Digital Logic Design Mohammad Fakih Numbering Systems

Digital Logic DesignMohammad Fakih

Numbering Systems

Digital Computer Systems

••

Digital systems consider discrete amounts ofdata.Examples–26 letters in the alphabet–10 decimal digits

• Larger quantities can be built from discrete values:––

Words made of letters: Football, UniversityNumbers made of decimal digits (e.g. 239875.32)

••

Computers operate on binary values (0 and1)Easy to represent binary values electrically–––

Voltages and currents.Can be implemented using circuitsCreate the building blocks of modern computers

1Digital Logic Design M.Fakih

Understanding Decimal Numbers

••

Decimal numbers are made of decimal digits: (0,1,2,3,4,5,6,7,8,9)Decimal number:– 8653 = 8x103 + 6x102 + 5x101 + 3x100

• What about fractions?––

97654.35 = 9x104 + 7x103 + 6x102 + 5x101 + 4x100 + 3x10-1 + 5x10-2

In formal notation -> (97654.35)10

• Why do we use 10 digits, anyway?

2Digital Logic Design – M.Fakih

Understanding Binary Numbers

• Binary numbers are made of binary digits (bits):– 0 and 1

• binary number:– (1011)2 = 1x23 + 0x22 + 1x21 + 1x20 = (11)10

• What about fractions?– (110.10)2 = 1x22 + 1x21 + 0x20 + 1x2-1 + 0x2-2

• Groups of eight bits are called a byte– (11001001) 2

bits• a nibble.Groups of four are called– (1101) 2


Conversion Between Number Bases

Decimal(base 10) Binary(base 2)

Hexadecimal(base16)

° Learn to convert between bases.


Decimal review

• a weightNumbers consist of a bunch of digits, each with

1100

610

21

. 31/10

71/100

51/1000

DigitsWeights

• These weights arethis:

all powers of the base, which is 10. We can rewrite

1102

6101

2100

. 310-1

710-2

510-3

DigitsWeights

• To find the decimal value of a number, multiply each digit by its weightand sum the products.

(1 x 102) + (6 x 101) + (2 x 100) + (3 x 10-1) + (7 x 10-2) + (5 x 10-3) = 162.375


Converting binary to decimal

• We can use the same trick to convert binary, or base 2, numbers todecimal. This time, the weights are powers of 2.– Example: 1101.01 in binary

123

122

021

120

. 02-1

12-2

Binary digits, or bitsWeights (in base 10)

– The decimal value is:

(1 x 23) + (1 x 22) + (0 x0

21) + (1+

x 20) + (0 x 2-1) + (1 x 2-2) ==8 + 4 + 1 + 0 + 0.25 13.25


Powers of 2: Useful abbreviations:20 = 1 24 = 16 28 = 256 K = 210 = 1,02421 = 2 25 = 32 29 = 512 M = 220 = 1,048,57622 = 4 26 = 64 210 = 1024 G = 230 = 1,073,741,82423 = 8 27 = 128

Converting decimal to binary

• To convert a decimal integer into binary, keep dividing by 2 until thequotient is 0. Collect the remainders in reverse order.

• To convert a fraction, keep multiplying the fractional part by 2 until itbecomes 0. Collect the integerExample: 162.375:

parts in forward order.•

162 / 281 / 240 / 220 / 210 / 2

5 / 22 / 21 / 2

========

814020105210

remremremremremremremrem

01000101

0.3750.7500.500

xxx

222

===

0.7501.5001.000

• So, 162.37510 = 10100010.0112


Base 16 is useful too

• The hexadecimal system uses 16 digits:0 1 2 3 4 5 6 7 8 9 A B C D E F

You can convert between base 10 and base16 using techniques like the ones we justshowed for converting between decimal andbinary.For our purposes, base 16 is most useful asa “shorthand” notation for binary numbers.

•

•

– Since 16 = 24, one hexadecimal digit isequivalent to 4 binary digits.

– It’s often easier to work withlike B4 instead of 10110100.

a number


Decimal Binary Hex0 0000 01 0001 12 0010 23 0011 34 0100 45 0101 56 0110 67 0111 78 1000 89 1001 910 1010 A11 1011 B12 1100 C13 1101 D14 1110 E15 1111 F

Binary and hexadecimal conversions

• Converting from hexadecimal to binary is easy: just replace each hexdigit with its equivalent 4-bit binary sequence.

261.3516 ==

20010

60110

10001

.

.3

0011516

01012

• To convert from binary to hex, make groups of 4 bits, starting from thebinary point. Add 0s to the ends of the number if needed. Then, justconvert each bit group to its corresponding hex digit.

10110100.0010112 ==

1011B

01004

.

.0010

211002

C16


Hex BinaryC 1100D 1101E 1110F 1111

Hex Binary8 10009 1001A 1010B 1011

Hex Binary4 01005 01016 01107 0111

Hex Binary0 00001 00012 00103 0011

Hex to Decimal

8163x


7162x C

161x 9

160x

(87C9)16 = (???)10

32,768 + 1792 + 192 + 9(87C9)16 = (34,761)10

Convert Decimal to Hex

Integer Part: Divide by the base,keep track of the remainder, and read up.

