Negative Binary Numbers

#Negative aka Signed Numbers in Two’s Complement

Signed numbers are numbers that can take on negative values. It’s called signed because it can take on a “negative sign” as well as a positive sign. This notation of representing negative numbers is called two’s complement.

To get the binary representation of a negative number, say -6, we have to:

1. Write the number positive version in binary

   6 in binary

   8	4	2	1
   0	1	1	0

2. Flip all the bits

   0 1 1 0 (6) becomes...
   _______
   1 0 0 1 (FLIPPED)

3. Add one to the flipped result

        
             1
         1 0 0 1 
       + 0 0 0 1
       _________
               0
       

        
             1
         1 0 0 1
       + 0 0 0 1
       _________
         1 0 1 0 (-6)
       

   Answer: -6 in binary is 1010.
   Let’s talk about two’s complement and what logistics drive 1010 = -6

#How Two’s complement encoding works

Notice: Only the most significant bit(left most highlighted in red) holds is of a negative weight.

Given a bit-width of 4, the decimal number -6₁₀ = 1010₂
Given a bit-width of 6, the decimal number -6₁₀ = 111010₂
Given a bit-width of 8, the decimal number -6₁₀ = 11111010₂

When representing negative numbers, the width matters, because only the most signifncant bit(left most as highlighted in red) holds the negative weight needed to split the positive and negative numbers as seen in the table way below. It’s important to notice that all negative numbers have a leading bit of 1.

Let’s try a few examples with variable bit-widths:

Check your understanding!

#Multiple Meanings

In English we have words like

“bark”:

1. A noise that a dog makes.
2. What trees are mare out of.

“sex”:

1. either of the two main categories (male and female) into which humans and most other living things are divided on the basis of their reproductive functions.
2. an activity

One word, multiple meanings!

Likewise in the world of bits, the bit pattern: 1101010₂ can mean:

1. The letter “j”
2. The number 106₁₀
3. The number -22₁₀
4. The hexadecimal number D2₁₆

Computers only see zeros and ones, its how and what we humans decide to interpret them that gives them meaning/value.

#Multiple interpretations of bits

Given the bit-width of 4:

-6₁₀ in binary is 1010₂, but wait...

10₁₀ in binary is 1010₂, also?!

They both use the same pattern of bits? What gives?!

Consider all the possibilities of unsigned and signed 4 bit numbers:

Again, as highlighted in red, the most significant bit(left most bit) in a signed number 2^bitwidth-1 takes on a negative weight (in this case it’s 8).

Now knowing that bit patterns can have 2 (or more) meanings, let’s go the other way around get some practice converting bit patterns to signed and unsigned decimal representations.

Check your understanding!

1110
FLIP
0001

     
          1
      0 0 0 1 
    + 0 0 0 1
    _________
            0
    
     
          1
      0 0 0 1
    + 0 0 0 1
    _________
      0 0 1 0 (2)