Computer Number Systems

Fundamentals of Number Systems

• Definition: A mathematical notation for representing quantities using a consistent set of symbols (digits) and rules.

• Positional System: The value of each digit depends on both the digit itself and its position (place) within the number. Each position corresponds to a power of the system's Base (Radix).

• Non-Positional System: Each symbol carries a fixed absolute value regardless of its position. Total value is determined by adding the symbol values.

• Bit: A single binary digit ($0$ or $1$), which is the fundamental unit of storage in digital computers.

Non-Positional Systems and Roman Numerals

• General Characteristics: No concept of zero as a placeholder; extremely cumbersome for arithmetic and computation.

• Roman Numerals Symbols: $I = 1$, $V = 5$, $X = 10$, $L = 50$, $C = 100$, $D = 500$, $M = 1000$.

• Subtractive Notation: Smaller values placed before larger values are subtracted (e.g., $IV = 4$, $XL = 40$, $CD = 400$, $CM = 900$).

• Limitations: Large numbers are unwieldy (e.g., $3999 = MMMCMXCIX$) and fractions are impossible to represent easily.

The Positional Formula

• General Expansion: $\text{Value} = d_n \times r^n + d_{n-1} \times r^{n-1} + \dots + d_1 \times r^1 + d_0 \times r^0$

• Fractional Expansion: $\dots + d_{-1} \times r^{-1} + d_{-2} \times r^{-2} + \dots$

• Variables: $r$ represents the base (radix) and $d$ represents the digit at a specific position.

Computer Number Bases

• Decimal (Base 10): Uses digits $0$-$9$. It serves as the universal reference point for human-computer interaction.

• Binary (Base 2): Fundamental computer language representing high ($1$) and low ($0$) voltage states.

• Octal (Base 8): Uses digits $0$-$7$. Historically used as shorthand in Unix/Linux systems. $1\text{ octal digit} = 3\text{ binary bits}$.

• Hexadecimal (Base 16): Uses digits $0$-$9$ and letters $A$-$F$ ($A=10, B=11, C=12, D=13, E=14, F=15$). Used for memory addresses, color codes ($\#FF8C00$), and MAC addresses. $1\text{ hex digit} = 4\text{ bits (a nibble)}$.

Conversion Methodologies

• To Decimal: Multiply each digit by its positional weight ($\text{base}^{\text{position}}$) and sum the results.

• Decimal to Integer: Successive division by the target base; read remainders from BOTTOM to TOP.

• Decimal to Fraction: Successive multiplication by the target base; read integer carries from TOP to BOTTOM.

• Binary Shortcuts:

  • To Octal: Group bits in threes from the right.

  • To Hex: Group bits in fours from the right.

• Key Limits:

  • $1\,\text{byte} = 8\,\text{bits}$. Max value: $11111111_2 = FF_{16} = 377_8 = 255_{10}$.

  • $2\,\text{bytes} = 16\,\text{bits}$. Max value: $FFFF_{16} = 65535_{10}$.

Questions & Discussion

• Q: Convert Roman Numeral $MCMXCIV$ to Decimal.

• A: $1000 + (1000 - 100) + (100 - 10) + (5 - 1) = 1994_{10}$.

• Q: What is the binary equivalent of $200_{10}$?

• A: Successive division yields $11001000_2$.

• Q: Convert $3F8_{16}$ to decimal.

• A: $(3 \times 256) + (15 \times 16) + (8 \times 1) = 1016_{10}$.

• Q: What is the octal equivalent of binary $10110111_2$?

• A: Grouped as $010|110|111 = 2|6|7$, resulting in $267_8$.

• Q: Convert the pixel color $\#FF8C00$ to decimal components.

• A: Red: $FF = 255_{10}$; Green: $8C = 140_{10}$; Blue: $00 = 0_{10}$.