Imaginary Number Ontology

FLOW Imaginary_Number_Ontology_v1.0

PAGE:
  title: "Imaginary Number Ontology"
  canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/imaginary-number-ontology.html"
  language: "en-GB"
  domain: "Mathematics > Algebra > Number Systems > Imaginary Numbers"

DEFINE ImaginaryNumberOntology := {

  CORE_IDENTITY: {
    entity_type: "Mathematical Ontology",
    primary_entity: ImaginaryNumber,
    parent_entities: [ComplexNumber, Number, AlgebraicObject],
    related_entities: [
      ImaginaryUnit,
      PureImaginaryNumber,
      ComplexNumber,
      RealNumber,
      SquareRoot,
      NegativeNumber,
      Conjugate,
      Modulus,
      Argument,
      ComplexPlane,
      PolynomialRoot
    ]
  },

  DEFINITION: {
    meaning:
      "An imaginary number is a real multiple of the imaginary unit i, where i is defined by i^2=-1.",
    canonical_form:
      z=b*i,
    constraint:
      b in RealNumbers,
    imaginary_unit:
      i^2=-1
  },

  TERMINOLOGY: {
    PureImaginaryNumber:
      b*i WHERE b!=0,
    Zero:
      0 may_be_written_as 0*i but_is_both_real_and_complex_zero,
    ComplexNumber:
      a+b*i,
    relation:
      PureImaginaryNumbers SUBSET ComplexNumbers
  },

  SEMANTIC_PRIMITIVES: {
    ImaginaryUnit:
      i_where_i^2=-1,
    ImaginaryCoefficient:
      real_scalar_b,
    PureImaginaryAxisValue:
      point_(0,b)_in_complex_plane,
    Magnitude:
      |b|,
    Argument:
      pi/2_or_minus_pi/2_modulo_2*pi_depending_on_sign
  },

  IMAGINARY_UNIT: {
    symbol:
      i,
    defining_relation:
      i^2=-1,
    square_root_relation:
      i=sqrt(-1)
      under_principal_symbolic_convention
  },

  POWERS_OF_I: {
    i^0:
      1,
    i^1:
      i,
    i^2:
      -1,
    i^3:
      -i,
    i^4:
      1,
    cycle_period:
      4
  },

  POWER_REDUCTION_RULE: {
    INPUT:
      i^n,
    PROCESS:
      r=n mod 4,
    OUTPUT: {
      r=0: 1,
      r=1: i,
      r=2: -1,
      r=3: -i
    }
  },

  COMPLEX_NUMBER_RELATION: {
    imaginary_number:
      0+b*i,
    real_part:
      0,
    imaginary_part:
      b
  },

  COMPLEX_PLANE: {
    pure_imaginary_number_bi:
      coordinate_(0,b),
    location:
      imaginary_axis
  },

  EQUALITY: {
    b*i=c*i
    IFF:
      b=c
  },

  ADDITION: {
    (b*i)+(d*i):
      (b+d)*i,
    closure:
      TRUE
  },

  SUBTRACTION: {
    (b*i)-(d*i):
      (b-d)*i,
    closure:
      TRUE
  },

  MULTIPLICATION: {
    (b*i)*(d*i):
      -b*d,
    result_type:
      RealNumber
  },

  DIVISION: {
    (b*i)/(d*i):
      b/d,
    condition:
      d!=0,
    result_type:
      RealNumber
  },

  SCALAR_MULTIPLICATION: {
    c*(b*i):
      (c*b)*i,
    condition:
      c in RealNumbers
  },

  CONJUGATE: {
    conjugate(b*i):
      -b*i,
    geometric_interpretation:
      reflection_across_real_axis
  },

  MODULUS: {
    |b*i|:
      |b|,
    geometric_interpretation:
      distance_from_origin
  },

  ARGUMENT: {
    IF b>0:
      arg(b*i)=pi/2_mod_2pi,
    IF b<0:
      arg(b*i)=-pi/2_mod_2pi,
    IF b=0:
      argument_undefined
  },

