Precalculus Foundations: Algebra and Geometry
The algebra and geometry that calculus quietly assumes: numbers built up carefully from the integers, equations solved and defended, the plane measured with distance and angle, the unit circle read as trigonometry, and curves finally organized as functions. One book, Lang's foundations shelf, read the way it was written.
You can compute with integers, fractions, and real numbers and name the laws you used, solve linear systems and quadratics and read the discriminant before you commit to a method, measure distance and angle in the plane and stand Pythagoras on solid ground, put coordinates on geometry and write the equation of a line, define sine and cosine on the unit circle, recognize a conic from its equation, and read the graph of a function the way calculus will soon ask you to.
Numbers and the line
Everything downstream leans on the rules for adding and multiplying integers; learning them as laws rather than habits is what makes the rest of the book provable.
Connectives, converses, and sets are the grammar every later proof is written in; mistaking an implication for its converse is the single most damaging reading error.
The rationals are the smallest system where division closes up, and the multiplicative inverse is the property that makes solving equations possible at all.
Positivity is the axiom that gives the line its direction, and inequalities are how that direction is used; calculus estimates live and die on this material.
Rational exponents unify powers and roots into one operation, and the square root is the tool the Pythagorean theorem and the quadratic formula both reach for.
Equations
Two unknowns, then three: elimination and substitution are the first genuinely algorithmic algebra, and the geometry of intersecting lines is hiding inside every solution.
Completing the square, the quadratic formula, and Vieta's formulas are one strategy wearing three coats, and the discriminant tells you how many real roots exist before you solve anything.
The plane
Distance and angle are the two primitive measurements of plane geometry, and the Pythagorean theorem is the first place the algebra of squares meets a picture.
The disc and its boundary are where pi enters mathematics honestly, through area and circumference arguments rather than by decree.
Coordinates turn geometry into algebra: the distance formula is Pythagoras restated, and the circle becomes an equation you can manipulate.
The line is the first curve to get an equation, and slope is the number calculus will later generalize into the derivative; this is where that story starts.
Trigonometry
Radians, sine, cosine, and tangent all read off one circle of radius one; defined this way they are ready for calculus, with no right triangle required.
Curves and functions
The three conics are the complete zoo of second-degree curves, and reading their equations closes the loop between the quadratics you solved and the plane you measured.
The function is the object calculus studies, and reading polynomial behavior off a graph is the skill every limit and derivative argument will assume you have.