Convert the following decimal values to a 16-bit 2's complement representation. (1*2Marks)i. -2342ii. 234

1. Break Down the Problem

We need to convert two decimal numbers, -2342 and 234, into their 16-bit 2's complement binary representations.

2. Relevant Concepts

• Positive Numbers: For positive decimal numbers, the conversion to binary can be done directly.
• Negative Numbers: For negative decimal numbers, convert the absolute value to binary, then find the 2's complement:
  1. Convert the number to binary.
  2. Invert the digits (change 0s to 1s and 1s to 0s).
  3. Add 1 to the inverted binary number.

3. Analysis and Detail

i. Converting -2342:

1. Find the binary of 2342:

   • 2342 in decimal to binary:
     • 2342 / 2 = 1171 remainder 0
     • 1171 / 2 = 585 remainder 1
     • 585 / 2 = 292 remainder 1
     • 292 / 2 = 146 remainder 0
     • 146 / 2 = 73 remainder 0
     • 73 / 2 = 36 remainder 1
     • 36 / 2 = 18 remainder 0
     • 18 / 2 = 9 remainder 0
     • 9 / 2 = 4 remainder 1
     • 4 / 2 = 2 remainder 0
     • 2 / 2 = 1 remainder 0
     • 1 / 2 = 0 remainder 1
   • Therefore, 2342 in binary is 100100011110.

2. Pad to 16 bits:

   • 0000100100011110

3. Invert the digits:

   • 1111011011100001

4. Add 1:

   • 1111011011100001 + 1 = 1111011011100010

Thus, -2342 in 16-bit 2's complement is 1111011011100010.

ii. Converting 234:

1. Find the binary of 234:

   • 234 in decimal to binary:
     • 234 / 2 = 117 remainder 0
     • 117 / 2 = 58 remainder 1
     • 58 / 2 = 29 remainder 0
     • 29 / 2 = 14 remainder 1
     • 14 / 2 = 7 remainder 0
     • 7 / 2 = 3 remainder 1
     • 3 / 2 = 1 remainder 1
     • 1 / 2 = 0 remainder 1
   • Therefore, 234 in binary is 11101010.

2. Pad to 16 bits:

   • 0000000111010110

Thus, 234 in 16-bit 2's complement is 0000000111010110.

4. Verify and Summarize

• The conversion for -2342 results in 1111011011100010.
• The conversion for 234 results in 0000000111010110.

Final Answer

1. -2342 in 16-bit 2's complement: 1111011011100010
2. 234 in 16-bit 2's complement: 0000000111010110