Number Bases

Historical Number Systems

• Aramaic numerals were used in Egypt around 500 BCE.

• Roman numerals were used during the Roman Empire and are still used for stylized dates.

• Arabic numerals (0-9) were developed by Indian mathematicians around 500 CE and introduced to Europe by Arabic scholars.

Roman Numerals Rules

• Symbols can be repeated up to three times in a row.

• A smaller symbol before a larger one means subtraction (e.g., IV = 4).

• Only one smaller symbol can be subtracted from a larger one.

• V (5), L (50), and D (500) are never subtracted.

Place Value Systems

• Roman numerals lack zero and place value, making arithmetic difficult.

• Positional notation (place value) uses powers of a base for easier arithmetic.

• Example: 3,241.98=3 × 10³ + 2 × 10² + 4 × 10¹ + 1 × 10⁰

Binary (Base 2) Numbers

• Uses only digits 0 and 1.

• Binary numbers follow the same place value rules as decimals.

• Example: 1102 = 1 × 2² + 1 × 2¹ + 0 × 2⁰ = 6

Computer Number Representation (8-bit)

• Bit = 1 (on) or 0 (off).

• Most significant bit (MSB) is the leftmost; least significant bit (LSB) is the rightmost.

• Example: 10100110₂ = 166 in decimal.

Conversion from Base 10 to Base 2

• Find the largest power of 2 less than or equal to the number.

• Subtract and repeat for the remainder until zero.

• Example: 93₁₀ = 1011101₂

Binary Integer Arithmetic

• Addition and subtraction follow place value rules.

• Carry bit is discarded if it exceeds word length

Hexadecimal (Base 16) Numbers

• Uses digits 0-9 and letters A-F (A=10, F=15).

• Example: A2E₁₆ = 10 × 16² + 2 × 16¹ + 14 × 16⁰ = 2606

Signed Binary Numbers

• n-bit word can represent integers in range [−2ⁿ⁻¹, 2ⁿ⁻¹ , -1]

• Sign magnitude representation: MSB indicates sign; remaining bits for magnitude.

• One's complement: flip all bits for negative numbers.

• Two's complement: flip bits and add 1 for negative numbers (most common).

Two's Complement Arithmetic

• Addition is normal; carry bit is discarded.

• Overflow detected if carry into sign bit differs from carry out.

• Two's complement negation is invert bits plus one.

Fixed Point Numbers (m.n notation)

• Fixed decimal (radix) point with mm integer and nn fractional bits.

• Range: [−2^m−1, 2^m−1 , -2^-n]

• Precision increases with more fractional bits.

• Represents numbers as: mantissa × base^exponent

• 32-bit (single precision): 1 sign bit, 8 exponent bits, 23 mantissa bits.

• Relative error ≈ 6 ×10⁻⁸⁶

• Range approx ±3.4×10³⁸