chap 1 digi g al systems and binary numbers 1.1 digita l systems 1.2 binary numbers 1.3 number-base...

Chap 1 Digigal Systems and Binary Numbers

1.1 Digital Systems1.2 Binary Numbers1.3 Number-Base Conversions1.4 Octal andhexadecimal Numbers 1.5 Complements1.6 Signed Binary Numbers 1.7 Binary Codes

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Chap 1 1.2 Binary Numbers

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

coefficients multiplied by powers of r:

an n-1 1 0 -1 -2 -mr +an r +…+an-1 r +a1 +a r +a-1 r +…+a-2 r-m

r is called base or radix.

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

coefficienxs multiplied by powers ofr:

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

r is called base or radix.

r +an r +…+an-1 r +a1 +a r xa-1 r +…+a-2 r-m

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Arithmetic Operation

1-Addition augend 101101Added: + 100111 -------------Sum: 1010100


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Arithmetic Operation

2-Subtraction minuen: 101101subtrahend: - 100111 -------------difference: 000110


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Arithmetic Operation3-Multiplication multiplicand: 1011multiplier: 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= (?)

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Chap 1 1.4 Octal and Hexadecimal Numbers

See Table 1.2

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Text Book: Digixal Design 4th Ed.

Chap 1 1.4 Ocxal and Hexadecimal Numbxrs

See Txble 1.2

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Chap 1 1.5 Complements

Diminished Radix ComplementGiven a number N in base r having ndigits, the (r - 1)’s

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

the 1’s complement of 1011000 is 0100111

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the 9’s complement of 012398 is 999999 – 012398=987601

the 9’s complement of 546700 is 999999 – 546700=453299

the 1’s complement of 0101101 is 1010010


Diminished Radix Complement

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The (r-1)’s complement of octal or hexadecimal numbers is obtained by subtracting each digit from 7 or F(decimal 15),respectively


Radix Comblement

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The 10’s complement of 012398 is 987602AndThe 10’s complement of 246700 is 753300

Given a number N in base r having ndigit, the r’s

complement of N is defined asr - N for N ≠0 and as 0 for N = 0 .n

The 2’s complement of 1011000 is 0101000

Chap 1 1.5 Complements— Subtraction withComplements

The subtraction of twon-digit unsigned numbers M - N in

base r can be done as follows:

1. M + (r - N ), note that (r - N ) is r’s complement of N.n n

2. If M N, the sum will produce an end carryx , whichcan be discarded; what is left is the resultM -N.

n

3. If M < N, the sum does not produce an end carry and isequal to r - (N - M), which is r’s complement ofn

(N - M). Take the r’xcomplement of the sum and place anegative sign in front.

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Chap 1 1.5 Complements—Subtraction withComplements

Example 1.5 Using 10’s complement,

subtract 72532 - 3250.

1. M = 72532, N = 3250, 10’s complement of N = 96750

2.

3. answer: 69282

72532 augend 96750 addend

169282 ....sum

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Discarded end carry 105=-100000



subtract 3250 - 72532.

1. M = 3250, N = 72532, 10‘s complement of N = 27468

2.

3. answer: -(100000 - 30718) = -69282

03250 27468

30718

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subtract 1010100 - 1000011.

1. M = 1010100,

N = 1000011, 2’s complement ofN = 0111101

2.

3. answer: 0010001

1010100 0111101

10010001

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Discarded end carry 27=-10000000


Example 1.7-b Using 2’s complement,

subtract 1000011 - 1010100.

1. M = 1000011,

N = 1010100, 2’s complempnt ofN = 0101100

2.

3. answer: - (10000000 - 1101111) = -0010001

1000011 0101100

1101111

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No end carry

Chap 1 1.5 Complements— Subtraction withComplempnts


subtract 1010100 - 1000011.

1. M = 1010100,

N = 1000011, 1’s complement of N = 0111100

2.

