Comprehensive Study Guide: Number Systems, Base Conversions, Signed Arithmetic, and Binary Codes

Positional Number Systems

• Base/Radix ($r$): Evaluates digits by positional weights $r^i$: $N = \sum a_i r^i$.

  • Decimal ($r=10$): Digits $\mathbf{\{0..9\}}$.

  • Binary ($r=2$): Digits $\mathbf{\{0, 1\}}$. Positional weights: integer $\dots, 2^2=4, 2^1=2, 2^0=1$; fractional $2^{-1}=0.5, 2^{-2}=0.25, \dots$.

  • Octal ($r=8$): Digits $\mathbf{\{0..7\}}$. Each digit equals 3 binary bits.

  • Hexadecimal ($r=16$): Symbols $\mathbf{\{0..9, A..F\}}$ ($A=10$ to $F=15$). Each symbol equals 4 binary bits.

Base Conversions

• Binary to Decimal: Sum each bit multiplied by its weight $2^i$.

  • Example: $(10110001)_2 = 128 + 32 + 16 + 1 = (177)_{10}$.

• Decimal to Binary:

  • Integer: Repeatedly divide by $2$; remainders yield bits from LSB to MSB. Example: (13)_{10} = (1101)_2$.\n - **Fractional:** Repeatedly multiply the fraction by 2; integer parts yield bits.\n- **Inter-conversion (Octal/Hex/Binary):** Group binary bits into sets of 3 for Octal or 4 for Hexadecimal starting from the binary point.\n\n### Signed Number Representation\n- **Sign Bit:** MSB indicates sign (0$= positive,$1 = negative).\n- **Formats:**\n - **Sign-Magnitude:** MSB is sign; remaining bits represent magnitude.\n - **1's Complement:** Invert all bits for negative numbers (0 \rightarrow 1, 1 \rightarrow 0).\n - **2's Complement:** \text{2's Complement} = \text{1's Complement} + 1.\n - **Direct Value Evaluation:** Assign MSB negative column weight -2^{n-1}$. Example:$11000110_2 = -128 + 64 + 4 + 2 = (-58){10}.\n\n### 2's Complement Arithmetic\n- **Addition:** Add all bits including the sign bit; **discard** any carry out of MSB.\n- **Subtraction:** Convert to addition with 2's complement of subtrahend: A - B = A + (-B).\n\n### Binary Codes\n- **Binary Coded Decimal (BCD):**\n - Encodes each decimal digit (0..9$) into a 4-bit binary group ($0000_2$to$1001_2).\n - Codes 1010_2$to$1111_2$are invalid. Example:$(374){10} = (0011\ 0111\ 0100){\text{BCD}}.\n- **Gray Code (Reflected Binary Code):**\n - Unweighted code where consecutive numbers differ by only **1 bit** to prevent hardware glitches.\n - **Binary to Gray:** G_n = B_n$and$G_i = B{i+1} \oplus B_i$. Example:$10110_2 = 11101_{\text{Gray}}.\n - **Gray to Binary:** B_n = G_n$and$B_i = B_{i+1} \oplus G_i$. Example:$11011_{\text{Gray}} = 10010_2$$.