How to Convert Negative Decimals to Binary: The Two's Complement Method

When dealing with negative decimal numbers, converting them to binary requires a specific technique to ensure correct representation in digital systems.

Introduction to Two's Complement

Two's complement is a method used to represent signed numbers in binary form. It is particularly useful in computer systems where negative values need to be handled efficiently. This guide provides a step-by-step process to convert negative decimal numbers into binary using the two's complement method.

Steps to Convert a Negative Decimal to Binary

Step 1: Convert the Absolute Value to Binary

The first step is to convert the positive version of the decimal number into binary.

Step 2: Find the Twos Complement

This is the actual conversion process that involves several sub-steps.

Invert the Digits

Changes all the 0s to 1s and all the 1s to 0s, also known as the ones' complement.

Add One

Add 1 to the least significant bit (LSB) of the inverted binary number to get the final two's complement.

Example: Convert -5 to Binary

Step 1: Convert the Absolute Value 5 to Binary

The binary representation of 5 is 0101 (assuming 4-bit representation).

Step 2: Find the Twos Complement

- Invert the Digits: 0101 becomes 1010.

- Add One: 1010 0001 1011.

So, the twos complement representation of -5 in a 4-bit binary system is 1011.

Important Notes

Bit Length: The bit length you choose, e.g., 8-bit, 16-bit, etc., affects the representation. Ensure you use enough bits to accommodate the range of values needed.

Overflow: If the number is too large to be represented in the chosen bit length, it will lead to an overflow, resulting in incorrect values.

Understanding Different Numerical Representations

It is important to note that while the two's complement method is the most common, there are other methods for representing numeric values, especially in the context of computer hardware. For example:

Sign and Magnitude

In this system, the leftmost bit represents the sign (0 for positive, 1 for negative), and the remaining bits represent the magnitude. For example, 7 in 4-bit sign and magnitude representation is 0111, and -7 is 1111.

One's Complement

Similar to sign and magnitude, but the negative numbers are represented by inverting all bits. For example, 7 in 4-bit one's complement representation is 0111, but -7 would be 1000.

Bias or Excess-N Notation

In this system, a bias value is added to the binary number. For example, a 4-bit biased representation might add 8 to the number. Thus, 7 would be 1111, and -7 would be 0001.

Mathematical vs. Computer Representation

It is crucial to distinguish between mathematical and computer representations. In mathematics, a number without a negative sign is positive. However, in computer systems, the lack of a negative sign might require the use of specific numerical systems such as two's complement.

Conclusion

In summary, to convert a negative decimal to binary:

Convert the positive value to binary. Invert the digits (ones' complement). Add one to get the twos' complement.

By following these steps, you can accurately represent negative decimal numbers in binary, ensuring correct handling in digital systems.