Unsigned integers represent only positive integers by using all the bits to represent the magnitude. Signed integers represent both positive and negative numbers by using the most significant bit to indicate the sign (0 for positive, 1 for negative) and the remaining bits to represent the magnitude. The 2's complement representation is commonly used to store negative numbers in computers, where the negative value is calculated by flipping the bits of the positive value and adding 1.

1. Unsigned Integers
• Represents positive integers only
• Example: ASCII character codes
• Not necessary to indicate a sign, so all 8 or 16 bits can be
used for the magnitude:
– 1 byte = 8 bits = 28 = 256 (0 to 255)
– 2 bytes = 16 bits = 216 = 65,536 (0 to 65,535)
– 4 bytes = 32 bits = 232
= 4,294,967,296 (0 to 4,294,967,295)

2. Signed Integers
 Represents positive and negative integers
… -2, -1, 0, 1, 2, …
 MSB (Most Significant Bit – leftmost bit) used to indicate
sign
0 = positive, 1 = negative
 One less bit is used for the magnitude, with one extra
negative value
1 byte = 8-1 bits = 27 (-128 to +127)
2 bytes = 16-1 bits = 215 (-32,768 to +32,767)
4 bytes = 32-1 bits = 231 (-2,147,483,648 to
+2,147,483,647 )

3. Signed Integers
 Let consider the binary:
I I 0 0 0 I I 0
Convert it to decimal number.
 Use positional representation as shown previously with one change in
the MSB
-27 * 1 = -128
26 * 1 = 64
25 * 0 = 0
24 * 0 = 0
23 * 0 = 0
22 * 1 = 4
21 * 1 = 2
20 * 0 = 0
-128 +(64+4+2)=58

4. Signed Integers
Some values of interest (8-bit example):
• +0
0 0 0 0 0 0 0 0
• +1
0 0 0 0 0 0 0 I
• -1
I 0 0 0 0 0 0 0
I I I I I I I I

5. Signed Integers
Some values of interest (8-bit example):
• Max positive value
0 I I I I I I I = 127
• Most negative value
I 0 0 0 0 0 0 0 = -128

6. 1’s & 2’s Complement
• 1’s complement form
– Formed by reversing (complementing) each bit
• 2’s complement form
– Formed by adding 1 to 1's complement
– Negative numbers are stored this way
• Additive inverse of a number
• Computer never has to subtract
• A – B = A + (-B)

7. Signed Integers (8-bit)
 Example: -9
1. 0000 1001b
2. 1111 0110b
+1b
1111 0111b
247
 Example: -32
1. 0010 0000b
2. 1101 1111b
+1b
1110 0000b
224

8. Class Exercise
• Convert the following negative decimal
numbers to 8 bit binary using the 2’s
complement
– -1, -3, -8, -17

9. Class Exercise
• Evaluate 25 + (- 5) in 8 bit binary
– Convert 25 to binary
– Convert -5 to 2’s complement
– Add together
– Check your answer by converting back to decimal