Negative numbers with Two’s Complement

Two’s complement is a method of representing negative numbers in a binary system, which is widely used in computer systems. In this article, we will discuss the basics of two’s complement and provide examples to help you understand how it works.

The most common way to represent negative numbers in a binary system is by using the two’s complement method. This method uses the leftmost bit of a binary number to represent the sign of the number. If the leftmost bit is 0, the number is positive, and if the leftmost bit is 1, the number is negative.

To find the two’s complement of a number, we first invert all the bits of the number (change 0s to 1s and 1s to 0s) and then add 1 to the resulting number. For example, let’s say we want to find the two’s complement of the binary number 0110. First, we invert all the bits: 1001. Then we add 1: 1010. So the two’s complement of 0110 is 1010.

To convert a two’s complement number back to its original value, we can again invert all the bits and add 1. For example, let’s say we have the two’s complement number 1010. First, we invert the bits: 0101. Then we add 1: 0110. So the original number is 0110.

Another way to represent a negative number in two’s complement is to consider the most significant bit (MSB) as the sign bit, if the MSB is 1 the number is negative, if its 0 it’s positive. To find the two’s complement of a negative number, we flip all the bits and add 1. For example, the binary representation of -5 is 1011, to find the two’s complement we flip the bits to 0100 and add 1, 0101.

Let’s say we want to convert the two’s complement binary number 1101 (which represents -3) to its decimal equivalent.

1. We start by identifying the sign bit, which is the leftmost bit. In this case, the sign bit is 1, so we know the number is negative.
2. Next, we invert all the bits, changing 1s to 0s and 0s to 1s. This gives us 0010.
3. We then add 1 to the inverted number, resulting in 0011.
4. Now, to find the decimal equivalent of the two’s complement number, we use the following formula: Decimal = -(2^(n-1) – (two’s complement number in decimal)) Where n is the number of bits in the binary number
5. Applying this formula, we get: Decimal = -(2^(4-1) – (0011 in decimal)) Decimal = -(2^3 – 3) Decimal = -(8 – 3) Decimal = -5

So the two’s complement binary number 1101 is equivalent to the decimal value of -5.

Two’s complement is widely used in computer systems because it simplifies the process of performing arithmetic operations, such as addition and subtraction. When adding or subtracting two’s complement numbers, the process is the same as for positive numbers, and the result is the correct representation of the final number.