RANGE: -(2ⁿ⁻¹) TO (2ⁿ⁻¹)-1
ERROR_DETECTION: PARITY_ACTIVE
ENCODING: EXCESS-3
STATE: SELF-COMPLEMENTING
ASKiIMAN™ DOOR 03 / THE DIGITAL MACHINE / REALM 03
--:-- · -- --- ----

Ontological Layer: Representation & Transformation

Binary Arithmetic & Codes

Mathematical Logic and Data Formatting

The Language of Operations

A number system dictates how a value is represented; arithmetic defines how it is manipulated. But a code goes further—it assigns specific binary patterns to symbols for transmission, error checking, or mechanical sensing. This realm bridges raw binary calculation with the structural codes required to make a digital system robust and functional.

NUMBER → BINARY REPRESENTATION → ARITHMETIC → SIGNED INTERPRETATION → ENCODING CODE → TRANSMISSION → ERROR CHECKING → DECODING

The 12 Computational Nodes

Ontological Mapping: Arithmetic & Code Theory

GROUP 01

Binary Arithmetic

101
+011
1000
  • 01.3 Binary Addition Rules
  • 01.4 Binary Subtraction
  • 01.7 Carry Generation
  • 01.8 Borrow Mechanisms
 GROUP 02

1's & 2's Complement

  • 02.1 Bit Inversion (1's Comp)
  • 02.3 2's Complement (+1)
  • 02.7 Subtraction via Addition
  • 02.8 End-Around Carry
 GROUP 03

Signed Representation

  • 03.3 Sign-Magnitude
  • 03.6 The Sign Bit Concept
  • 03.8 Negative Number Forms
  • 03.11 Signed Overflow
 GROUP 04

Arithmetic Hardware

  • 04.1 Half Adder Logic
  • 04.2 Full Adder Structure
  • 04.3 Ripple Carry Dynamics
  • 04.5 Adder-Subtractor Design
 GROUP 05-06

BCD & Excess-3

9 → 1001
10 → 1010
Ex-3 → + 0011
  • 05.2 8421 Binary-Coded Decimal
  • 05.5 Invalid BCD Codes (>9)
  • 05.7 BCD Correction (+6)
  • 06.4 Self-Complementing Codes
 GROUP 07

Gray Code

  • 07.2 Unit-Distance Property
  • 07.3 Binary-to-Gray Conversion
  • 07.5 Reflected Binary property
  • 07.7 Sensor/Encoder Applications
 GROUP 08-09

Alphanumeric Codes

  • 08.1 Weighted vs Non-Weighted
  • 09.1 Character Encoding
  • 09.2 ASCII (7-bit / 8-bit)
  • 09.7 Control Characters
 GROUP 10

Error Codes

  • 10.3 Data Redundancy
  • 10.5 Even vs 10.6 Odd Parity
  • 10.8 Block Parity Checks
  • 10.10 Hamming Code (Correction)
 GROUP 11

Code Conversion

  • 11.1 Binary ↔ BCD Logic
  • 11.2 Binary ↔ Gray Conversion
  • 11.4 Decimal ↔ Excess-3
  • 11.5 Code-Translation Logic
 GROUP 12-13

Numerical Limits

  • 12.1 Fixed-Width Arithmetic
  • 12.3 Sign Extension
  • 13.2 Signed Range Boundaries
  • 13.3 n-Bit Representation Limits
 GROUP 14-17

Errors & Applications

  • 14.4 Overflow Detection
  • 14.5 Invalid BCD Handling
  • 15 Displays & Rotary Encoders
  • 17 Code-Based System Design
 FINAL PROTOCOL

18-19 — Representation & Error Lab

A - B = A + (~B + 1)
Range = -(2ⁿ⁻¹) to (2ⁿ⁻¹)-1
P_even = b₀ ⊕ b₁ ⊕ b₂ ⊕ b₃
[1] COMPLEMENT CONVERSION [2] SIGNED INTERPRETATION [3] BCD CORRECTION [4] PARITY CHECK [5] OVERFLOW VERIFICATION

Arithmetic hardware is blind; it adds bits without knowing what they mean. The engineer must manage the representation convention—knowing when a '1' is a magnitude, a negative sign, an invalid BCD state, or a parity flag indicating a corrupted transmission.

SIMULATION RUNNING