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Sistem – Sistem Bilangan,
Operasi dan kode
ENDY SA
Program Studi Teknik Elektro
Fakultas Teknik
Universitas Muhammadiyah Prof. Dr. HAMKA
Program Studi T. Elektro
FT - UHAMKA
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Tujuan Topik Bahasan


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


Mengulas kembali sistem bilangan desimal.
Menghitung dalam bentuk bilangan biner.
Memindahkan dari bentuk bilangan desimal
ke biner dan dalam biner ke dalam desimal.
Penggunaan operasi aritmatika pada
bilangan biner.
Menentukan komplemen 1 dan 2 dari sebuah
bilangan biner.
Dan lain – lainnya……..
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Pendahuluan


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Sistem Biner dan Kode – kode digital
merupakan dasar untuk komputer dan
elektronika digital secara umum.
Sistem bilangan biner seperti desimal,
hexadesimal dan oktal juga dibahas pada
bagian ini.
Operasi aritmatika dengan bilangan biner
akan dibahas untuk memberikan dasar
pengertian bagaimana komputer dan jenis –
jenis perangkat digital lain bekerja.
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Sistem Bilangan
 Desimal
0
~9
 Biner
0
~1
 Oktal
0
~7
 Hexadesimal
0
~F
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Bilangan Desimal

Dalam setiap bilangan desimal terdiri dari 10
digit, 0 sampai dengan 9
Contoh:
Ungkapkan bilangan
desimal 2745.214
sebagai penjumlahan
nilai setiap digit.
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Bilangan Biner


