Let's convert the following negative decimal numbers to binary using two's complement representation:

a) -19

b) -84

To convert a negative decimal number to binary in two's complement, we follow these steps:

1. Find the binary representation of the positive number.
2. Extend the number to the desired number of bits (usually 8 or 16 bits, depending on the problem).
3. Invert all the bits (one's complement).
4. Add 1 to the result (this gives us the two's complement).

I'll start with the first number, -19.

1. Convert +19 to binary:

   • 19 in decimal = 10011 in binary.

2. Extend to 8 bits:

   • 19 = 00010011 (8 bits).

3. Invert all bits:

   • Inverting 00010011 gives us 11101100.

4. Add 1 to the result:

   • 11101100 + 1 = 11101101.

So, the two's complement representation of -19 is 11101101.

Now, let's do the same for -84:

1. Convert +84 to binary:

   • 84 in decimal = 1010100 in binary.

2. Extend to 8 bits:

   • 84 = 01010100 (8 bits).

3. Invert all bits:

   • Inverting 01010100 gives us 10101011.

4. Add 1 to the result:

   • 10101011 + 1 = 10101100.

So, the two's complement representation of -84 is 10101100.

Final answers:

• a) -19 = 11101101 (two's complement in 8 bits)
• b) -84 = 10101100 (two's complement in 8 bits)

Would you like any further explanation or details on these conversions?

Here are some related questions to explore:

1. How would you convert -19 to binary using sign and magnitude representation?
2. What would the two's complement of -19 be in a 16-bit representation?
3. How do you convert a positive decimal number to binary?
4. Can you explain how to perform a bitwise NOT operation?
5. How do you handle overflow when adding binary numbers?

Tip: When converting negative numbers to two's complement, always ensure you start with the binary representation of the positive version and apply the inversion and addition of 1.