What Is 1's complement

Content on WhatAnswers is provided "as is" for informational purposes. While we strive for accuracy, we make no guarantees. Content is AI-assisted and should not be used as professional advice.

Last updated: April 11, 2026

Quick Answer: 1's complement is a binary representation method for signed integers where negative numbers are obtained by inverting all bits of the positive number, creating a system with both positive and negative zero representations that was used in early computers but largely replaced by two's complement by the 1980s.

Key Facts

1's complement inverts all bits to represent negative numbers: 0s become 1s and 1s become 0s
An 8-bit 1's complement number can represent values from -127 to +127, with a range of -(2^7 - 1) to 2^7 - 1
1's complement creates two representations of zero (00000000 and 11111111 in 8-bit systems), causing computational inefficiency
Early computers like the IBM 1130 and UNIVAC used 1's complement throughout the 1950s-1970s
Two's complement became the universal industry standard because it eliminates dual zero and simplifies arithmetic operations

Overview

1's complement (pronounced "ones complement") is a binary representation method used in computing to represent signed integers—both positive and negative numbers. It was one of the earliest approaches to handling negative numbers in computer systems, developed during the pioneering era of computing in the 1950s and 1960s when computer scientists were exploring different methods for arithmetic operations.

In 1's complement representation, negative numbers are created by inverting all bits of their positive counterpart. For example, if positive 5 is represented as 00000101 in an 8-bit system, negative 5 would be 11111010. This simple inversion operation was appealing to early hardware designers, but it proved to have significant limitations. Despite its historical importance and initial adoption in mainframe computers like the IBM 1130 and certain UNIVAC systems, 1's complement is rarely used in modern computing, having been superseded by two's complement.

How It Works

1's complement operates through a simple but distinctive mechanism that relies on bit inversion to represent negative values:

Bit Inversion Process: To find the 1's complement of a binary number, every bit is flipped—every 0 becomes a 1, and every 1 becomes a 0. This operation is also called the bitwise NOT operation or logical complement, and it can be performed quickly in hardware using simple logic gates.
Sign Bit Function: The leftmost bit (most significant bit) serves as the sign bit in 1's complement representation. A 0 in this position indicates a positive number, while a 1 indicates a negative number, allowing representation of both positive and negative values.
Range of Representable Values: For an n-bit number, 1's complement can represent values from -(2^(n-1) - 1) to 2^(n-1) - 1. With 8 bits, the range is -127 to +127; with 16 bits, it ranges from -32,767 to +32,767.
Dual Zero Representation: A unique and problematic characteristic of 1's complement is that it has two ways to represent zero: positive zero (00000000) and negative zero (11111111) in 8-bit systems. This redundancy creates computational challenges and inefficient use of available bit space.
Arithmetic Operations: Addition and subtraction in 1's complement require special handling with an end-around carry mechanism where overflow carries are added back to the least significant bit, complicating the arithmetic logic unit design.

Key Comparisons

1's complement differs significantly from two's complement, the system that became the industry standard for modern computing:

Feature	1's Complement	Two's Complement
Method of Negation	Flip all bits to get negative	Flip all bits, then add 1
Range for 8-bit	-127 to +127	-128 to +127
Zero Representation	Two representations (00000000 and 11111111)	Single unambiguous zero (00000000)
Arithmetic Simplicity	Complex, requires end-around carry	Straightforward binary addition
Hardware Efficiency	Less efficient, requires more gates	More efficient, simpler circuitry
Current Status	Obsolete, not used in practice	Universal standard for all modern systems

Why It Matters

Historical Significance: 1's complement represents a crucial chapter in computing history, demonstrating how early engineers solved the challenge of representing negative numbers in binary before discovering better solutions that became industry standards.
Educational Value: Learning about 1's complement helps computer science students understand the evolution of number systems and the engineering trade-offs between different binary representations used in computer architecture and processor design.
Legacy System Knowledge: Some older computer systems, specialized embedded applications, and mainframe systems may still use or reference 1's complement, making it important for engineers maintaining or updating legacy code and systems.
Technical Understanding: By studying 1's complement alongside two's complement, developers gain deeper insights into how modern processors handle signed integer arithmetic and why certain architectural decisions were adopted industry-wide.

The transition from 1's complement to two's complement in the 1970s and 1980s was driven by practical necessity and efficiency gains. Engineers recognized that two's complement offered significant advantages: more efficient arithmetic operations without special carry logic, elimination of the problematic dual zero representation, better utilization of available bit space, and simpler hardware implementation requiring fewer logic gates. This standardization proved transformative for computing, reducing arithmetic logic unit complexity and enabling faster computation across all systems. Today, virtually all processors, programming languages, and digital systems worldwide use two's complement for signed integer representation, making 1's complement primarily a subject of historical and educational interest rather than practical application in contemporary software development.