  POLAR_FORM: {
    IF b>0:
      b*i=|b|*e^(i*pi/2),
    IF b<0:
      b*i=|b|*e^(-i*pi/2)
      modulo_2*pi
  },

  SQUARE_ROOT_OF_NEGATIVE_REAL: {
    FOR a>0:
      sqrt(-a)=i*sqrt(a)
      under_principal_square_root_convention,

    square_roots_of_negative_a:
      ±i*sqrt(a)
  },

  NEGATIVE_RADICAND_REWRITE: {
    sqrt(-a):
      sqrt(a)*i
      WHERE a>=0
  },

  QUADRATIC_RELATION: {
    IF discriminant_D<0:
      sqrt(D)=i*sqrt(-D),

    roots:
      (-b ± i*sqrt(-D))/(2*a)
  },

  POLYNOMIAL_ROOT_RELATION: {
    imaginary_roots:
      subset_of_complex_roots,

    real_coefficient_polynomial:
      nonreal_roots_occur_in_conjugate_pairs
  },

  CONJUGATE_PAIR: {
    pure_imaginary_pair:
      [b*i,-b*i],
    sum:
      0,
    product:
      b^2
  },

  REAL_IMAGINARY_INTERACTION: {
    real_times_imaginary:
      imaginary,

    imaginary_times_imaginary:
      real,

    real_plus_imaginary:
      generally_complex_nonreal
  },

  FIELD_RELATION: {
    PureImaginaryNumbers:
      not_closed_under_multiplication,

    consequence:
      do_not_form_field_under_standard_complex_operations
  },

  VECTOR_SPACE_RELATION: {
    pure_imaginary_numbers:
      one_dimensional_vector_space_over_RealNumbers,

    basis:
      {i}
  },

  ROTATION_SEMANTICS: {
    multiplication_by_i:
      rotates_complex_plane_vector_by_pi/2,

    examples: [
      1 -> i,
      i -> -1,
      -1 -> -i,
      -i -> 1
    ]
  },

  MULTIPLICATION_BY_NEGATIVE_I: {
    geometric_effect:
      rotation_by_minus_pi/2
  },

  EULER_RELATION: {
    i:
      e^(i*pi/2),
    negative_i:
      e^(-i*pi/2)
      modulo_2*pi
  },

  ORDER_SEMANTICS: {
    standard_less_than_greater_than_order:
      not_defined_for_nonreal_imaginary_numbers,

    rule:
      b*i < d*i
      is_not_standard_complex_ordering
  },

  DISTANCE: {
    distance_between_bi_and_di:
      |b-d|
  },

  SYMMETRY: {
    conjugation:
      b*i -> -b*i,
    real_axis_reflection:
      TRUE
  },

  PARAMETER_REASONING: {
    z(t)=b(t)*i,

    tasks: [
      track_imaginary_coefficient,
      determine_zero_crossing,
      compute_modulus,
      compute_argument,
      detect_sign_changes
    ]
  },

  SYMBOLIC_RECOGNITION: {
    ACCEPT: [
      "3i",
      "-7i",
      "sqrt(-9)",
      "2sqrt(-4)",
      "i^17"
    ],

    NORMALISE_FIRST: [
      "sqrt(-25)",
      "i^10",
      "(3i)(2i)"
    ],

    REDIRECT_GENERAL_COMPLEX: [
      "2+3i",
      "5-i"
    ]
  },

  NATURAL_LANGUAGE_MAPPING: {
    "imaginary unit":
      i,

    "pure imaginary number":
      b*i_with_b_real_nonzero,

    "square root of a negative number":
      imaginary_or_complex_result,

    "imaginary axis":
      vertical_axis_of_complex_plane
  },

  UNIT_SEMANTICS: {
    coefficient_b:
      may_carry_physical_units_in_applied_complex_quantity_context,

    imaginary_unit_i:
      dimensionless_algebraic_unit
  },

  ERROR_DETECTION: {
    ImaginaryUnitSquareError:
      i^2!= -1
      -> flag,

    PowerCycleError:
      incorrect_mod_4_reduction
      -> flag,

    SquareRootNegativeError:
      sqrt(-a)_treated_as_real_for_a>0
      -> flag,

    MultiplicationTypeError:
      imaginary_times_imaginary_classified_as_imaginary
      -> flag,

