Show the binary addition and subtraction of 125 (decimal) and 200 (decimal).

To perform binary addition and subtraction of the decimal numbers 125 and 200, we first need to convert these decimal numbers into binary (base-2) form. Then, we can apply binary addition and subtraction rules to find the results. Let's go through each step:

Binary Addition of 125 and 200:

1. Convert Decimal Numbers to Binary:

   • Decimal 125 in binary: (125_{10} = 1111101_2)
     • Breakdown: (125_{10} = 64 + 32 + 16 + 8 + 4 + 1 = 1111101_2)
   • Decimal 200 in binary: (200_{10} = 11001000_2)
     • Breakdown: (200_{10} = 128 + 64 + 8 = 11001000_2)

2. Perform Binary Addition:

   • Align the binary numbers:

        1111101   (125)
      + 11001000  (200)
      ----------
     

   • Perform binary addition (from right to left), including any carries:

        1111101   (125)
      + 11001000  (200)
      ----------
       10000101   (325)
     

   • Therefore, (125{10} + 200{10} = 325_{10}), and (1111101_2 + 11001000_2 = 10000101_2).

Binary Subtraction of 125 and 200:

1. Convert Decimal Numbers to Binary:

   • Use the same binary representations from above:
     • (125_{10} = 1111101_2)
     • (200_{10} = 11001000_2)

2. Perform Binary Subtraction:

   • Align the binary numbers (subtract 200 from 125):

        1111101   (125)
      - 11001000  (200)
      ----------
     

   • Perform binary subtraction (from right to left), borrowing as needed:

        1111101   (125)
      - 11001000  (200)
      ----------
       10110101   (Subtraction result in binary)
     

   • Therefore, (125{10} – 200{10} = -75_{10}), and (1111101_2 – 11001000_2 = 10110101_2) (in two's complement form representing -75).

Summary:

• Binary Addition: (125{10} + 200{10} = 325_{10}), and (1111101_2 + 11001000_2 = 10000101_2).
• Binary Subtraction: (125{10} – 200{10} = -75_{10}), and (1111101_2 – 11001000_2 = 10110101_2).

In conclusion, we have demonstrated how to perform binary addition and subtraction of decimal numbers 125 and 200 by converting them into binary form and applying binary arithmetic rules. Binary addition results in the decimal sum of 325, represented as (10000101_2), while binary subtraction results in the decimal difference of -75, represented as (10110101_2) in two's complement form. Binary arithmetic is fundamental in digital electronics and computer systems for performing mathematical operations efficiently using binary representation.