What is the Signed 1’s Complement Form?
The signed 1’s complement form is a method for representing signed numbers in binary, where the sign and magnitude are encoded differently from unsigned numbers. In this system, positive numbers are represented as their standard binary equivalents, with the most significant bit (MSB) set to 0. Negative numbers are formed by taking the bitwise or 1’s complement (inverting all bits) of the positive number’s binary representation, resulting in an MSB of 1. For an n-bit number in 1’s complement form, one bit indicates the sign, and the remaining n-1 bits represent the magnitude or its complement. This approach simplifies certain arithmetic operations but introduces complexities compared to other representations.
Representing Numbers in 4-Bit 1’s Complement Form
In a 4-bit 1’s complement representation, one bit is reserved for the sign, leaving three bits for the magnitude. This allows numbers from \(\mathsf{-7}\) to \(\mathsf{+7}\) to be represented. Positive numbers are written directly in binary with a \(\mathsf{0}\) sign bit (e.g., \(\mathsf{+5}\) is \(\mathsf{0101}\)). To represent a negative number, take the positive number’s binary form, invert all bits, and keep the MSB as \(\mathsf{1}\) (e.g., \(\mathsf{-5}\) is the complement of \(\mathsf{0101}\), which is \(\mathsf{1010}\)). A key drawback is that, like signed magnitude, 1’s complement has two representations for zero: \(\mathsf{+0 (0000)}\) and \(\mathsf{-0 (1111)}\). This dual zero representation can complicate arithmetic and programming, making 1’s complement less common in modern systems.
Range of Numbers Stored in n-Bits
The range of numbers in 1’s complement form for n bits is determined by the n-1 bits available for the magnitude. The range spans from \(\mathsf{-(2^{n-1} – 1)}\) to \(\mathsf{+(2^{n-1} – 1)}\). For example:
- For \(\mathsf{4}\) bits (\(\mathsf{n=4}\)), the range is \(\mathsf{-7}\) to \(\mathsf{+7}\).
- For \(\mathsf{8}\) bits (\(\mathsf{n=8}\)), the range is \(\mathsf{-127}\) to \(\mathsf{+127}\).
This range is identical to signed magnitude but differs from 2’s complement, which supports one additional negative number. A comparison of ranges across different signed number representations is available in the referenced figure.
Advantages and Limitations of 1’s Complement Form
Advantages
- Simplified Negation: Converting a positive number to its negative equivalent is straightforward, requiring only a bitwise inversion.
- Symmetric Range: The range for positive and negative numbers is balanced, making it intuitive for certain applications.
Limitations
- Dual Zero Representation: The existence of \(\mathsf{+0}\) and \(\mathsf{-0}\) creates ambiguity, complicating arithmetic operations and software design.
- End-Around Carry: Addition in 1’s complement requires handling an end-around carry, where a carry-out from the MSB is added back to the least significant bit, adding complexity.
- Limited Modern Use: Due to its inefficiencies and the superior handling of arithmetic in 2’s complement, 1’s complement is rarely used in contemporary computing.
Why Understand 1’s Complement?
While 1’s complement is less prevalent in modern digital systems, studying it provides valuable insight into the evolution of binary number representations. It serves as a bridge to understanding more advanced systems like 2’s complement, which is the standard in today’s computing. Exploring 1’s complement enhances your understanding of how signed numbers are managed in binary arithmetic, a critical concept in computer science and digital electronics. For further details, visit the related pages:
- Signed Magnitude Form: Learn about the simplest signed number representation.
- Signed 2’s Complement Form: Understand the modern standard for binary arithmetic.
- Addition and Subtraction in 2’s Complement: Explore arithmetic operations in detail.
- Overflow in Signed Arithmetic: Dive into overflow detection and prevention.
By mastering the 1’s complement form, you’ll gain a deeper appreciation for the design choices in binary number systems and their impact on computational efficiency.