1616 2,172 rem

remremrem

91278

16 = C Read up16

0

34,76110 = 87C916


8135

34,761

Questions

100101012 = ? (decimal)


Questions

85710 = ?16


Questions

BED16 = ?2


Addition


Binary Addition

••

Binary addition is very simple.This is best shown in an example of adding two binary numbers…

1

01

1 1

1

1

11

1

10

1

11

carries

11+

---------------------1 0 1 0 1 0 0


Binary addition example worked out

••

Some terms are given hereExercise: what are these numbers equivalent to in decimal?

The initial carryin is implicitly 0

1 111

101

011

(Carries)(Augend)(Addend)

10+

1 1 0 0 1 (Sum)

most significantbit (MSB)

least significantbit (LSB)


Subtraction


Binary Subtraction

° We can also perform subtraction (with borrows in place ofcarries).

° Let’s subtract (10111)2 from (1001101)2…

110

100

borrows0 10 0 10

1 0 01

10

11

01

11-

------------------------1 1 0 1 1 0


Negative Numbers

Subtract by adding

73-3538

73+65138

10’s complement

Ignore carry


Two’s Complement Shortcuts• Algorithm – Simply complement each bit and then add 1 to

result.the

– Finding the 2’s complement of (01100101)2complement…

and of its 2’s

N = 0110010110011010

[N] = 1001101101100100

1---------------01100101

+ 1 +---------------

10011011


This is called one’scomplement

Negative Numbers

Subtract by

0 1 0 0 1 0 0 1

1 1 0 1 1 1 0 1

adding

7335

2’s

0100100100100011

11011100 flip

+1

-----------

11011101

0 0 1 0 0 1 1 0 comp

-35


73-3538

Table 2.2Positive and Negative Binary Numbers


Signed decimal Hex Binary Unsigned decimal-128-127-126

………-3-2-10123

…

…125126127

808182………FDFEFF00010203……

7D7E7F

100000001000000110000010………11111101111111101111111100000000000000010000001000000011………011111010111111001111111

128129130

………

253254255

0123

………

125126127

Signed Numbers

4-bit: 8H = -8 toto

7H = +71000 0111

8-bit: 80H = -128 to 7FH=

+127

16-bit: 8000H7FFFH

==-32,768+32,767

to

32-bit: 80000000H7FFFFFFFH

==-2,147,483,648+2,147,483,647

to


Questions

What is the two’s00101100?

complement of

What hex number represents thedecimal number -40?


Multiplication


Binary Multiplication

• Binary multiplication is much the same as decimal multiplication,except that the multiplication operations are much simpler…

1 01

10

11

10X

-----------------------0001

0101

010

01

0101

001

-----------------------1 1 1 0 0 1 1 0


ASCII CODES


ASCII

• The most commonly used code for representing letters, numeralsand punctuation characters (alphanumeric data)

• Each character is represented with a 7-bit string; for example:‘3’ = 00110011‘ ’ = 00100000

(hex 33)(hex 20)

• An 8-bit extension of ASCII has also been defined


ASCII Code

••

American Standard Code for Information InterchangeASCII is a 7-bit code, frequently used with an 8th bit for errordetection (more about that in a bit).


Character ASCII (bin) ASCII (hex) Decimal

A 1000001 41 65

B 1000010 42 66

C 1000011 43 67

…

Z

a

…

1

‘

BCD


Binary Coded Decimal

• Binary coded decimal (BCD) represents each decimal digit with four bits– Ex. 0011 0010 1001 = 329BCD

3 2 9••

This is NOT the same as 0011001010012

Why do this? Because people think in decimal.


Digit BCD Code Digit BCD Code

0 0000 5 01011 0001 6 01102 0010 7 01113 0011 8 10004 0100 9 1001

Putting It All Together

°°

BCD not very efficient

Used in earlycomputers (40s, 50s)Used to encodenumbers for seven-segment displays.

°

° Easier to read?


Gray code


Gray code


0123456789101112131415

0000000100100011010001010110011110001001101010111100110111101111

Decimal Binary Gray code

0000000100110010011001110101010011001101111111101010101110011000

Gray code is an un-weighted code that has asingle bit change betweenone code word and thenext in a sequence. Graycode is used to avoidproblems in systemswhere an error can occurif more than one bitchanges at a time.

Floatingpoint


Express the speed of light, c, in single precision floating pointnotation. (c = 0.2998 x 109)

Floating point notation is capable of representing verylarge or small numbers by using a form of scientificnotation. A 32-bit single precision number is illustrated.

S E (8 bits) F (23 bits)

Sign bit Magnitude with MSB droppedBiased exponent (+127)

In scientific notation, c = 1.0001 1101 1110 1001 0101 1100 0000 x 228.

0 10011011 001 1101 1110 1001 0101 1100

In binary, c = 0001 0001 1101 1110 1001 0101 1100 00002.

S = 0 because the number is positive. E = 28 + 127 = 15510 = 1001 10112.F is the next 23 bits after the first 1 is dropped.In floating point notation, c =

Floating point

Digital Logic Design – M.Fakih 37

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Digital Logic Design Mohammad Fakih Numbering Systems