Sistem Bilangan biner merupakan cara lain
untuk melambangkan kuantitas, dimana 1
(HIGH) dan 0 (LOW).
Sistem bilangan biner mempunyai nilai basis
2 dengan nilai setiap posisi dibagi dengan
faktor 2:
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Contoh :
Konversikan seluruh bilangan biner 1101101 ke
desimal
Hasil:
Nilai :
2 6 25 24 23 22 21 20
Biner :
1 1 0 1 1 0 1
1101101 = 26 + 25 + 23 + 22 + 20
Coba ini!!
1111001
= 64 + 32 + 8 + 4 + 1 = 109
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Bilangan
Desimal
Bilangan Biner
22 21 20
0
0
0
0
0
0 0 0
1
0
0
0
1
0 0 1
2
0
0
1
0
0 2 0
3
0
0
1
1
0 2 1
4
0
1
0
0
4 0 0
5
0
1
0
1
4 0 1
6
0
1
1
0
4 2 0
23 22 21 20
8 0 0 0
7
0
1
1
1
4 2 1
8
1
0
0
0
8 0 0 1
9
1
0
0
1
8 0 2 0
10
1
0
1
0
8 0 2 1
11
1
0
1
1
8 4 0 0
8 4 0 1
12
1
1
0
0
13
1
1
0
1
8 4 2 0
14
1
1
1
0
8 4 2 1
15
1
1
1
1
Program Studi T. Elektro
FT - UHAMKA
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Aplikasi Digital
Ilustrasi sebuah penggunaan hitungan
biner sederhana.
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Konversi Desimal ke Biner
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Metode Sum-of-Weight.
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Pengulangan pembagian dengan Metode
bilangan 2.
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Konversi fraksi desimal ke biner.
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Metode Sum-of-Weight
1
0
0
1
Example:
Convert the following decimal
numbers to binary:
Bilangan
desimal
9
sebagai The decimal
number 9, for example,
can be expressed as the
sum of binary weight of:
Program Studi T. Elektro
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a) 12 b) 25 c) 58 d) 82
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1100 11001 111010 101001011
Repeated Division by 2 Method
A systematic method of converting whole numbers from decimal to
binary is the repeated division-by-2 process.
Remainder
Convert the
decimal number
12 to binary
Stop when the
whole-number
quotient
is 0
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12
6
2
6
3
2
3
1
2
1
0
2
0
0
1
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1
1100
MSB
LSB
Convert decimal number
12
39 to binary?
Converting Decimal Fractions
to Binary
0.625 = 0.5 + 0.125 = 2-1 + 2-3 = 0.101
Carry
MSB
LSB
. 101
Stop
0.625 x 2 = 1.25
when the
fractional
0.25 x 2 = 0.50
part is all
zeros
0.50 x 2 = 1.00
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1
0
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1
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Binary Arithmetic
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Binary arithmetic is essential in all digital
computers and in many other types of digital
systems.
Addition, Subtraction, Multiplication, and
Division
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Binary Addition
The four basic rules for adding binary digits (bits) are as follows:
0+0=0
sum of 0 with a carry of 0
0+1=1
sum of 1 with a carry 0f 0
1+0=1
sum of 1 with a carry of 0
1+ 1 = 10
sum of 0 with a carry 0f 1
Program Studi T. Elektro
FT - UHAMKA
1
1
0
1
1
+0
0
1
1
0Slide - 2
Carry
0
Try This:
11 + 11 = ??
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Binary Subtraction
The four basic rules for subtracting bits are as follows:
0–0=0
1–1=0
1–0=1
10 – 1 = 1
0 – 1 with a borrow of 1
1 1 – 0 1 = ??
Try This:
11
- 01
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10
1 0 1 – 0 1 1 = ???
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Binary Multiplication
The four basic rules for multiplying bits are as follows:
0X0=0
0X1=0
1X0=0
1X1=1
1 1 X 1 1 = ??
11
Try This:
X 11
1 1 1 X 1 0 1 = ??
11
+1 1
1001
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Binary Division
Division in binary follows the same procedure as division in
decimal.
1 1 0 ÷ 11 = ??
10
Try This:
11 1 1 0
1 1 0 ÷ 10 = ??
11
000
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1’s and 2’s Complements of
Binary Numbers
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The 1’s and 2’s Complements of Binary
Numbers are very important because they
permit the representation of negative
numbers.
The method of 2’s compliment arithmetic is
commonly used in computers to handle
negative numbers
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FT - UHAMKA
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Finding the 1’s Complement
The 1’s complement of a binary number is found by
changing all 1s to 0s and all 0s to 1s.
Example:
1 0 1 1 0 0 1 0 (Binary Number)
0 1 0 0 1 1 0 1 (1’s Complement)
NOT Gate
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Finding the 2’s Complement
The 2’s complement of a binary number is found by
adding 1 to the LSB of the 1’s complement
Find the 2’s complement of 10110010
+
10110010
(Binary number)
01001101
(1’s complement)
1
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01001110
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(Add 1)
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Alternative Method to find 2’s
Complement
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Start at the right with the LSB and write the
bits as they are up and including the first 1
Take the 1’s complements of the remaining
bits
10111000
(Binary Number)
01001000
(2’s Complement)
1’sProgram
Complements
of original bits
Studi T. Elektro
FT - UHAMKA
These
bits stay the same
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Try This:
10010001
0110111122
Signed Numbers
Digital systems, such as the computer, must be able to
handle both positive and negative numbers. A signed binary
number consists of both sign and magnitude information. The
sign indicates whether a number is positive or negative and
the magnitude is the value of the number. There three forms
in which signed integer (whole) numbers can be represented
in binary:
1. Sign-Magnitude
2. 1’s Complement
3. 2’s Complement
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The Sign Bit
The left-most bit in a signed binary number is the sign bit,
which tells you whether the number is positive or negative.
Sign-Magnitude Form
When a signed binary number is represented in signmagnitude, the left-most bit is the sign bit and the remaining
bits are the magnitude bits. The magnitude bits are in true
(uncomplemented) binary for both positive and negative
numbers.
Decimal number, +25 is expressed as an
8-bit signed binary number using signmagnitude form as:
Program Studi T. Elektro
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Sign Bit
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00011001
Magnitude Bit
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1’s Complement Form
Positive numbers in 1’s complement form are represented
the same way as the positive sign-magnitude numbers.
Negative numbers, however, are the 1’s complements of the
corresponding positive numbers. Example: The decimal
number -25 is expressed as the 1’s complement of +25
(00011001) as (11100110)
2’s Complement Form
In the 2’s complement form, a negative number is the 2’s
complement of the corresponding positive number
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FT - UHAMKA
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Express the decimal number -39 in sign-magnitude, 1’s
complement and 2’s complement
00100111
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00100111
>>>
10100111
00100111
>>>
11011000
00100111
>>>
11011001
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The Decimal Value of Signed
Numbers
Decimal Value of positive and negative
numbers in the sign-magnitude form are
Sign-Magnitude: determined by summing the weights in all
the magnitude bit positions where there
are 1s and ignoring those positions where
there are zeros.
Determine the decimal value of this signed binary
number expressed in sign magnitude: 1 0 0 1 0 1 0 1
26 25 24 23 22 21 20
0 0 1 0 1 0 1 >> 16 + 4 + 1 = 21
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FT - UHAMKA
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The sign bit is 1: Therefore, the decimal number is -21
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The Decimal Value of Signed
Numbers
1’s Complement: Decimal values of negative numbers are
determined by assigning a negative value
to the weight of the sign bit, summing all
the weight where there are 1s and adding
1 to the result
Determine the decimal values of this signed binary
numbers expressed in 1’s complement
00010111
11101000
-27 26 25 24 23 22 21 20
0 0 0 1 0 1 1 1
16 + 4 + 2 + 1 = +23
Program Studi T. Elektro
FT - UHAMKA
-27 26 25 24 23 22 21 20
1 1 1 0 1 0 0 0
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-128 + 64 + 32 + 8 = -24 + 1 =28-23
The Decimal Value of Signed
Numbers
2’s Complement:
The weight of the sign bit in a negative
number is given a negative value
Determine the decimal values of this signed binary
numbers expressed in 1’s complement
01010110
10101010
-27 26 25 24 23 22 21 20
0 1 0 1 0 1 1 0
64 + 16 + 4 + 2 = +86
Program Studi T. Elektro
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-27 26 25 24 23 22 21 20
1 0 1 0 1 0 1 0
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-128 + 32 + 8 + 2 = -86
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Arithmetic Operations with
Signed Number
In this section we will learn how signed
numbers are added, subtracted, multiplied and
divided. This section will cover only on the 2’s
complement arithmetic, because, it widely
used in computers and microprocessor-based
system .
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Addition
00000111
+0 0 0 0 0 1 0 0
7+4
00001011
The Sum is Positive and is therefore in true binary
Discard
Carry
00001111
+1 1 1 1 1 0 1 0
1
Program Studi T. Elektro
FT - UHAMKA
15 + (-6)
00001001
The Final Carry is Discarded.
2
The SumSlide
is -Positive
and is therefore in true binary31
Addition
00010000
+1 1 1 0 1 0 0 0
16 + (-24)
11111000
The Sum is Negative and is therefore in
2’s complement form
Discard
Carry
11111011
+11110111
1
Program Studi T. Elektro
FT - UHAMKA
-5 + (-9)
11110010
The Final Carry is Discarded.
The Sum is Negative and is therefore in
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2’s complement form
Subtraction
To subtract two signed
numbers, take the 2’s
Complement of the subtrahend and ADD. Discard any
final carry bit
00001000 - 00000011
8 – 3 = 8 + (-3) = 5
Discard Cary
Program Studi T. Elektro
FT - UHAMKA
00001000
+ 11111101
1 00000101
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2’s Complement
Difference
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Multiplication
The numbers in a multiplication are the multiplicand, the
multiplier and the product. Direct Addition and Partial
Products are two basic methods for performing multiplication
using addition.
8 X 3 = 24
8 + 8 + 8 = 24
(Decimal)
00001000
+ 00001000
00010000
+ 00001000
00011000
Standard Procedure
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FT - UHAMKA
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Division
The division operation in computers is accomplished using
subtraction. Since subtraction is done with an adder, division
can also be accomplished with an adder. The result of a
division is called the quotient.
Step 1:
Determine the SIGN BIT for both DIVIDEND and DIVISOR
Step 2:
Subtract the DIVISOR from the DIVIDEND using 2’s Complement addition to get
the first partial remainder and ADD 1 to quotient. If ZERO or NEGATIVE the
division is complete.
Step 3:
Subtract the divisor from the partial remainder and ADD 1 to the quotient. If the
result
is POSITIVE repeat Step 2 or If Slide
ZERO
or NEGATIVE the division is
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FT - UHAMKA
complete.
Hexadecimal Numbers
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
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Most digital systems deal with groups of bits in
even powers of 2 such as 8, 16, 32, and 64 bits.
Hexadecimal uses groups of 4 bits.
Base 16
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
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16 possible symbols
0-9 and A-F
Allows for convenient handling of long binary
strings.
Program Studi T. Elektro
FT - UHAMKA
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Hexadecimal Numbers

