# Seven-Round Algebra and Probability Foundations: Powers, Identities, Functions, and Finite Models Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Variables and algebraic expressions](/en/mko/mko-variable-expression/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-variable-expression) [Concept guides](/en/derivations/?q=mko-variable-expression) - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number) - Stage 1 · required ## [Equation](/en/mko/mko-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-equation) [Concept guides](/en/derivations/?q=mko-equation) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Linear equation in one variable](/en/mko/mko-linear-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-equation) [Concept guides](/en/derivations/?q=mko-linear-equation) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 1 · required ## [Event](/en/mko/mko-event/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-event) [Concept guides](/en/derivations/?q=mko-event) - Stage 1 · required ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-ratio-proportion) [Concept guides](/en/derivations/?q=mko-ratio-proportion) - Stage 2 · optional ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) This entry covers natural-number powers of real bases and principal square roots of nonnegative real numbers, keeping powers, root values, and solution sets distinct. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 3 · optional ## [Univariate polynomials over the rationals](/en/mko/mko-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) A polynomial is finitely supported rational-coefficient data; combine like powers before reading coefficients or degree. [Related questions](/en/explore/?mko=mko-polynomial) [Concept guides](/en/derivations/?q=mko-polynomial) - Stage 4 · optional ## [Polynomial factorization over the rationals](/en/mko/mko-factorization/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values. [Related questions](/en/explore/?mko=mko-factorization) [Concept guides](/en/derivations/?q=mko-factorization) - Stage 5 · optional ## [Quadratic equations over the reals](/en/mko/mko-quadratic-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) For a real equation with a nonzero quadratic coefficient, the discriminant classifies its distinct real roots. The equation, root set, and repeated roots are kept distinct. [Related questions](/en/explore/?mko=mko-quadratic-equation) [Concept guides](/en/derivations/?q=mko-quadratic-equation) - Stage 6 · optional ## [Affine functions and degree-one functions](/en/mko/mko-linear-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) A real function f(x)=mx+b is affine. If m≠0 it has degree one; if m=0 it is constant. [Related questions](/en/explore/?mko=mko-linear-function) [Concept guides](/en/derivations/?q=mko-linear-function) - Stage 7 · optional ## [Graphs of real-valued functions](/en/mko/mko-function-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output. [Related questions](/en/explore/?mko=mko-function-graph) [Concept guides](/en/derivations/?q=mko-function-graph) - Stage 8 · optional ## [Probability on finite sample spaces](/en/mko/mko-probability/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261003%2F) Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes. [Related questions](/en/explore/?mko=mko-probability) [Concept guides](/en/derivations/?q=mko-probability) Path source & statuspath-pulse-foundations-20261003 · candidate [Original paths JSON](/data/architecture/learning-paths.json)