Signed Bin Numbers (2's complement + sign-and-magnitude)

Can either be represented with

1. Sign-and-magnitude
   • There is a separate bit that determines the sign for the rest of the number
   • 1 = negative. = 0 positive.
   • For an 8-bit number, the range would be from $[-255,255]$
2. Signed numbers (2’s complement)
   • For an $n$-bit number, the $2^n$ term is negated. This is the Most significant bit (MSB)
     • For an $8-$bit number, the number $10000000$ is actually $-256$
   • Every other digit is positive

Represent -18 as a 6-bit number via Sign and magnitude + two's complement (signed)

For sign-and-magnitude, the sign would be 1 (it’s negative). The other 5 bits would be 10010

For 2’s complement, it would be 101110 100000 = -32 01110 = 14 -32 + 14 = -18, as required!

Shortcut to converting a binary number to two’s complement

Taking the 2’s complement of a number negates it!

Idea: $\text{2’s complement}=\text{1’s complement}+1$

To get 1’s complement, you invert every bit Then you just add 1

Corollary: $X+2^{\prime }\mathrm{scomplement}(X)=0$

So to be clear, the 2’s complement of $X$ is just $-X$

Shorter shortcut

Start going through each number from the right side. Once you hit a 1 from the right side, you keep that number and flip everything left of it.

Common negative values

For an $n$-bit number:

• -1 is always gonna be 111111...
• $2^n-1=0111\dots 11$
• $-2^n=1000\dots 11$
• $0=000\dots 00$