# Seven-Round Foundation Expansion: Numbers, Functions, Counting, and Coordinates Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. [← Return to results](/en/paths/) This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Set](/en/mko/mko-set/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set) [Concept guides](/en/derivations/?q=mko-set) - Stage 1 · required ## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number) - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) 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 ## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-counting-principle) [Concept guides](/en/derivations/?q=mko-counting-principle) - Stage 1 · required ## [Factorial](/en/mko/mko-factorial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-factorial) [Concept guides](/en/derivations/?q=mko-factorial) - Stage 1 · required ## [Points, lines, and planes](/en/mko/mko-point-line-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-point-line-plane) [Concept guides](/en/derivations/?q=mko-point-line-plane) - Stage 2 · optional ## [Fraction](/en/mko/mko-fraction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Treat p/q as a rational-number representation with integer numerator and denominator and a nonzero denominator. [Related questions](/en/explore/?mko=mko-fraction) [Concept guides](/en/derivations/?q=mko-fraction) - Stage 3 · optional ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Interpret the numerical ratio a:b as a/b and a proportion as equality between two valid ratios. [Related questions](/en/explore/?mko=mko-ratio-proportion) [Concept guides](/en/derivations/?q=mko-ratio-proportion) - Stage 4 · optional ## [Domain, codomain, and range](/en/mko/mko-domain-codomain-range/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Distinguish the domain A, declared codomain B, and actually attained range f[A] of a total function f:A→B. [Related questions](/en/explore/?mko=mko-domain-codomain-range) [Concept guides](/en/derivations/?q=mko-domain-codomain-range) - Stage 5 · optional ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Evaluate g∘f by applying f first and verify that the intermediate value is an allowed input of g. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) - Stage 6 · optional ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Use zero-based real sequences a:N→R whose terms retain their index order rather than becoming an unordered set. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 7 · optional ## [Permutations and combinations](/en/mko/mko-permutation-combination/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Distinguish ordered permutations from unordered combinations when selecting distinct objects without replacement. [Related questions](/en/explore/?mko=mko-permutation-combination) [Concept guides](/en/derivations/?q=mko-permutation-combination) - Stage 8 · optional ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20260930%2F%3Ffrom%3D%252Fen%252Fpaths%252F) Represent plane points by ordered real pairs after fixing an origin, perpendicular oriented axes, and units. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) Path source & statuspath-pulse-foundations-20260930 · candidate [Original paths JSON](/data/architecture/learning-paths.json)