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OCME · Bilingual mathematics & derivations

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.

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This is a suggested learning sequence, not a universal logical dependency or a formal proof.

1. Stage 1 · required

   Rational number

   Existing canonical prerequisite for the new objects in this pulse.

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2. Stage 1 · required

   Arithmetic operations

   Existing canonical prerequisite for the new objects in this pulse.

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3. Stage 1 · required

   Function mapping

   Existing canonical prerequisite for the new objects in this pulse.

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4. Stage 1 · required

   Set

   Existing canonical prerequisite for the new objects in this pulse.

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5. Stage 1 · required

   Natural number

   Existing canonical prerequisite for the new objects in this pulse.

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6. Stage 1 · required

   Real number

   Existing canonical prerequisite for the new objects in this pulse.

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7. Stage 1 · required

   Basic counting principles

   Existing canonical prerequisite for the new objects in this pulse.

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8. Stage 1 · required

   Factorial

   Existing canonical prerequisite for the new objects in this pulse.

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9. Stage 1 · required

   Points, lines, and planes

   Existing canonical prerequisite for the new objects in this pulse.

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10. Stage 2 · optional

    Fraction

    Treat p/q as a rational-number representation with integer numerator and denominator and a nonzero denominator.

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11. Stage 3 · optional

    Ratio and proportion

    Interpret the numerical ratio a:b as a/b and a proportion as equality between two valid ratios.

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12. Stage 4 · optional

    Domain, codomain, and range

    Distinguish the domain A, declared codomain B, and actually attained range f[A] of a total function f:A→B.

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13. Stage 5 · optional

    Function composition

    Evaluate g∘f by applying f first and verify that the intermediate value is an allowed input of g.

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14. Stage 6 · optional

    Sequence

    Use zero-based real sequences a:N→R whose terms retain their index order rather than becoming an unordered set.

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15. Stage 7 · optional

    Permutations and combinations

    Distinguish ordered permutations from unordered combinations when selecting distinct objects without replacement.

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16. Stage 8 · optional

    Coordinate plane

    Represent plane points by ordered real pairs after fixing an origin, perpendicular oriented axes, and units.

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