Question

question 3 (10 pt). in a 6 - bit system, use two’s complement scheme, please: - represent 7₁₀ in binary (base 2) - represent -7₁₀ in binary (base 2) - by taking the two’s complement - perform the addition of these two values in binary - check the resulting sum to validate the result in decimal (base 10). specifically, convert the result to decimal to validate.

Explanation:

Step1: Convert 7 to binary

To convert a decimal number to binary, we use the division - by - 2 method.
$7\div2 = 3$ remainder $1$
$3\div2=1$ remainder $1$
$1\div2 = 0$ remainder $1$
Reading the remainders from bottom - to - top, $7_{10}=000111_2$ in a 6 - bit system.

Step2: Find two's complement of 7 to represent - 7

First, find the one's complement of 000111. Flip all the bits: 111000.
Then, add 1 to the one's complement: $111000 + 1=111001$. So, $- 7_{10}=111001_2$ in a 6 - bit two's complement system.

Step3: Perform binary addition

Add 000111 and 111001:

  000111
+ 111001
--------
 1000000

Since we are working in a 6 - bit system, we discard the left - most bit (the carry - out). So the result is 000000.

Step4: Convert the result to decimal

The binary number 000000 in decimal is $0\times2^5+0\times2^4+0\times2^3+0\times2^2+0\times2^1+0\times2^0 = 0_{10}$. And $7+( - 7)=0$, which validates our result.

Answer:

The binary representation of $7_{10}$ is 000111, the binary representation of $-7_{10}$ is 111001, the result of the binary addition is 000000, and the decimal validation shows the sum is 0.