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. I also made a practice page to help you remember what you've learned. You can now follow me on twitter.

When I'm not writing about maths or making videos, I'm an online maths tutor.

If you like my work, please support me on patreon.

Changelog

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
2016-07-29 Ptolemy's theorem (cyclic quadrilaterals)
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
2015-11-05 Babylonian quadratics
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

Previously completed (before I started keeping a changelog)

Fibonacci's rabbit problem
Decimal division question about ratios between Fibonacci numbers
Archimedes and the gold crown
Decimal division question about pi
Equivalent fractions
Draft start of proportion/multiplication with real numbers section
Intro to algebra chapter
Applying formulas: Heron's formula
Expanding $(a+b)^3$
An algebra question involving the golden ratio
An algebra question involving the golden ratio and the Fibonacci sequence
Solving cubic equations
Finding the sum of a sequence
Proof that root two is irrational
Geometric algebra
Trig question about an equilateral triangle in a square
How Madhava calculated pi
Solving equations of the form $A\sin x + B\cos x = C$
How many faces, edges, and vertices does a dodecahedron have? (Using division in combinatorics)
First recorded example of the Fibonacci sequence
Derivation of Snell's law (calculus minimisation question)
Madhava's arctangent series
Symbols appendix

Contents

Practice | | Support me on Patreon



Creative Commons License
The How and Why of Mathematics by J L Gorham is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.