Question

Find the 2 's complement of the following binary numbers: (a) 1001010 (b) 111001 (c) 10000010 (d) 111110001

Short answer

(a) 0110110, (b) 000111, (c) 01111110, (d) 000001111

Step-by-step solution

Understand the concept of 2's complement

The 2's complement of a binary number is obtained by inverting all the bits (turning all 0s to 1s and all 1s to 0s) and then adding 1 to the least significant bit (LSB). This operation converts the number's sign in binary form.

Invert the bits of (a) 1001010

First, invert each bit of the binary number 1001010 to get 0110101.

Add 1 to the LSB of the inverted (a)

Adding 1 to the LSB of 0110101 gives us the 2's complement: 0110110.

Invert the bits of (b) 111001

First, invert each bit of the binary number 111001 to get 000110.

Add 1 to the LSB of the inverted (b)

Adding 1 to the LSB of 000110 gives us the 2's complement: 000111.

Invert the bits of (c) 10000010

First, invert each bit of the binary number 10000010 to get 01111101.

Add 1 to the LSB of the inverted (c)

Adding 1 to the LSB of 01111101 gives us the 2's complement: 01111110.

Invert the bits of (d) 111110001

First, invert each bit of the binary number 111110001 to get 000001110.

Add 1 to the LSB of the inverted (d)

Adding 1 to the LSB of 000001110 gives us the 2's complement: 000001111.

Key concepts

Understanding Binary Numbers

Binary numbers form the core of computing systems, only utilizing digits 0 and 1. Each digit within a binary number is referred to as a bit, and these bits represent successive powers of two. The farthest right bit is the least significant, holding the value of 2⁰ or 1, and as we move left, each bit represents a value that is twice the previous one. So, a binary number such as 1001010 represents the decimal number 74, calculated as 1x2⁶ + 0x2⁵ + 0x2⁴ + 1x2³ + 0x2² + 1x2¹ + 0x2⁰.

Bit Inversion Explained

Bit inversion is a straightforward process where every 0 in the binary string is replaced with a 1, and every 1 is replaced with a 0. It's crucial in calculations such as finding the 2's complement of a binary number. For instance, inverting the bits of the binary number 111001 results in 000110. This process is sometimes also referred to as a bitwise NOT operation. It's the first step in changing the sign of a binary number, preparing it for the addition that completes the 2's complement process.

Visualizing Bit Inversion:

• Original: 111001
• Inverted: 000110

The Role of the Least Significant Bit (LSB)

The LSB is the rightmost bit in a binary number and holds the lowest place value. In the context of 2's complement, after bit inversion, you add 1 to the LSB. This step is critical because it flips the sign of the number. For example, the LSB in the inverted string 0110101 is the last '1'. When you add '1' to this, you get 0110110. The addition may cause a ripple effect where subsequent bits change value if there is a carry out, but it starts with the LSB.

Critical Points to Remember About LSB:

• The LSB represents the digit with the least value in a binary number.
• Addition in binary works similarly to decimal addition, with 1+1 resulting in a carry-over.

Sign Conversion in Binary through 2's Complement

The 2's complement technique is essential for sign conversion in binary numbers, allowing computers to represent and process negative numbers. This method negates a number and is hugely significant in computer arithmetic. If a binary number starts with a '1', it is often denoted as negative in computer systems, which use the 2's complement representation for negative values. For example, the 2's complement of 111110001 is obtained by first inverting the bits to get 000001110, then adding 1 to the LSB, resulting in the negative binary equivalent 000001111.

How Sign Conversion Works:

• Positive to negative: Perform the 2's complement operation on a binary number.
• Negative to positive: Also, use 2's complement to revert back to a positive number.