Convert -52 (base 10) to binary 8-bit two’s complement.
The correct answer and explanation is :
To convert -52 in decimal (base 10) to binary in 8-bit two’s complement form, we follow these steps:
Step 1: Convert the absolute value to binary
Start by converting the positive value of 52 into binary.
- First, divide 52 by 2 and write down the remainder for each division:
- 52 ÷ 2 = 26 remainder 0
- 26 ÷ 2 = 13 remainder 0
- 13 ÷ 2 = 6 remainder 1
- 6 ÷ 2 = 3 remainder 0
- 3 ÷ 2 = 1 remainder 1
- 1 ÷ 2 = 0 remainder 1
Reading the remainders from bottom to top, the binary representation of 52 is 110100.
Since we need 8 bits, we pad the result with leading zeros to make it 8 bits:
00110100
Step 2: Find the two’s complement
To represent a negative number in two’s complement, we:
- Invert all the bits (flip 1s to 0s and 0s to 1s).
- Add 1 to the inverted binary number.
Inverting the bits:
The binary representation of +52 is: 00110100.
Inverting each bit gives: 11001011.
Adding 1:
Now, add 1 to the inverted binary:
- 11001011
- 1
= 11001100
So, the 8-bit two’s complement representation of -52 is 11001100.
Explanation of Two’s Complement:
In two’s complement notation, positive numbers are represented as their regular binary equivalents, while negative numbers are represented by taking the two’s complement of their positive binary counterpart. The two’s complement method is widely used in digital systems because it simplifies arithmetic operations, particularly subtraction. It allows for a single circuit to handle both addition and subtraction by treating negative numbers as their complements. When adding a number and its two’s complement, the result is always zero, which is essential for binary arithmetic in computer systems.
Thus, the 8-bit two’s complement representation of -52 is 11001100.