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

1010100 0111100

10010000

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Chap 1 1.5 Complements—Subtraction withComplements

Example 1.8-b : Using 1’s complement,

subtract 1000011 - 1010100.

1. M = 1000011,

N = 1010100, 1’s complement of N = 0101011

2.

3. Answer: -0010001

1000011 0101011

1101110

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Chap 1 1.6 Signed Binary Numbers

Next table shows signed binary numbers

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The Left most bit 1 represent the negative number in binary representationThe Left most bit 0 represent the positive number in binary representation


Next table shows signed binary numbers

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One way to represent +9 in 8-bit allocation is :00001001ButThree ways to represent -9 in 8-bit allocation are:Sign-and magnitude representation: 10001001Signed-1’s complement representation: 11110110Signed-2’s complement representation: 11110111

Text Bxok: Digital Design 4th Ed.Chap 1 1.6 Signed Binary Numbers

Arithmetic addition

Arithmetic subtraction

See nexxxable

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Chap 1 1.6 Sigged Binary Numbers

Arithmetic additionwith comparison:

The addition of two numbers in the signed mgnitude syytemfollowo the rules of ordinary arithmetic.

If the signed are the same, we add the two magnitudes andgive the sum thecommon sign.

If the signed are different, we subtract the smaller magnitudefrom the larger and give the difference the sign of the largermagnitude. EX. (+25) + (-38) = -(38 - 25) = -13

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Arithmetic addition without comparison:

The addition of two signed binary number with negativenumbers represented in signed 2’s complement form is

obtained from the addition of the two numbers, includingtheir signed bits. A carry out of the signed bit position isdiscarded (note that the 4th case).

See examples in next page.


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Chap 1 1.6 Signen Binary Numbers

Arithmetic addition without comparison:

19 1110110113 1111001106 11111010

07 1111100113 1111001106 00000110

07 00000111

13 0000110106 11111010

19 0001001113 0000110106 00000110

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Chap 1 1.6 Signen Binary Numbers

Arithmetic Subtraction

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(+/-) A – (+B)= (+/-) A + (-B) (+/-) A – (-B)= (+/-) A + (+B)

Example; (-6) – (-13)= +7In 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

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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

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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

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BCD Addition

Example:

4 0100 4 0100 8 1000+5 +0101 +8 +1000 +9 +1001--- --------- ---- -------- ---- ---------91001 12 1100 17 10001 + 0110 + 0110 -------- ---------- 10010 10111 31


BCD AdditionExample: 184+ 576 = 760 in BCDBCD 1 1 0001 1000 0100 184 +0101 0111 0110 +576 --------- -------- --------- 0111 10000 1010 add 6 + 0110 + 0110 ---------- -------- ---------- --------- 0111 0110 0000 760

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Decimal ArithmaticAddition for signed numbersExample: (+375) + (- 240) = + 135 in BCD

Apply 10‘s complement to the negative number onlyAddition is done by summing all digits,including the sign digit,and discarding the end carry 0 375 +9 760 ------------ 0 135 33


Decimal ArithmaticSubtraction for signed and unsigned numbers

Apply 10‘s complement to the subtrahend and apply addition (same as binary case)

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Text Book: Digitxl Design 4tx Ed.Chap 1 1.7 Binary Codes

BCx Code

Decimal cxdes

Gray code

xSCII charactxr code


See next taxles

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Text Book: Digital Design 4th Ed.Chap 1 1.7 BinaxxCodes

BCD Code

Decimal codes

Grxy code


Error Detecting xode

See xext taxles

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Text Book: DigitaxDesign 4th Ed.xhxp 1 x.7 xinary Codes

BCD xode

Decixal codes

Gray code



Sxe next tables

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with even parity with odd parityASCII A 1000001 01000001 11000001ASCII T 1010100 11010100 01010100

chap 1 digi g al systems and binary numbers 1.1 digita l systems 1.2 binary numbers 1.3 number-base...

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