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CHAPTER 1 Digital Systems and Binary Numbers

Chap 1 Digigal Systems and Binary Numbers

1.1 Digital Systems

1.2 Binary Numbers

1.3 Number-Base Conversions

1.4 Octal and hexadecimal Numbers

1.5 Complements

1.6 Signed Binary Numbers

1.7 Binary Codes

Chap 1 1.2 Binary Numbers

In general, a number expressed in a base-r system

has

coefficients multiplied by powers of r

a

?r a

?r a

?r a

a

?r a

?r a

?r

n

n-1

1

-1

-2

-m

n n-1 1 0 -1 -2 -m

r is called base or radix.

In generax, a number expressed in a base-r sysxem

hax

coefficienxs multiplied by powers of r

a

?r a

?r a

?r a

a

?r xa

?r a

?r

n

n-1

1

-1

-2

-m

n n-1 1 0 -1 -2 -m

r is called base or radix.

Chap 1 1.2 Binary Numbers

Arithmetic Operation 1-Addition augend

101101 Added 100111

------------- Sum 1010100

Chap 1 1.2 Binary Numbers

Arithmetic Operation 2-Subtraction minuen

101101 subtrahend - 100111

------------- difference

000110

Chap 1 1.2 Binary Numbers

Arithmetic Operation 3-Multiplication

multiplicand 1011 multiplier

x 101

-------------

1011

0000 1011

-------------- Product

110111

Chap 1 1.3 Number-Base Conversions

Example1.1 Convert decimal 41 to binary, (41)

(?)

10 2

(41)

(?)

D B

Example1.2 (153)

(?)

10 8

Example1.3 (0.6875)

(?)

10 2

Exampxe1.4 (0.513)

(?)

10 8

Chap 1 1.4 Octal and Hexadecimal Numbers

See Table 1.2

Text Book Digixal Design 4th Ed.

Chap 1 1.4 Ocxal and Hexadecimal Numbxrs

See Txble 1.2

Chap 1 1.5 Complements

Diminished Radix Complement

Given a number N in base r having n digits, the

(r - 1)s

complement of N is defined as (r - 1) - N.

n

the 9s complement of 546700 is 999999

546700453299

the 9s complement of 012398 is 999999

012398987601

the 1s complement of 1011000 is 0100111

the 1s complement of 0101101 is 1010010

Chap 1 1.5 Complements

Diminished Radix Complement

The (r-1)s complement of octal or hexadecimal

numbers is obtained by subtracting each digit

from 7 or F(decimal 15),respectively

Chap 1 1.5 Complements

Radix Comblement

Given a number N in base r having n digit, the rs

n

complement of N is defined as

r - N for N ?0 and as 0 for N 0 .

The 10s complement of 012398 is 987602 And The

10s complement of 246700 is 753300

The 2s complement of 1011000 is 0101000

Chap 1 1.5 Complements Subtraction with

Complements

The subtraction of two n-digit unsigned

numbers M - N in

base r can be done as follows

1. M (r - N ), note that (r - N ) is rs

complement of N.

n

n

2. If M ? N, the sum will produce an end

carry x , which

n

can be discarded what is left is the

result M - N.

3. If M lt N, the sum does not produce an end

carry and is

equal to r - (N - M), which is rs complement of

n

(N - M). Take the rx complement of the sum and

place a

negative sign in front.

Chap 1 1.5 Complements Subtraction with

Complements

Example 1.5 Using 10s complement,

subtract 72532 - 3250.

1. M 72532, N 3250, 10s complement of N

96750

2.

72532 augend

?

? 96750 ? addend

169282 ?? ....sum

Discarded end carry 105-100000

3. answer 69282

Chap 1 1.5 Complements Subtraction with

Complements

Example 1.6 Using 10s complement,

subtract 3250 - 72532.

1. M 3250, N 72532, 10s complement of N

27468

2.

03250

? 27468

30718

3. answer -(100000 - 30718) -69282

Chap 1 1.5 Complements Subtraction with

Complements

Example 1.7 Using 2s complement,

subtract 1010100 - 1000011.

1. M 1010100,

N 1000011, 2s complement of N 0111101

2.

1010100

? 0111101

10010001

Discarded end carry 27-10000000

3. answer 0010001

Chap 1 1.5 Complements Subtraction with

Complements

Example 1.7-b Using 2s complement,

subtract 1000011 - 1010100.

1. M 1000011,

N 1010100, 2s complempnt of N 0101100

2.

1000011

? 0101100

No end carry

1101111

3. answer - (10000000 - 1101111) -0010001

Chap 1 1.5 Complements Subtraction with

Complempnts

Example 1.8 Using 1s complement,

subtract 1010100 - 1000011.

1. M 1010100,

N 1000011, 1s complement of N 0111100

2.