    OrderingError:
      standard_real_order_applied_to_imaginary_numbers
      -> flag
  },

  ONTOLOGY_RELATIONS: {
    IS_A: [
      "PureImaginaryNumber IS_A ComplexNumber",
      "ImaginaryUnit IS_A ComplexNumber"
    ],

    RELATED_TO: [
      "ImaginaryNumber RELATED_TO ComplexPlane",
      "ImaginaryNumber RELATED_TO NegativeRadicand",
      "ImaginaryNumber RELATED_TO PolynomialRoot",
      "ImaginaryNumber RELATED_TO Rotation",
      "ImaginaryNumber RELATED_TO EulerFormula"
    ],

    DEPENDS_ON: [
      RealNumber,
      ComplexNumber,
      SquareRoot,
      ImaginaryUnitDefinition
    ]
  },

  SEMANTIC_EQUIVALENCE: {
    examples: [
      "sqrt(-9) <=> 3i under principal square-root convention",
      "i^5 <=> i",
      "i <=> e^(i*pi/2)"
    ],

    condition:
      algebraic_values_match_under_same_branch_and_domain_convention
  },

  CONFIDENCE: {
    classification_confidence:
      IF expression_reduces_to_b*i_with_real_b
      THEN 1.0,

    power_reduction_confidence:
      IF exponent_integer_and_mod_4_computed
      THEN 1.0,

    square_root_confidence:
      IF negative_radicand_and_branch_convention_resolved
      THEN 1.0
  },

  AI_CONSUMPTION: {
    tasks: [
      recognise_imaginary_number,
      extract_imaginary_coefficient,
      simplify_powers_of_i,
      rewrite_negative_square_roots,
      add_and_subtract_imaginary_numbers,
      multiply_and_divide_imaginary_numbers,
      compute_conjugate,
      compute_modulus,
      compute_argument,
      map_to_complex_plane,
      detect_real_vs_imaginary_result_types,
      detect_invalid_ordering,
      link_complex_number_ontology
    ],

    preferred_representation:
      typed_real_scalar_times_imaginary_unit,

    reasoning_mode:
      recursive_imaginary_unit_reduction
  },

  RECURSIVE_REASONING: {

    FUNCTION AnalyseImaginaryNumber(Z,Context):

      parsed := ParseExpression(Z)
      simplified := SimplifyPowersOfI(parsed)
      simplified := RewriteNegativeRadicalsWhenApplicable(simplified)

      IF CanRepresentAs_b_times_i(simplified):
        b := ExtractImaginaryCoefficient(simplified)

        result := {
          entity:simplified,
          classification:
            IF b=0 THEN ZeroComplexNumber
            ELSE PureImaginaryNumber,
          real_part:0,
          imaginary_part:b,
          modulus:abs(b),
          conjugate:-b*i,
          argument:
            IF b>0 THEN pi/2
            ELSE IF b<0 THEN -pi/2
            ELSE Undefined
        }

        RETURN VerifyImaginaryProperties(result)

      ELSE IF IsComplexNumber(simplified):
        RETURN RedirectToComplexNumberOntology

