Twos complement: Negative numbers in binary

Introduction

This tutorial explains how to represent negative numbers in binary using the two's complement method. Understanding two's complement is crucial for working with binary arithmetic, especially in computer science and digital electronics, as it simplifies the implementation of arithmetic operations.

Step 1: Understand the Basics of Binary Representation

Before diving into two's complement, it’s essential to grasp how binary numbers work.

• Binary System: Uses only two digits, 0 and 1.
• Positive Numbers: Represented the same way as in decimal, with increasing powers of 2.

For example, the binary number 1011 represents:

• 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰ = 8 + 0 + 2 + 1 = 11

Step 2: Explore the Need for Negative Numbers

In computing, we need to represent negative numbers alongside positive ones. Here are the common techniques:

1. Sign Bit: Uses one bit to indicate the sign (0 for positive, 1 for negative), which is inefficient for arithmetic.
2. Ones Complement: Inverts all bits to represent negative numbers, but it has issues like two representations for zero.

Step 3: Learn About Two's Complement

Two's complement is the most popular method for representing negative numbers in binary, offering a straightforward approach to arithmetic.

How to Calculate Two's Complement

1. Start with a Positive Binary Number: For example, consider 5, which is 0101 in binary.
2. Invert the Digits: Change all 0s to 1s and all 1s to 0s.
   • 0101 becomes 1010.
3. Add 1 to the Inverted Number:
   • 1010
   • + 0001
   • Result: 1011

The two's complement of 5 is 1011, which represents -5.

Step 4: Verify the Two's Complement

To ensure the correctness of two's complement representation:

• Convert Back to Decimal: If you have a two's complement binary number, like 1011:
  1. Invert the digits: 0100
  2. Add 1:
     • 0100
     • + 0001
     • Result: 0101 (which is 5)
  3. Conclude that 1011 represents -5.

Step 5: Practical Applications of Two's Complement

Two's complement allows for seamless addition and subtraction of binary numbers, regardless of their sign.

Example of Addition

To add 5 (0101) and -5 (1011):

• Align the numbers:

  0101
+ 1011
------
  10000

• The result is 10000. The leftmost bit (carry) is disregarded, leaving 0000, which confirms that 5 + (-5) = 0.

Conclusion

Two's complement provides an efficient way to represent negative numbers in binary and simplifies arithmetic operations. Remember these key points:

• Two's complement is commonly used in computing for negative number representation.
• The process involves inverting the bits and adding one.
• It allows for straightforward addition and subtraction of signed binary numbers.

Next steps could include practicing with different numbers or exploring how two's complement is implemented in programming languages.