Digital Logic & Design

Lecture 02

Recap

Last lecture discussion Decimal Number Systems Caveman Base 5 Number System Binary Number System Number System Conversion

Today’s lecture discussion Decimal-Binary Conversion

Binary to Decimal Conversion

Sum-of-Weights Expression base number & weights Sum terms Paper and pencil method

Sum of non-zero terms Mental Arithmetic, quick method Sum of weights of non-zero terms

Binary to Decimal Conversion

Sum-of-Weights 100112

(1 x 24) + (0 x 23) + (0 x 22) + (1 x 21)

+ (1 x 20) Terms 16, 0, 0, 2 and 1 19

Binary to Decimal Conversion

Add weights of non-zero terms Weights increase/decrease by power of 2 100112 = 16 + 2 + 1 = 19

1011.1012 = 8 + 2 + 1 + 1/2 + 1/8

= 11 + 5/8

= 11.625

Decimal to Binary Conversion

Sum-of-Weights method used in reverse Highest binary weight less than the decimal

number Subsequent smaller weights that add up to

decimal number Repeated division by 2

Paper and pencil method Number repeatedly divided by 2

Decimal to Binary Conversion using Sum-of-Weights

number Weight Result after subtraction Binary Bit

392 256 392-256 = 136 1 b8

136 128 136-128 = 8 1 b7

8 64 0 b6

8 32 0 b5

8 16 0 b4

8 8 8-8 =0 1 b3

0 4 0 b2

0 2 0 b1

0 1 0 b0

Decimal to Binary Conversion

number Quotient after division Remainder after division

392 196 0 (b0)

196 98 0 (b1)

98 49 0 (b2)

49 24 1 (b3)

24 12 0 (b4)

12 6 0 (b5)

6 3 0 (b6)

3 1 1 (b7)

1 0 1 (b8)

Binary-Decimal fraction conversion

Binary to Decimal Conversion Sum-of-Weights method Weights decrease by a factor of 2 0.11012 weights ½, ¼, 1/16 Sum up to 0.8125

Decimal to Binary Conversion Repeated Multiplication by 2 example

Decimal-Binary fraction conversion

Decimal to Binary Conversion Repeated multiplication by 2

Number Mult. By 2 Integer

0.8125 1.625 1 (b-1)

0.625 1.250 1 (b-2)

0.250 0.500 0 (b-3)

0.500 1.000 1 (b-4)

Binary Arithmetic

Binary Addition Binary Subtraction Binary Multiplication Binary Division

Binary Addition

Four Basic rules for binary addition

1st digit 2nd digit Sum Carry

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

Addition of multiple binary numbers

Binary Addition

Carry 1 10 1

1st Number 1 0 1 1

2nd Number 1 1 0

3rd Number 1 0 0 0

4th Number 1 1

Result 1 1 1 0 0

Binary Subtraction

Four Basic rules for binary subtraction

1st digit 2nd digit Difference Borrow

0 0 0 0

0 1 1 1

1 0 1 0

1 1 0 0

Binary Subtraction

Borrow 1

1st Number 1 0 1 1

2nd Number 1 1 0

Result 1 0 1

Binary Multiplication

Four Basic rules for binary multiplication

1st digit 2nd digit Product

0 0 0

0 1 0

1 0 0

1 1 1

Example of Binary Multiplication

Binary Multiplication

1101 (13)

x 101 (5)

1st product term 1101

2nd product term 0000

3rd product term 1101

Product 1000001 (65)

Multiplication by shifting left

Decimal 29 shifted left by one digit 290 Shift left 1 digit is multiply by 10

Binary 111012 (29) shifted left by one bit

1110102 (58) Shift left 1 bit is multiply by 2

Binary Division

10

101 | 1101

101

011

000

11

Division by shifting right

Decimal 29 shifted right by one digit 2.9 Shift left 1 digit is divide by 10

Binary 111012 (29) shifted left by one bit

1110.12 (14.5) Shift left 1 bit is divide by 2

Signed and Unsigned Numbers

Unsigned Binary Numbers Signed Binary Numbers

Most significant bit represents sign 0 represents a positive number 1 represents a negative number

2’s Complement form

1’s complement form 2’s complement form

Binary number 01101 (13)

1’s complement 10010

+ 1

2’s complement 10011 (-13)

Addition and Subtraction with 2’s Complement

0101 +5 0101 +50010 +2 1110 -20111 +7 0011 +3

1011 -5 1011 -51110 -2 0010 +21001 -7 1101 -3

Range of Numbers

Maximum Range Number of digits Decimal number example Binary number example Overflow

Range of Binary Numbers

DecimalNumber

Sign Magnitude2n-1, 2n-1 -1

2’s Complement2n-1, 2n-1 -1

Unsigned2n, 2n - 1

0 0000 0000 000

1 0001 0001 001

2 0010 0010 010

3 0011 0011 011

4 0100 0100 100

5 0101 0101 101

6 0110 0110 110

7 0111 0111 111

Range of Binary Numbers

DecimalNumber

Sign Magnitude2n-1-1, -(2n-1 -1)

2’s Complement2n-1, -2n-1

Unsigned

-8 1000

-7 1111 1001

-6 1110 1010

-5 1101 1011

-4 1100 1100

-3 1011 1101

-2 1010 1110

-1 1001 1111

Summary

Binary to Decimal Conversion Sum-of-Weights Sum of non-zero terms

Decimal to Binary Conversion Sum-of-Weights (in reverse) Repeated Division by 2

Summary

Binary to Decimal fraction conversion Sum-of-Weights

Binary to Decimal fraction conversion Repeated Multiplication by 2

Summary

Binary Addition Binary Subtraction Binary Multiplication

Multiplication by shift left operation Binary Division

Division by shift right operation

Summary

Unsigned Binary Signed Binary

Sign Bit 2’s Complement 1’s Complement Range of Binary Numbers

Lecture No. 2

Number Systems