Unsigned and Signed Numbers Representation in Binary Number System
Last Updated :
08 Apr, 2025
The binary number system uses only two digits, 0 and 1, to represent all data in computing and digital electronics. Understanding unsigned and signed numbers is important for efficient data handling and accurate computations in these fields.
- The binary system forms the foundation of all digital systems, enabling devices to process and store data.
- Unsigned numbers represent only positive values while signed numbers handle both positive and negative values using methods like two’s complement.
- Mastering these concepts is essential for programming, error-free calculations and optimizing system performance.
- Its real-world applications include computer arithmetic, embedded systems and digital signal processing.
Unsigned Numbers
Unsigned numbers are numeric values that represent only non-negative quantities (zero and positive values). In the binary system, unsigned numbers are represented using only the magnitude of the value, with each bit contributing to its total. The smallest unsigned number is always zero (all bits set to 0), while the maximum value depends on the number of bits used.
Range of Unsigned Numbers in Binary System:
The range depends directly on the bit length. For an n-bit number:
- Minimum value: 0
- Maximum value: 2n - 1
Range of unsigned numbers: 0 to 2n-1
Example:
- 4-bit Number: Ranges 0 to 15.
- 8-bit Number: Ranges 0 to 255.
- 16-bit Number: Ranges 0 to 65,535.
Use Cases of Unsigned Binary Representation:
Unsigned binary numbers serve fundamental roles across computing systems where negative values aren't needed.
- Every byte in RAM gets a unique unsigned address. A 32-bit system can access 4GB of memory (0 to 4,294,967,295).
- RGB color values (0-255) use 8-bit unsigned numbers per channel. Image dimensions and pixel counts also rely on unsigned ranges.
- Microprocessors use unsigned values for status flags and control signals where only positive states exist.
- Packet sizes, port numbers and IP header fields often use unsigned integers to prevent negative interpretations.
- Sensor readings (like temperature or light intensity) and timer counts utilize unsigned formats when negative measurements are impossible.
Signed Numbers
Signed numbers represent both positive and negative values in computing by allocating one bit (typically the MSB) as a sign indicator. In binary representation, they enable negative value storage through three primary methods:
The 2's complement method dominates modern systems due to its arithmetic efficiency and absence of redundant zero representations.
Sign-Magnitude Representation
This method uses the leftmost bit as a sign flag (0 for positive, 1 for negative) while remaining bits store the absolute value. It creates two zero representations (+0 and -0) and complicates arithmetic operations due to separate sign handling.
Range of Sign-Magnitude Representation
-2n-1 - 1 to 2n-1 - 1
Example: In 8-bit system
- Decimal number +25 is represented as 00011001 (Sign bit 0, magnitude 25)
- Decimal number -25 is represented as 10011001 (Sign bit 1, magnitude 25)
Sign - Magnitude RepresentationWhy Sign-Magnitude Fails in Practice
- Wastes a representable value by encoding both +0 (0000) and -0 (1000), creating unnecessary complexity in comparisons and arithmetic.
- Basic calculations fail because the sign bit requires separate handling. Adding positive and negative versions of the same number doesn't produce zero.
- Requires extra circuitry to manage sign bits during calculations, slowing down processors compared to complement systems.
- Effectively loses one bit of precision since the sign bit doesn't contribute to magnitude, unlike modern systems that use all bits for value representation.
1's Complement Representation
In 1's complement, negative numbers are created by flipping all bits of the positive counterpart. The leftmost bit still indicates sign (0 for positive, 1 for negative). This method also suffers from dual zero representations (0000 as +0 and 1111 as -0) but it simplifies subtraction through bit inversion.
Range of 1's Complement Representation
-2n-1 - 1 to 2n-1 - 1
Example: In 8-bit system
- Decimal number +18 is represented as 00010010
- Decimal number -18 is represented as 11101101 (All bits flipped)
Critical Flaws of 1's Complement
- Maintains two zero forms (0000 and 1111) wasting a valid number representation and complicating equality checks.
- Forces an extra addition step when carry bits overflow making arithmetic slower than 2's complement.
- Basic math operations need sign-bit checks creating hardware complexity modern systems avoid.
- While better than sign-magnitude its quirks made 2's complement the universal standard for binary arithmetic.
2's Complement Representation
2's complement forms negative numbers by inverting all bits of the positive value and adding 1. The leftmost bit serves as the sign indicator while enabling single zero representation. It simplifies hardware design by allowing identical addition and subtraction operations.
2's ComplementRange of 2's Complement Representation
-2n-1 to 2n-1 - 1
Example: In 8-bit system
- Decimal number +20 is represented as 00010100
- Decimal number -20 is represented as 11101100 (Invert bits: 11101011, then add 1)
Limitations of 2's Complement
- In a fixed-bit system, 2's complement has a limited range. For an n-bit system, it represents values from -2(n-1) to 2(n-1) - 1, restricting the number of values that can be processed.
- The range for positive numbers is smaller than negative numbers. Negative values range from -2(n-1) to -1, while positive values range from 0 to 2(n-1) - 1, creating an imbalance.
- Overflow occurs when arithmetic operations exceed the representable range, leading to incorrect results.
- Operations like addition and subtraction become complex with sign extension, especially for numbers of different sizes.
- The zero in 2's complement has only one form, while negative numbers have a mirrored representation (e.g., -1 is 111...1), complicating algorithms like multiplication.
Note:
Key Differences Between Unsigned and Signed Number Representations
Feature | Unsigned Numbers | Sign-Magnitude | 1's Complement | 2's Complement |
|---|
Representation | Only positive values | Sign bit + absolute value | Bit inversion of positives | Bit inversion + 1 |
|---|
Zero Values | Single zero (00...0) | Two zeros (+0/-0) | Two zeros (+0/-0) | Single zero (00...0) |
|---|
Range (8-bit) | 0 to 255 | -127 to +127 | -127 to +127 | -128 to +127 |
|---|
Hardware Complexity | Simplest | Moderate (sign handling) | Moderate (end-around carry) | Most efficient |
|---|
Arithmetic Operations | Straightforward | Requires sign checks | Needs carry adjustment | Uniform handling |
|---|
Modern Usage | Common for counters/addresses | Rarely used | Obsolete | Industry standard |
|---|
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