Concepts

The ideas themselves — what a thing is, why it works, and where it shows up.

• AdditionThe arithmetic operation of combining quantities, including its commutative and associative properties and its extension to matrices and complex numbers.
• Differential EquationsEquations that relate a function to its own derivative, the main families they are sorted into, and how separation of variables solves the simplest of them.
• DivisibilityWhat it means for one number to divide another, defined through multiplication rather than division, with worked proofs of the basic properties.
• Euler's ConstantThe irrational number e, defined as a limit, notable for being the base of the natural logarithm and its own derivative.
• FractalsAn introduction to fractals, self-similar shapes with finite area but infinite perimeter.
• FunctionsHow functions map inputs to outputs, from notation and the formal set-theoretic definition through inverse, injective, surjective, and bijective functions.
• InductionProving a statement for every natural number by establishing a base case and a step that carries each case to the next, with the well-ordering property behind it and a worked example.
• IntegralsThe operator representing the area or volume bounded by a function, covering definite and indefinite integrals and the notation for both.
• LogarithmsThe inverse exponential operator
• PEMDAS & BODMASMnemonic devices to help remember elemtary order of operations.
• Proof by ContradictionThe proof technique of assuming the negation of what you want to prove and following it until something breaks, with a worked example showing there is no largest integer.
• Pythagorean TheoremThe relationship between the legs and the hypotenuse of a right triangle, and the integer triples that satisfy it.
• Set TheoryUnderstanding how to group things and the properties that follow.
• Sets of NumbersExploring different sets of numbers and introductory set theory.
• Sigma NotationExploring different sets of numbers and introductory set theory.
• The Quadratic FormulaThe formula for finding the x-intercepts of a second-degree polynomial (quadratic equation).