Cartesian product of two sets A and B is the set of ordered pairs (a, b). Named after René Descartes, who developed it in analytic geometry. Not commutative and not associative unless one set is empty. Cardinality of product equals product of cardinalities of input sets
Cartesian coordinates specify points uniquely using signed distances from perpendicular axes. Origin is point where axes meet, represented by (0,0) coordinates. System can be extended to three dimensions with three perpendicular axes. Named after René Descartes, who published it in 1637
Exercises cover exhaustion method for curve areas and set properties. Proofs of field and order axioms are included. Summation notation and induction proofs are covered
Geometry studies shapes, spatial relationships, and properties of space. Euclidean geometry codified in Elements by Euclid around 300 BCE. Analytic geometry introduced by Descartes with rectangular coordinates. Projective geometry developed by Desargues for non-projecting figures. Differential geometry initiated by Gauss for surveying problems
Analytical geometry combines algebra and geometry for two-dimensional space. Coordinate plane divides space into four quadrants with origin at (0,0). Points are represented by coordinates (x,y) with x-axis and y-axis
Linear equation describes unique line passing through two points. Equations can be in standard form (Ax + By + C = 0) or slope-intercept form (y = ax + b). Three-dimensional space requires additional vector equation