Convert from hex to decimal by
multiplying each hex digit by its
positional weight.
Example:
16316  1 (16 )  6  (16 )  3 (16 )
 1 256  6 16  3 1
 35510
2
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0
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Hexadecimal Numbers



Convert from decimal to hex by using the
repeated division method used for decimal to
binary and decimal to octal conversion.
Divide the decimal number by 16
The first remainder is the LSB and the last is
the MSB.

Note, when done on a calculator a decimal
remainder can be multiplied by 16 to get the
result. If the remainder is greater than 9, the
letters A through F are used.
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Hexadecimal Numbers

Example of hex to binary conversion:
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Hexadecimal Numbers
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Hexadecimal Numbers
Hexadecimal is useful for representing
long strings of bits.
 Understanding the conversion process
and memorizing the 4 bit patterns for each
hexadecimal digit will prove valuable later.

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BCD
Binary Coded Decimal (BCD) is another
way to present decimal numbers in binary
form.
 BCD is widely used and combines
features of both decimal and binary
systems.
 Each digit is converted to a binary
equivalent.

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BCD

To convert the number 87410 to BCD:
8
7
4
1000 0111 0100 = 100001110100BCD



Each decimal digit is represented using 4
bits.
Each 4-bit group can never be greater than 9.
Reverse the process to convert BCD to
decimal.
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BCD
BCD is not a number system.
 BCD is a decimal number with each digit
encoded to its binary equivalent.
 A BCD number is not the same as a
straight binary number.
 The primary advantage of BCD is the
relative ease of converting to and from
decimal.

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Alphanumeric Codes


Represents characters and functions found on a
computer keyboard.
ASCII – American Standard Code for Information
Interchange.



Seven bit code: 27 = 128 possible code groups
Table 2-4 lists the standard ASCII codes
Examples of use are: to transfer information between
computers, between computers and printers, and for
internal storage.
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Thank You
“ Buku yang selalu dibaca tidak akan
mengumpul habuk dan debu. Berjinaklah
dengan buku kerana ia adalah teman yang
paling berguna menimba ilmu “
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FT - UHAMKA
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