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How to Calculate Binary of Negative Number

Reviewed by Calculator Editorial Team

Negative numbers in binary are represented using the two's complement method, which is the standard way computers store signed integers. This guide explains how to calculate the binary representation of negative numbers, including the formula, step-by-step process, and practical examples.

What is Negative Binary?

In binary, numbers are represented using only 0s and 1s. However, representing negative numbers requires a special method because binary is inherently positive. The most common method is two's complement, which is used in most modern computing systems.

The two's complement method allows a binary number to represent both positive and negative values within the same number of bits. This is achieved by using the most significant bit (MSB) as a sign bit, where 0 represents positive and 1 represents negative.

Two's Complement Method

The two's complement method involves three main steps to convert a negative decimal number to binary:

  1. Convert the absolute value of the number to binary.
  2. Invert all the bits (change 0s to 1s and 1s to 0s).
  3. Add 1 to the inverted binary number.

Formula: Two's complement of a negative number N is calculated as:

1. Binary of |N| (absolute value of N)

2. Invert all bits

3. Add 1 to the inverted binary

This process ensures that the binary representation of a negative number can be correctly interpreted by computers.

How to Calculate Negative Binary

Step-by-Step Process

  1. Determine the number of bits: Choose the number of bits you want to use for the representation. Common choices are 8, 16, or 32 bits.
  2. Convert the absolute value to binary: Convert the absolute value of your negative number to binary using the standard binary conversion method.
  3. Pad with leading zeros: If necessary, pad the binary representation with leading zeros to reach the desired number of bits.
  4. Invert all bits: Flip all the bits in the binary number (change 0s to 1s and 1s to 0s).
  5. Add 1: Add 1 to the inverted binary number. This is done by performing binary addition.

Note: The two's complement method works best when the number of bits is fixed. For example, an 8-bit representation can handle numbers from -128 to 127.

Examples

Let's look at a few examples to understand how to calculate the binary representation of negative numbers.

Example 1: Convert -5 to 8-bit binary

  1. Absolute value: 5
  2. Binary of 5: 00000101 (8-bit)
  3. Invert bits: 11111010
  4. Add 1: 11111011

The 8-bit two's complement representation of -5 is 11111011.

Example 2: Convert -10 to 8-bit binary

  1. Absolute value: 10
  2. Binary of 10: 00001010 (8-bit)
  3. Invert bits: 11110101
  4. Add 1: 11110110

The 8-bit two's complement representation of -10 is 11110110.

Decimal Number 8-bit Binary Two's Complement
-1 00000001 11111111
-2 00000010 11111110
-3 00000011 11111101
-4 00000100 11111100
-5 00000101 11111011

FAQ

What is the difference between one's complement and two's complement?
One's complement inverts all bits of the positive binary number, while two's complement inverts all bits and then adds 1. Two's complement is more commonly used because it has a unique representation for zero and simplifies arithmetic operations.
Can two's complement represent zero?
Yes, two's complement represents zero as all zeros. This is different from one's complement, where zero can be represented in two ways (all zeros and all ones).
What is the range of numbers that can be represented with 8 bits using two's complement?
With 8 bits, the range is from -128 to 127. The most significant bit is the sign bit, and the remaining 7 bits represent the magnitude.
How do I convert a negative binary number back to decimal?
To convert a negative binary number in two's complement back to decimal, you can invert the bits, add 1, and then convert the result to decimal with a negative sign.