Sistem –
Sistem Bilangan, Operasi dan kode
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- Tujuan Topik Bahasan
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Mengulas kembali sistem bilangan desimal.
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Menghitung dalam bentuk bilangan
biner.
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Memindahkan dari bentuk bilangan
desimal ke biner dan dalam biner ke dalam desimal.
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Penggunaan operasi aritmatika pada
bilangan biner.
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Menentukan komplemen 1 dan 2 dari sebuah
bilangan biner Dan lain – lainnya……..
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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.
- - 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:
Contoh :
Konversikan
seluruh bilangan biner 1101101 ke desimal
Hasil:
Nilai : 26 25 24 23 22 21 20
Biner
: 1 1 0 1 1
0 1
1101101 = 26 +
25 + 23 + 22 + 20
= 64 + 32 + 8 + 4 + 1 = 109
- - Konversi Desimal ke Biner
Metode
Sum-of-Weight.
Pengulangan
pembagian dengan Metode bilangan 2.
Konversi fraksi desimal ke biner.
- - Metode Sum-of-Weight
Example:
Bilangan
desimal 9 sebagai
Convert
the following decimal
The decimal
number 9, for
numbers
to binary:
example,
can be expressed as
a)
12 b) 25 c) 58 d) 82
the sum of
binary weight of: 1100 11001 111010
1010010
Binary
Arithmetic
l Binary
arithmetic is essential in all digital computers and in many other types of
digital systems.
l Addition,
Subtraction, Multiplication, and Division
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
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
Binary Multiplication
The four basic rules for multiplying bits are as follows:
0 X 0 = 0
0 X 1 = 0
1 X 0 = 0
1 X 1 = 1
Binary Division
Division in binary follows the same procedure as division in decimal.
1’s and 2’s Complements of Binary Numbers
l The
1’s and 2’s Complements of Binary Numbers are very important because they
permit the representation of negative numbers.
l The method
of 2’s compliment arithmetic is commonly used in computers to handle negative
numbers
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)
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
2's Complement = (1's Complement) + 1
Alternative Method to find
2’s Complement
l Start
at the right with the LSB and write the bits as they are up and including the
first 1
l Take the
1’s complements of the remaining bits
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:
- Sign-Magnitude
- 1’s Complement
- 2’s Complement
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.
0 = Positive Number and 1 = Negative Number
Sign-Magnitude Form
When a signed binary number is represented in sign-magnitude, 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
sign-magnitude form as:
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
Example:
Express the decimal number -39
in sign-magnitude, 1’s complement and 2’s complement
00100111
Sign-Magnitude: 00100111
>>> 10100111
1's Complement: 00100111
>>> 11011000
2's
Complement: 00100111
>>> 11011001
The Decimal Value of Signed
Numbers
Sign-Magnitude: Decimal Value of positive and negative numbers in
the sign-magnitude form are determined by summing the weights in all the
magnitude bit positions where there are 1s and ignoring those positions where there
are zeros.
Example: 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
The sign bit is 1: Therefore, the decimal number is -21
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
Example: Determine
the decimal values of this signed binary numbers expressed in 1’s complement
01010110 10101010
-27 26 25 24 23 22 21 20 -27
26 25 24 23 22 21 20
0
1 0 1 0 1 1
0
1 0 1 0 1 0
1 0
64 + 16 + 4 + 2 = +86
-128 + 32 + 8 + 2
= -86
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 .
Addition
Both
Number Positive: 0 0 0 0 0 1 1 1 7
+ 4
+ 0 0 0 0
0 1 0 0
0
0 0 0 1 0 1 1
The
Sum is Positive and is therefore in true binary
Positive
Number with
Discard 0
0 0 0 1 1 1 1 15
+ (-6)
Magnitude
Larger
Carry
+1 1 1 1 1 0 1 0
Negative
Number:
1 0
0 0 0 1 0 0 1
The
Final Carry is Discarded.
The Sum is Positive and is therefore in true binary
Negative
Number
with 0
0 0 1 0 0 0 0 16
+ (-24)
Magnitude
Larger
than +1
1 1 0 1 0 0 0
Positive
Number: 1
1 1 1 1 0 0 0
The
Sum is Negative and is therefore in
2’s complement form
Both
Number Negative: Discard
1 1 1 1 1 0 1 1 -5
+ (-9)
Carry
+ 1 1 1 1 0 1 1 1
1 1 1 1 1 0 0 1 0
The Final Carry is Discarded. The Sum is Negative and is therefore in 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
Example: 0
0 0 0 1 0 0 0 - 0 0 0 0 0 0 1 1
8
– 3 = 8 + (-3) = 5
Solution:
0 0 0 0 1 0 0 0
+ 1 1 1 1 1 1
0 1
> 2’s Complement
Discard Cary 1 0 0 0 0 0 1 0 1
> Difference
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.
Direct
Addition: 8 X 3 = 24
0 0 0 0 1 0 0 0
8
+ 8 + 8 = 24 + 0 0 0 0 1 0 0 0
(Decimal) 0
0 0 1 0 0 0 0
+ 0 0 0 0 1 0 0 0
0
0 0 1 1 0 0 0
Partial
Product: Standard Procedure
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 ZERO or NEGATIVE the division
is complete.
Hexadecimal
Numbers
l Most
digital systems deal with groups of bits in even powers of 2 such as 8, 16, 32,
and 64 bits.
l Hexadecimal
uses groups of 4 bits.
l Base
16
l 16
possible symbols
l 0-9
and A-F
l Allows
for convenient handling of long binary strings.
l Convert
from hex to decimal by multiplying each hex digit by its positional weight.
Example:
l Convert
from decimal to hex by using the repeated division method used for decimal to
binary and decimal to octal conversion.
l Divide
the decimal number by 16
l The
first remainder is the LSB and the last is the MSB.
l 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.
l Example
of hex to binary conversion:
l Hexadecimal
is useful for representing long strings of bits.
l Understanding
the conversion process and memorizing the 4 bit patterns for each hexadecimal
digit will prove valuable later.
BCD
l Binary
Coded Decimal (BCD) is another way to present decimal numbers in binary form.
l BCD
is widely used and combines features of both decimal and binary systems.
l Each
digit is converted to a binary equivalent.
l To
convert the number 87410 to BCD:
l
8
7 4
l
1000 0111 0100 = 100001110100BCD
l Each
decimal digit is represented using 4 bits.
l Each
4-bit group can never be greater than 9.
l Reverse
the process to convert BCD to decimal.
l BCD
is not a number system.
l BCD
is a decimal number with each digit encoded to its binary equivalent.
l A
BCD number is not the same as a straight binary number.
l The
primary advantage of BCD is the relative ease of converting to and from
decimal.
Alphanumeric
Codes
l Represents
characters and functions found on a computer keyboard.
l ASCII
– American Standard Code for Information Interchange.
l Seven
bit code: 27 = 128 possible code groups
l Table
2-4 lists the standard ASCII codes
l Examples
of use are: to transfer information between computers, between computers
and printers, and for internal storage.
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