4.0 Data Representation - Integers

Tags: #◇Computer Science #◇CCICOMP #◇Problem Solving #◇M2

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◇Overview

Data representation involves encoding information into binary form so it can be reliably transported and processed by computer circuitry. Any numeric representation format balances size, range, accuracy, ease of manipulation, and standardization. Integers can be represented as unsigned (only non-negative values) or signed (capable of representing both positive and negative values). While various encoding schemes exist for signed integers, modern CPUs almost universally use the 2's complement scheme. Since data types use fixed-width bit strings, calculations that exceed the format's capacity result in overflow, producing mathematically illogical values.

◇Important Formulas

Unsigned Integers Value Evaluation: An $n$-bit unsigned integer $b_{n-1}\dots b_1b_0$ has the decimal value:

$\text{Value} = \sum_{i=0}^{n-1} b_i 2^i$ Range: $0$ to $2^n - 1$. For example, an $8$-bit unsigned integer ranges from $0$ to $255$. Zero Extension: To expand the width of an unsigned integer, simply pad the left side (most significant bits) with $0$s.

Signed Integers

Signed integers allocate the Most Significant Bit (MSB) as the sign bit: $0$ for positive, $1$ for negative. Sign-and-Magnitude: The MSB is the sign, and the remaining bits represent the absolute magnitude of the number in standard binary. 1's Complement: To represent a negative number, start with the positive unsigned binary value and invert every bit ($0 \to 1$, $1 \to 0$). 2's Complement: To represent a negative number, invert every bit of the positive unsigned binary value and add $1$.
Range: $-(2^{n-1})$ to $2^{n-1} - 1$. E.g., for $8$-bit: $-128$ to $127$.
Only uses one representation of $0$ (unlike 1's complement and sign-and-magnitude). Sign Extension: To expand the width of a signed integer, replicate the sign bit to the left (e.g., $1011 \to 11111011$ for negatives, $0101 \to 00000101$ for positives).

◇Worked Examples

1. Converting to 2's Complement

Represent $-25_{10}$ as an $8$-bit signed integer in 2's complement.

1. Determine absolute value in binary: $25_{10} = 00011001_2$ (with leading zeros to make $8$ bits).
2. Invert bits (1's complement): $11100110_2$.
3. Add $1$: $11100110_2 + 1 = 11100111_2$.
   Result: $11100111_2$.

2. Converting from 2's Complement

Find the decimal equivalent of the $8$-bit 2's complement number $11100111_2$.

1. Check MSB: $1$ means negative.
2. Invert bits: $00011000_2$.
3. Add $1$: $00011000_2 + 1 = 00011001_2$.
4. Convert to decimal: $00011001_2 = 25_{10}$.
5. Apply sign: $-25_{10}$.

3. Overflow Condition

Add the $8$-bit signed values $90_{10}$ and $40_{10}$. $90_{10} = 01011010_2$ $40_{10} = 00101000_2$ Addition: $01011010_2 + 00101000_2 = 10000010_2$
Notice the MSB is now $1$. In an $8$-bit signed 2's complement system, $10000010_2$ is interpreted as $-126_{10}$. Because $90 + 40 = 130$, which exceeds the maximum $8$-bit positive range of $127$, the magnitude overflowed into the sign bit, causing an illogical negative result.

◇References & Navigation

• Previous Topic: ◇Introduction to Number Systems
• Next Topic: ◇Data Representation - Real Numbers
• Related Notes: ◇Integer Representation & Fixed-Point Arithmetic, ◇Data Representation - Characters and Boolean