Storing Negative Integers

Section 5.3 Storing Negative Integers

Now we know how positive integers can be stored in the computer. How about negative integers? There actually are several different options. The following video describes the signed magnitude representation of negative numbers:

Investigate 5.5.

Does the binary number 10000001 represent the same decimal number in both unsigned binary and signed magnitude binary?

Hint.

Try converting the binary number to decimal, following the rules of each binary representation method.

We have been representing positive integers as binary numbers, what about negative integers?

Several strategies are possible:

The most obvious is known as signed magnitude, which uses the most significant bit (MSB) for the sign: 0 for + and 1 for -.

For an 8-bit binary number:

This allows for a range from -127 to 127 to be represented.

For an n-bit binary number: \(-(2^{n-1}-1)\) to \((2^{n-1}-1)\)

Subsection 5.3.1 Adding in Binary

How do you add binary numbers?

Recall addition in the decimal system:

                   1
   79              79      
+ 106           + 106
______   -->    ______   
  185             185

Notice how we "carry" the one from the ones digits (shown on the right above)? We add similarly in binary, following these rules:

1 + 0 = 01
1 + 1 = 10

Here’s an example of adding in binary (with "carrying" the ones shown on the right again):

                       1  111
  01001111             01001111     
+ 01101010           + 01101010
___________   -->    ___________   
  10111001             10111001

We can check that this binary addition aligns with decimal addition, as 79 + 106 = 185 (as shown above).

Subsection 5.3.2 Adding in Signed Magnitude

How about adding numbers in binary when using the signed magnitude representation of integers?

Adding two positive numbers:

  00001101   (13 decimal)
+ 01100100   (100 decimal)
___________  
  01110001   (113 decimal)

Adding a positive and a negative number:

  00001101   (13 decimal)
+ 11100100   (-100 decimal)
___________  
  11110001   (-113 decimal)

Addition does not work with signed magnitude numbers!

Check Your Understanding 5.3.3 Check Your Understanding

1.

What is the 8-bit representation of the decimal number -63, if the signed magnitude representation is used?

Subsection 5.3.4 Binary Two’s Complement

We noticed in the last video that arithmetic is not straight forward in the signed magnitude representation of negative numbers. We therefore introduce a different way to store negative integers, namely the binary two’s complement representation. While this representation may seem cumbersome at first, the point is really to fix the problems we observed with binary addition when negative numbers are involved. The binary two’s complement representation allows for the same process to be used in adding integers internally inside the computer, regardless of whether the numbers are positive or negative. This process is typically hard-wired in the computer’s processor which makes for super fast execution time.

Addition does not work with signed magnitude numbers.

Solution: use an alternative for representing signed integers known as binary 2’s complement.

Idea: Store negative integers in such a way that when summed with its complement (positive number) the result is zero.

Rules for writing the 2’s complement of a number:

Write down the binary representation of the magnitude
Positive integers stay the same
Negative integers:
1. Change all 0s to 1s and all 1s to 0s
2. Add 1

Subsection 5.3.5 2’s Complement Example

Decimal:

     13
+  -100
________
    -87

Step 1: magnitude

13    -->  00001101
-100  -->  01100100

Steps 2 and 3a: binary complement

00001101  -->  stays the same
01100100  -->  10011011

Steps 2 and 3b: add 1

00001101  -->  stays the same
10011011  -->  10011100

Add!

  00001101  
+ 10011011 
___________
  10101001

Did it work? Is this the 2’s complement of -87?

We need a way to decode a 2’s complement number...

Subsection 5.3.6 Decoding 2’s Complement Numbers

All 2’s complement numbers that are negative have MSB ’set’ (negative) -- shown in blue
Add values of the places which are zero: (64 + 16 + 4 + 2) -- shown in pink
Add one to the result

So the binary 2’s complement number 10101001 is:

-(64 + 16 + 4 + 2 + 1) = -87

So, addition with 2’s complement integers works!

Whats the range of numbers you can represent when using the binary 2’s complement?

8-bit 2’s complement numbers range: -128 to 127

n-bit 2’s complement numbers: \(-(2^{n-1})\) to \((2^{n-1}-1)\)

Food for thought: why is \(2^{n-1}\) not represented in 2’s complement?

Check Your Understanding 5.3.7 Check Your Understanding

1.

What is the 8-bit binary 2’s complement representation of the number 63 (careful: this is a positive number... what do positive numbers look like in 2’s complement?)

2.

What is the 8-bit binary 2’s complement representation of the number -63

Subsection 5.3.8 Counting in Binary Two’s Complement

Finally, let’s take a look at what happens when counting beyond the largest possible number in binary 2’s complement.

More food for thought: start at 0, keep adding 1. What happens?

Decimal     Binary Two's Complement
   0              00000000
   1              00000001
   2              00000010
   3              00000011
   .                  .
   .                  .
   .                  .
 126              01111110
 127              01111111
       +1 ?????

What is the decimal value of this 2’s complement number?

           1111111     <-- carrying the ones
  127      01111111
+   1    + 00000001
______   ___________ 
  ???      10000000

MSB is ’set’, so the number is negative
Add values of the places which are zero: 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127
Add 1 to the result

Thus, the result is -128.

Check Your Understanding 5.3.9 Check Your Understanding

1.

What is the decimal value of the 8-bit binary 2’s complement number 10101010?

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