Question

Convert the following binary numbers into signed decimal numbers:

(a) 10000000
(b) 00110011
(c) 10010010
(d) 10001001

Answer 1

The conversion from binary to signed decimal requires interpreting the first bit as the sign bit and converting the remaining bits. The numbers (a) 10000000, (b) 00110011, (c) 10010010, and (d) 10001001 convert to -128, +51, -110, and -111 respectively, using two's complement notation.

Step-by-step explanation:

The question involves converting binary numbers into signed decimal numbers. The process includes interpreting the first bit as the sign bit (where 0 represents positive and 1 represents negative) and then converting the remaining bits into their decimal equivalent. Let's go through each given binary number:

1. 10000000: The first bit is '1', so the number is negative. The rest of the bits are '0000000', which converts to 0 in decimal. However, with 8-bit binary numbers, this represents -128 due to two's complement notation.
2. 00110011: The first bit is '0', so the number is positive. The conversion of '0110011' gives 1*32 + 1*16 + 0*8 + 0*4 + 1*2 + 1*1 = 51.
3. 10010010: The first bit is '1', indicating a negative number. To find its positive counterpart, we invert the digits and add 1 resulting in '01101110' which is 64 + 32 + 8 + 4 + 2 = 110, hence the signed decimal is -110.
4. 10001001: Starting with '1', it's negative. Inverting and adding 1 to '0001001' results in '01101111', which is 64 + 32 + 8 + 4 + 2 + 1 = 111, making the signed decimal -111.

To note, negative numbers in binary under the two's complement system require inverting all digits and adding one to get the positive value that is then negated to express the signed decimal.