•  Data Representation
•  Two's Complement
•  ASCII

Binary Numbers

Digital devices use binary numbers.

Decimal numbers are arranged with the most significant digit on left side and least significant digit on right.

A 16-bit binary number:

 

1.3.3    Storage sizes

Standard sizes in bits:

 

Ranges of Unsigned Integers:

 

Data Representation

Must know number of bits used to represent data.

In the following for convenience we'll use just 4 bits to represent numbers.

• Unsigned - All numbers are positive.

 

• Unsigned 4 bit numbers

  Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
  Unsigned	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111

Bits	Range	Decimal	Binary
4	0 to 24-1	0 to 15	00002 to 11112
8	0 to 28-1	0 to 255	000000002 to 111111112
16	0 to 216-1	0 to 65535	00000000000000002 to 11111111111111112
n	0 to 2n-1

 

• Signed
  • 

  Both positive and negative numbers, a specific representation must be selected.

  Normally the leftmost bit is used as a sign bit. 0 implies positive or + and 1 implies negative or -.

 

• Subtraction - Subtraction can be done using addition: 3 - 5 = 3 + (-5) = -2
   
• Signed Magnitude - The leftmost bit is the sign, the remaining bits are the magnitude. Problem: cannot do arithmetic.

0011 = 3            0101 = 5                3      3      0011    
1011 = -3           1101 = -5              -5   +(-5)    +1101  
                                           -2     -2     10000  =  -0

• One's Complement - Negative numbers only, convert from positive to negative by inverting all bits (e.g. invert 1 to 0, 0 to 1). Can do arithmetic.

0011 = 3            0101 = 5                3      3      0011     
1100 = -3           1010 = -5              -5   +(-5)    +1010  
                                           -2     -2      1101   =  -2

• Two's Complement  - Negative numbers only, convert from positive to negative by forming one's complement and adding 1. Can do arithmetic.

      0011 = 3            0101 =  5              3      3      0011        
1's   1100 = -3           1010 = -5             -5   +(-5)    +1011  
        +1                       +1             -2     -2      1110   = -2
2's   1101                     1011

 

Question - Using 6-bit values:

1. Represent -17 as 2's complement.
    
2. Represent 10 as 2's complement.
    
3. 10+(-17) using 2's complement representation.
    
4. Show that the 2's complement result of Question 3 is correct by converting to decimal.
    

Signed magnitude, One's and Two's Complement 4-bit numbers

Decimal	Signed Magnitude	One's Complement	Two's Complement
0	0000	0000	0000
1	0001	0001	0001
2	0010	0010	0010
3	0011	0011	0011
4	0100	0100	0100
5	0101	0101	0101
6	0110	0110	0110
7	0111	0111	0111
-0	1000	1111	None
-1	1001	1110	1111
-2	1010	1101	1110
-3	1011	1100	1101
-4	1100	1011	1100
-5	1101	1010	1011
-6	1110	1001	1010
-7	1111	1000	1001
-8	None	None	1000

 

Bits	Range	Decimal	Binary
4	-24-1 to 24-1-1	-8 to 7	10002 to 01112
8	-28-1 to 28-1-1	-128 to 127	100000002 to 011111112
16	-216-1 to 216-1-1	-32768 to 32767	10000000000000002 to 01111111111111112
n	-2n-1 to 2n-1-1

ASCII

Characters are represented by a code number, the character A is number code 65₁₀ or 41₁₆, or 01000001₂.

Character 7 is number code 55₁₀ or 37₁₆, or 00110111₂.

Character A is in row 4 and column 1 so is code 41₁₆.

Character M is in row D₁₆ and column 4 so is code D4₁₆.

ASCII chart

1.4    Boolean Operations

Boolean operations

 

Operation Precedence

1. Not
2. And
3. Or

 

Truth table for ØX v Y