Question

a. Suppose you have the following 2 bytes: 01011100 and 01100101 . What is the 1 s complement of the sum of these 2 bytes? b. Suppose you have the following 2 bytes: 11011010 and 01100101 . What is the \(1 \mathrm{~s}\) complement of the sum of these 2 bytes? c. For the bytes in part (a), give an example where one bit is flipped in each of the 2 bytes and yet the 1 s complement doesn't change.

Short answer

a. 00111110 b. 00000000 c. Flip to 01011101 & 01100100, 1's complement unchanged: 00111110

Step-by-step solution

Convert Binary to Decimal

First, let's convert each byte from binary to decimal for easier addition. For part (a):- For 01011100, calculate: \(0 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 92\)- For 01100101, calculate: \(0 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 101\)For part (b):- For 11011010, calculate: \(1 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 218\)

Add the Decimal Values

Add the decimal values obtained from the binary conversion.For part (a), add 92 and 101:\[92 + 101 = 193\]For part (b), add 218 and 101:\[218 + 101 = 319\]

Convert Sum to Binary

Convert the decimal sum back to binary, as we're looking for the 1's complement which is originally binary. For part (a): - 193 in binary is 11000001. For part (b): - Since 319 exceeds 8 bits, convert to binary: - 319 in binary is 00000001 00111111 (splitting it into two bytes because it's more than one byte as 8 bits can only handle up to 255 in decimal) - Consider only the lower 8 bits which is 11111111.

Calculate 1's Complement

To find the 1's complement, convert each binary digit: 0 becomes 1, and 1 becomes 0. For part (a): - Starting with 11000001, the 1's complement is 00111110. For part (b): - After considering only the lower 8 bits: 11111111 inverts to 00000000.

Flipping Bits

Now, for part (c), we need to find an example where one bit is flipped in each byte, but the sum's 1's complement remains the same: - Original bytes were 01011100 and 01100101, sum was 193 with 1's complement 00111110. - If we flip the least significant bits: - Change 01011100 to 01011101. - Change 01100101 to 01100100. - New sum becomes: 01011101 (93) + 01100100 (100) = 193. - 1's complement is still 00111110.

Key concepts

1's Complement

The concept of 1's complement is a crucial idea in computer networking and binary arithmetic. It is a method used to represent negative numbers in binary systems. In the 1's complement system, each bit of a binary number is inverted. This means every 0 becomes a 1 and every 1 becomes a 0. This operation is simple yet powerful, providing an easy way to perform binary subtraction using addition.

When calculating the 1's complement of a binary number, just flip each bit. For instance, the binary number 1100 would have a 1's complement of 0011. This method is not just limited to calculations of negative numbers but is also widely adopted in error detection and data manipulation.

To understand how 1's complement affects operations, let's look at a simple example. Say we have two binary numbers to add: 1010 and 1110. After addition, inverting all bits of the result yields the 1's complement, allowing us to identify any potential overflow or provide a basis for subtraction.

Binary Arithmetic

Binary arithmetic is fundamental in computer systems, manipulating data at the most basic level. Unlike decimal arithmetic, which is based on ten digits (0-9), binary arithmetic uses only two: 0 and 1. This might seem limited, but it's ideal for computers that use binary logic.

In binary addition, you follow simple rules similar to those used in decimal addition. When two bits are added, a sum and a carry may result as follows:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0 (with carry 1)

These rules enable us to perform operations such as addition, subtraction, multiplication, and division on binary numbers efficiently.

Suppose we have two bytes, like 01011100 and 01100101. To add them, align the bits and perform addition using the described rules. Converting these into decimal helps us understand their sums, namely 92 and 101 respectively, leading to an overall sum of 193. This conversion back and forth between binary and decimal facilitates easier manipulation, especially in systems that use binary logic for arithmetic operations.

Bit Manipulation

Bit manipulation is a technique employed to alter or evaluate bits' values directly within binary numbers. It gains importance in a myriad of applications, ranging from low-level hardware programming to high-level algorithm optimization.

Common operations in bit manipulation include actions like flipping (changing 1 to 0 or 0 to 1), setting (changing to 1), clearing (changing to 0), and shifting (moving bits left or right). Each of these operations allows programmers to control and optimize how data is processed.

In our exercise, consider flipping bits in given bytes, like altering 01011100 to 01011101. This involves changing a specific bit to achieve the desired adjustment. Bit manipulation is crucial when checking bit positions, setting configurations, or optimizing arithmetic operations. Through bit flipping, we can change the least significant bits yet achieve the same 1's complement result. This demonstrates the precision and control that bit manipulation offers in binary arithmetic and computer networking.