Mathematical Foundations

Each kind of number is forced into existence by an equation the previous kind cannot solve, and the chain runs from counting to the continuum.

The short list of laws that licence every manipulation, why subtraction and division are not fundamental, and why dividing by zero is barred.

Solving by inverse operations, deriving the quadratic formula by completing the square, and reading the discriminant before doing any work.

The rule that gives one output per input: domain, range, composition, and the condition a rule must meet before it can be run backwards.

Coordinates make a function visible: distance, the straight line, the parabola read off its vertex form, and what shifting and stretching do to an equation.

Degree, division with remainder, the factor theorem, and why the roots of a polynomial and the shape of its graph are the same information twice.

Extending repeated multiplication to every real exponent, the number compound interest converges on, and why every exponential eventually beats every polynomial.

The inverse of the exponential: how it turns multiplication into addition, how it solves for an unknown exponent, and why so many measurement scales are logarithmic.

Ratios fixed by an angle alone, the exact values worth knowing, and the sine and cosine rules that reach triangles with no right angle in them.

Escaping the triangle: definitions valid for every angle, radians, the wave graphs, and the identities that fall out of the circle.

Arithmetic and geometric patterns, closed forms for their sums, why an infinite series can have a finite total, and the loans they price.

Direct argument, contrapositive, contradiction and induction, put to work on the two claims this course left open.