Converting Signed 8-Bit Binary Number to Decimal

Question:

How do we convert the signed 8-bit binary number 1001 1001 to decimal?

Answer:

The signed 8-bit binary number 1001 1001 represents a negative decimal value. By calculating its magnitude as a positive number and then applying the two's complement rule, we find that 1001 1001 is equivalent to -25 in decimal.

Explanation:

To convert the signed 8-bit binary number 1001 1001 to decimal, we first determine the sign of the number. In signed binary, the left-most bit is the sign bit. Since it's a 1, this indicates a negative number. The remaining 7 bits represent the magnitude of the number. We calculate the magnitude by converting the binary digits to decimal, ignoring the sign bit for now.

The binary number 001 1001 (excluding the sign bit) converts to decimal as follows:

Multiply each binary digit by its corresponding power of 2, starting from the right and moving left.

(0 × 2^6) + (0 × 2^5) + (1 × 2^4) + (1 × 2^3) + (0 × 2^2) + (0 × 2^1) + (1 × 2^0) equals 0 + 0 + 16 + 8 + 0 + 0 + 1.

Add up the results to get the magnitude: 16 + 8 + 1 = 25.

Since the original binary number is negative, we apply the two's complement rule to find the negative decimal equivalent. Inverting the binary digits of the magnitude and adding 1 gives us:

(1110 0110 + 1) = 1110 0111

This binary corresponds to the decimal number -25. Hence, the signed 8-bit binary number 1001 1001 is -25 in decimal.