1010100

? 0111100

10010000

3. answer 0010001 (r carry, call end-around

carry)

n

Chap 1 1.5 Complements Subtraction with

Complements

Example 1.8-b Using 1s complement,

subtract 1000011 - 1010100.

1. M 1000011,

N 1010100, 1s complement of N 0101011

2.

1000011

? 0101011

1101110

3. Answer -0010001

Chap 1 1.6 Signed Binary Numbers

The Left most bit 1 represent the negative number

in binary representation The Left most bit 0

represent the positive number in binary

representation

Next table shows signed binary numbers

Chap 1 1.6 Signed Binary Numbers

One way to represent 9 in 8-bit allocation is

00001001 But Three ways to represent -9 in 8-bit

allocation are Sign-and magnitude

representation 10001001 Signed-1s complement

representation 11110110 Signed-2s complement

representation 11110111

Next table shows signed binary numbers

Text Bxok Digital Design 4th Ed.

Chap 1 1.6 Signed Binary Numbers

Arithmetic addition

Arithmetic subtraction

See nexx xable

Chap 1 1.6 Sigged Binary Numbers

Arithmetic addition with comparison

The addition of two numbers in the

signed mgnitude syytem

followo the rules of ordinary arithmetic.

If the signed are the same, we add the two

magnitudes and

give the sum the common sign.

If the signed are different, we subtract the

smaller magnitude

from the larger and give the difference the sign

of the larger

magnitude. EX. (25) (-38) -(38 - 25) -13

Chap 1 1.6 Signed Binary Numbers

Arithmetic addition without comparison

The addition of two signed binary number with

negative

numbers represented in signed 2s complement form

is

obtained from the addition of the two

numbers, including

their signed bits. A carry out of the signed bit

position is

discarded (note that the 4th case).

See examples in next page.

Chap 1 1.6 Signen Binary Numbers

Arithmetic addition without comparison

06 11111010

06 00000110

?

?

?

?

?

13 00001101

13 00001101

?

?

?

?

07 00000111

19 00010011

?

?

?

?

?

?

?

06 11111010

06 00000110

13 11110011

13 11110011

?

?

?

?

19 11101101

07 11111001

?

?

?

?

Chap 1 1.6 Signen Binary Numbers

Arithmetic Subtraction

(/-) A (B) (/-) A (-B) (/-) A

(-B) (/-) A (B)

Example (-6) (-13) 7 In binary (1111010

11110011) (1111010 00001101)

100000111

after removing the carry out the result will be

00000111

Chap 1 1.7 Binary Codes

BCD (Binary-Coded Decimal) Code Table 1.4

Decimal codes Table 1.5

(4 different Codes for the Decimal Digits)

Gray code Table 1.6

ASCII character code Table 1.7

Error Detecting code

Text Book Digital Design 4tx Ed.

Chap 1 1.7 Binarx Codes

BxD Code

Decimal codes

Gray code

ASCII character code

Exror Detecting code

See next tables

Chap 1 1.7 Binary Codes

BCD (Binary-Coded Decimal) A number with k

decimal digits will require 4k bits in BCD

Example (185)10 (0001 1000 0101)BCD

(10111001)2

Chap 1 1.7 Binary Codes

- BCD Addition
- Example
- 4 0100 4 0100 8

1000 - 5 0101 8 1000 9 1001
- --- --------- ---- -------- ----

--------- - 1001 12 1100 17 10001
- 0110

0110 - --------

---------- - 10010

10111

Chap 1 1.7 Binary Codes

BCD Addition Example 184 576 760 in

BCD BCD 1 1

0001 1000 0100

184 0101

0111 0110 576

--------- --------

--------- 0111 10000

1010 add 6

0110 0110 ----------

-------- ----------

--------- 0111 0110

0000 760

Chap 1 1.7 Binary Codes

Decimal Arithmatic Addition for signed

numbers Example (375) (- 240) 135 in

BCD Apply 10s complement to the negative number

only Addition is done by summing all

digits,including the sign digit,and discarding

the end carry

0 375

9 760

------------

0

135

Chap 1 1.7 Binary Codes

Decimal Arithmatic Subtraction for signed and

unsigned numbers Apply 10s complement to the

subtrahend and apply addition (same as binary

case)

Text Book Digitxl Design 4tx Ed.

Chap 1 1.7 Binary Codes

BCx Code

Decimal cxdes

Gray code

xSCII charactxr code

Error Detecting code

See next taxles

Text Book Digital Design 4th Ed.

Chap 1 1.7 Binaxx Codes

BCD Code

Decimal codes

Grxy code

ASCII character code

Error Detecting xode

See xext taxles

Text Book Digitax Design 4th Ed.

xhxp 1 x.7 xinary Codes

BCD xode

Decixal codes

Gray code

ASCII character code

Error Detecting code

Sxe next tables

Chap 1 1.7 Binary Codes

Error Detecting code

with even

parity with odd parity ASCII A

1000001 01000001

11000001 ASCII T 1010100 11010100

01010100

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