Converting negative decimal to binary

Solution 1:

You're confused because you forgot that there must be something that distinguishes positive numbers from negative ones.

Let's say you want to store non-negative numbers on 8 bits.

• 00000000 is 0,
• 00000001 is 1,
• 00000010 is 2,
• 00000011 is 3,
• 00000100 is 4,
• ...
• 11111111 is 255

So you can store numbers in range 0-255 on 8 bits. 255 = 2⁸ - 1. (2 is the base of binary system, 8 is the number of bits, 1 is subtracted because we want to count 0 in)

Now, let's say you want to store negative numbers too. How can we achieve that? We can dedicate one bit for sign. If this bit is 0 then we interpret other 7 bits as a positive number, otherwise as a negative number. It's best to use most significant bit for sign because it makes some operations easier.

• Trivial approach: Just read a number as-is:

  • 00000001 == 1 and 10000001 == -1
  • 01000010 == 66 and 11000010 == -66
  • 01111111 == 127 and 11111111 == -127

• Ones' complement: For any number x, negating its binary representation yields binary representation of -x. This means that:

  • 00000001 == 1 and 11111110 == -1
  • 01000010 == 66 and 10111101 == -66
  • 01111111 == 127 and 10000000 == -127

• Two's complement: For any number x, negating its binary representation and adding 1 yields binary representation of -x. This means that:

  • 00000001 == 1 and 11111111 == -1
  • 01000010 == 66 and 10111110 == -66
  • 01111111 == 127 and 1000001 == -127
  • 10000000 == -128

Why is two's complement the best?

• Because it has the widest range: -128...127, while trivial approach and ones' complement have -127...127
• Zero is well defined:
  • In two's complement only 00000000 is zero
  • In trivial approach both 00000000 and 10000000 are zero
  • In ones' complement both 00000000 and 11111111 are zero
• Addition and subtraction is identical as with unsigned numbers, so CPU doesn't need additional instructions for adding signed numbers.

Note that if we dedicate most significant bit for sign bit, then we can't convert number to binary without knowing how many bits we will need. For example is we have 4 bits, then the number -5 in trivial approach is 1101, on 7 bits it would be 1000101. 0001101 (4-bit -5 padded with zeros to 7 bits length) is actually 13 (most significant bit is 0, so it's positive).

I won't do the homework for you, but I can give you general tips:

To convert -x to N bits long two's complement representation:

1. Convert -x to binary using two's complement.
2. Left-pad it with zeros up to N-1 length.
3. Add the negative sign bit on the left side.

I think you can figure out the rest from this answer. If you have any more questions, leave a comment.