-10-05 Example of calculating square roots
2015-10-01 Missionaries and cannibals/jealous husbands river crossing problem
2015-09-27 Shortest distance between a point and a line
2015-09-26 Solving a cubic equation with trigonometry
2015-09-24 More work on Pythagoras' theorem
2015-09-22 Partial fraction decomposition
2015-09-12 Pythagoras' theorem
2015-09-09 The family river crossing problem
2015-08-28 The wolf, the goat, and the cabbage, Rhombus area using Pythagoras, Expanding cubes the old-fashioned way
2015-08-18 How many three-character codes are there?, added Puzzles chapter
2015-08-09 Made some changes to Real and Imaginary section
2015-08-08 Bike rack (Pythagoras)
2015-07-25 How far is the horizon?, Diagonal of a cube

#### Contents

• Introduction
• 1 Puzzles
• 1.1 Misc
• 1.2 River-crossing problems (permaquestions: 1.2.1 1.2.2 1.2.3)
• 1.3 Logic Puzzles
• 1.4 Graph theory
• 1.5 Games
• 1.6 3D Visualisation
• 2 Numbers
• 2.1 What are Numbers for?
• 2.1.1 Counting
• 2.1.2 Measuring
• 2.1.3 Identifying
• 2.2 Hindu-Arabic Numerals
• 2.2.1 The Digits
• 2.2.2 Place Value
• 2.2.3 Decimals
• 2.2.4 Names of Numbers
• 2.2.5 Ordinal Numbers
• 2.2.6 Comparing Numbers
• 2.2.7 Rounding
• 2.3 Time
• 2.4 Roman Numerals
• 2.5 Tallying
• 4 Multiplying and Dividing
• 5 Exponents and Logarithms
• 5.1 Repeated Multiplication (permaquestions: 5.1.12)
• 5.2 Exponential Growth and Decay
• 5.2.1 Logarithmic Scales
• 5.2.2 Slide Rules and Logarithm Tables
• 5.2.3 Euler’s Number
• 6 Data
• 6.1 Representing Data
• 6.1.1 Tables
• 6.1.2 Scatter Plots
• 6.1.3 Bar Charts
• 6.1.4 Line Graphs
• 6.1.5 Pie Charts
• 6.2 Interpreting Data
• 7 Elementary Algebra
• 8 Sets and Functions
• 8.1 Sets
• 8.1.1 Unions and Intersections
• 8.1.2 Subsets and Supersets
• 8.1.3 Intervals
• 8.1.4 Set-Builder Notation
• 8.1.5 Cardinality
• 8.1.6 Cartesian Product
• 8.2 Functions
• 8.2.1 Domain and Range
• 8.2.2 Maps in General
• 8.2.3 Composition of Functions
• 8.2.4 Inverse Functions
• 8.2.5 Equation-Solving Revisited
• 8.2.6 Polynomial Division
• 9 Proof
• 9.1 Logic
• 9.2 Rigour
• 9.3 Symmetry (permaquestions: 9.3.1)
• 9.4 Axioms
• 9.5 Using Definitions and Theorems
• 9.6 Techniques (permaquestions: 9.6.7)
• 9.6.1 Direct Proof
• 9.6.2 Counterexample
• 9.6.3 Exhaustion
• 9.6.5 Induction
• 9.7.1 Logic
• 10 Geometry
• 10.1 The Greek Alphabet
• 10.2 Points and Lines (permaquestions: 10.2.1)
• 10.3 Angles
• 10.4 Two-Dimensional Shapes (permaquestions: 10.4.2)
• 10.4.1 Polygons
• 10.4.3 Triangles
• 10.4.4 Ellipses
• 10.5 Three-Dimensional Objects
• 10.6 Transformations (permaquestions: 10.6.2 10.6.3)
• 10.6.1 Translation
• 10.6.2 Scaling
• 10.6.3 Rotation
• 10.6.4 Reflection
• 10.6.5 Congruence and Similarity
• 10.6.6 Symmetry
• 10.7 Perimeter
• 10.7.1 Circumference
• 10.8 Area (permaquestions: 10.8.1 10.8.2 10.8.3 10.8.5)
• 10.8.1 Surface Area
• 10.9 Volume (permaquestions: 10.9.2)
• 11 Trigonometry and Coordinate Geometry
• 12 Linear Algebra
• 12.1 Linear Maps
• 12.2 Matrices
• 12.3 Matrix Multiplication
• 12.4 Square Matrices (permaquestions: 12.4.3)
• 12.4.1 Eigenvalues
• 12.5 Linear Equations
• 13 Combinatorics
• 13.1 Systematic Counting
• 13.2 A Quicker Way (permaquestions: 13.2.1 13.2.2 13.2.3)
• 13.3 Factorials
• 13.4 Permutations (permaquestions: 13.4.3)
• 13.5 Combinations (permaquestions: 13.5.2)
• 13.6 Graph Theory
• 13.6.1 Trees
• 13.6.2 Planar Graphs
• 14 Probability
• 14.1 Classical Definition (permaquestions: 14.1.1 14.1.2)
• 14.2 Physical Probability
• 14.3 Evidential Probability
• 15 Calculus
• 16 Complex Numbers
• 16.1 Rotation and Scaling
• 16.2 Real and Imaginary
• 16.3 Multiplication
• 16.4 Complex Conjugates
• 16.5 Division
• 16.6 Roots and Branches
• 16.7 Euler’s Formula
• A Units
• A.1 SI Units
• A.1.1 Base Units
• A.1.2 Derived Units
• A.2 Other Units
• A.3 Prefixes
• A.3.1 SI Prefixes
• A.3.2 IEC Binary Prefixes
• B Symbols
• C Extras
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