Understanding Binary Number Representations

This problem requires us to convert a given 4-bit binary number, 1100, into its decimal equivalent using three different signed representations: signed magnitude, 1's complement, and 2's complement. We will then sum these decimal values and finally convert the result into a 6-bit 2's complement representation.

Decimal Value of P (Signed Magnitude Representation)

In signed magnitude representation, the most significant bit (MSB) indicates the sign of the number: '0' for positive and '1' for negative. The remaining bits represent the magnitude of the number.

• The given 4-bit binary number is 1100.
• The MSB is '1', which means the number is negative.
• The remaining bits, 100, represent the magnitude.
• Converting the magnitude 100 to decimal:
  • $1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 4 + 0 + 0 = 4$
• Since the number is negative, P = -4.

Decimal Value of Q (1's Complement Representation)

In 1's complement representation, if the MSB is '0', the number is positive and its decimal value is directly calculated from the binary. If the MSB is '1', the number is negative. To find its magnitude, we take the 1's complement of the number (invert all bits) and then find the decimal value of the inverted result.

• The given 4-bit binary number is 1100.
• The MSB is '1', indicating a negative number.
• To find the magnitude, we take the 1's complement of 1100.
  • Inverting all bits (0 becomes 1, 1 becomes 0): 1100 becomes 0011.
• Converting 0011 to decimal:
  • $0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 0 + 0 + 2 + 1 = 3$
• Since the original number was negative, Q = -3.

Decimal Value of R (2's Complement Representation)

In 2's complement representation, if the MSB is '0', the number is positive. If the MSB is '1', the number is negative. To find its magnitude, we take the 2's complement of the number (invert all bits and then add 1 to the result) and then find the decimal value of this result.

• The given 4-bit binary number is 1100.
• The MSB is '1', indicating a negative number.
• To find the magnitude, we take the 2's complement of 1100.
  1. Invert all bits: 1100 becomes 0011.
  2. Add 1 to the inverted result: 0011 + 1 = 0100.
• Converting 0100 to decimal:
  • $0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 0 + 4 + 0 + 0 = 4$
• Since the original number was negative, R = -4.

Calculating the Sum (P + Q + R)

Now, we sum the decimal values of P, Q, and R.

• P = -4
• Q = -3
• R = -4
• Sum = P + Q + R = $(-4) + (-3) + (-4) = -11$.

6-bit 2's Complement Representation of the Sum

We need to represent the sum, -11, in 6-bit 2's complement form.

• First, find the 6-bit binary representation of positive 11.
  • The decimal number 11 can be written as $8 + 2 + 1$, which corresponds to $2^3 + 2^1 + 2^0$.
  • In 6-bit binary, +11 is 001011.
• To get -11 in 2's complement, we take the 2's complement of 001011.
  1. Invert all bits of 001011: 110100.
  2. Add 1 to the inverted result: 110100 + 1 = 110101.

Therefore, the 6-bit 2's complement representation of (P + Q + R) is 110101.

The final 6-bit 2's complement representation of (P + Q + R) is 110101.