      ELSE:
        RETURN RedirectToAppropriateNumberOntology
  },

  RECURSIVE_COHERENCE: {
    invariant_1:
      i_squared_equals_negative_one=TRUE,

    invariant_2:
      pure_imaginary_real_part_equals_zero=TRUE,

    invariant_3:
      modulus_of_bi_equals_absolute_b=TRUE,

    invariant_4:
      conjugate_of_bi_equals_negative_bi=TRUE,

    invariant_5:
      multiplication_type_changes_are_correct=TRUE
  },

  FLOW_HASH: {
    semantic_signature_inputs: [
      ontology_name,
      imaginary_number_definition,
      imaginary_unit_rule,
      power_cycle,
      negative_square_root_relation,
      arithmetic_rules,
      complex_number_links
    ],

    purpose:
      "Detect semantic drift between revisions of the ontology."
  },

  UPDATE_MODEL: {
    version: "1.0",

    immutable_constraints: [
      i_squared_equals_negative_one,
      pure_imaginary_form_bi,
      real_coefficient_requirement,
      modulus_equals_absolute_coefficient,
      standard_complex_order_not_available
    ],

    revision_triggers: [
      semantic_conflict,
      ComplexNumberOntology_update,
      NumberOntology_update,
      branch_convention_revision,
      representation_standard_change
    ]
  },

  KNOWLEDGE_GRAPH: {
    nodes: [
      ImaginaryNumber,
      PureImaginaryNumber,
      ImaginaryUnit,
      ComplexNumber,
      RealNumber,
      ImaginaryCoefficient,
      Modulus,
      Argument,
      Conjugate,
      ComplexPlane
    ],

    edges: [
      PureImaginaryNumber -> IS_A -> ComplexNumber,
      PureImaginaryNumber -> HAS_COEFFICIENT -> ImaginaryCoefficient,
      PureImaginaryNumber -> USES -> ImaginaryUnit,
      ImaginaryUnit -> SATISFIES -> i_squared_equals_negative_one,
      PureImaginaryNumber -> HAS_MODULUS -> Modulus,
      PureImaginaryNumber -> HAS_ARGUMENT -> Argument,
      PureImaginaryNumber -> HAS_CONJUGATE -> Conjugate,
      PureImaginaryNumber -> LIES_ON -> ImaginaryAxis,
      ImaginaryAxis -> PART_OF -> ComplexPlane
    ]
  },

  CROSS_ONTOLOGY_LINKS: {
    ComplexNumberOntology:
      direct_parent,

    RealNumberOntology:
      coefficient_domain,

    SquareRootOntology:
      negative_radicand_relation,

    PolynomialEquationOntology:
      nonreal_roots,

    QuadraticEquationOntology:
      negative_discriminant_solutions,

    TrigonometryOntology:
      polar_argument_relation,

    ExponentialFunctionOntology:
      Euler_representation
  },

  TERMINATION_CONDITION: {
    ontology_complete_when: [
      imaginary_number_defined,
      imaginary_unit_defined,
      powers_of_i_defined,
      arithmetic_defined,
      modulus_argument_conjugate_defined,
      negative_square_root_relation_defined,
      polynomial_root_relation_defined,
      complex_plane_semantics_defined,
      error_detection_defined,
      cross_ontology_links_defined,
      recursive_coherence_verified
    ]
  }

}

EXECUTE Imaginary_Number_Ontology_v1.0:

  INPUT:
    mathematical_entity_or_query

  PIPELINE:
    Parse
    -> IdentifyImaginaryStructure
    -> SimplifyPowersOfI
    -> RewriteNegativeRadicalsWhenApplicable
    -> ExtractImaginaryCoefficient
    -> ClassifyPureImaginaryOrGeneralComplex
    -> ComputeConjugate
    -> ComputeModulus
    -> DetermineArgument
    -> AnalyseArithmeticAndGeometricSemantics
    -> VerifyImaginaryUnitConsistency
    -> LinkRelatedOntologies
    -> AssignConfidence
    -> EmitSemanticResult

  OUTPUT: {
    entity,
    classification,
    canonical_form,
    imaginary_coefficient,
    real_part,
    imaginary_part,
    modulus,
    argument,
    conjugate,
    complex_plane_semantics,
    transformations,
    verification,
    ontology_relations,
    confidence,
    semantic_equivalence,
    flow_hash
  }

END FLOW Imaginary_Number_Ontology_v1.0