Converting negative decimal to binary

I've been trying to convert the number -441 to binary, but I don't really know how I can accomplish this.

I first converted 441 to binary which is: 110111001 Then I'm supposed to take the compliment of this number which is: 001000110 And then I'd have to add one which would result in: 001000111

The exercise says that I have to give the binary representation in 10 bit and 16 bit, and so I though I could just put a zero before the number and that's it, but after I lot of searching I discovered that I'm supposed to put a ONE before the number, why is this the case?

How would I go about converting -441 to a 16 bit number?

Thank you.

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.