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 Slide - 2 1

Tujuan Topik Bahasan

 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……..

Program Studi T. Elektro FT - UHAMKA Slide - 2 2

Pendahuluan

 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. Program Studi T. Elektro FT - UHAMKA Slide - 2 3

Sistem Bilangan

  

Desimal Biner Oktal



0 ~ 9



0 ~ 1



0 ~ 7



Hexadesimal

Program Studi T. Elektro FT - UHAMKA Slide - 2 

0 ~ F

4

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.

Program Studi T. Elektro FT - UHAMKA Slide - 2 5

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: Program Studi T. Elektro FT - UHAMKA Slide - 2 6

Contoh :

Konversikan seluruh bilangan biner 1101101 ke desimal Hasil: Nilai : 2 6 2 5 2 4 2 3 2 2 2 1 2 0 Biner : 1 1 0 1 1 0 1 1101101 = 2 6 + 2 5 + 2 3 + 2 2 + 2 0 = 64 + 32 + 8 + 4 + 1 = 109 Program Studi T. Elektro FT - UHAMKA Slide - 2 Coba ini!!

1111001 7

Bilangan Desimal

0 1 2 3 4 5 2 3 2 2 2 1 2 0 8 0 0 0 8 0 0 1 8 0 2 0 6 7 8 9 10 8 0 2 1 11 8 4 0 0 8 4 0 1 12 13 8 4 2 0 14 8 4 2 1 15 Program Studi T. Elektro FT - UHAMKA 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 Bilangan Biner 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 Slide - 2 0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 2 2 2 1 2 0 0 0 0 0 0 1 0 2 0 0 2 1 4 0 0 4 0 1 4 2 0 4 2 1 8

Aplikasi Digital

Ilustrasi sebuah penggunaan hitungan biner sederhana.

Program Studi T. Elektro FT - UHAMKA Slide - 2 9

Konversi Desimal ke Biner

 Metode Sum-of-Weight.

 Pengulangan pembagian dengan Metode bilangan 2.  Konversi fraksi desimal ke biner.

Program Studi T. Elektro FT - UHAMKA Slide - 2 10

Metode Sum-of-Weight

1 0 0 1

Bilangan sebagai desimal The 9 decimal number 9, for example, can be expressed as the sum of binary weight of: Program Studi T. Elektro FT - UHAMKA Slide - 2 Example: Convert the following decimal numbers to binary: a) 12 b) 25 c) 58 d) 82 1100 11001 111010 1010010 11

Repeated Division by 2 Method

A systematic method of converting whole numbers from decimal to binary is the

repeated division-by-2

process.

Remainder

0 12  6 2 Convert the decimal number 12 to binary 6 2  3 3  1 Stop when the 2 whole-number 1 quotient is 0 Program Studi T. Elektro FT - UHAMKA 2  0 0 1 Slide - 2 1 1100

MSB

39 to binary?

LSB

Converting Decimal Fractions to Binary

0.625 = 0.5 + 0.125 = 2 -1 + 2 -3 =

0.101

Carry Stop when the fractional part is all FT - UHAMKA 0.625 x 2 = 1.25

0.25 x 2 = 0.50

0.50 x 2 = 1.00

1 0 Slide - 2 1 MSB . 1 0 1 LSB 13

Binary Arithmetic

 Binary arithmetic is essential in all digital computers and in many other types of digital systems.

 Addition, Division Subtraction, Multiplication, and 14 Program Studi T. Elektro FT - UHAMKA Slide - 2

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 1 + 0 = 1 1+ 1 = 10 sum of 1 with a carry 0f 0 sum of 1 with a carry of 0 sum of 0 with a carry 0f 1 Program Studi T. Elektro FT - UHAMKA 1 0 + 0 1 1 1 0 0 Slide - 2 1 1 0 Carry Try This: 11 + 11 = ??

15

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 = ??

1 1 - 0 1 Program Studi T. Elektro FT - UHAMKA 1 0 Slide - 2 Try This: 1 0 1 – 0 1 1 = ???

16

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 1 1 X 1 1 = ??

1 1 X 1 1 Try This: 1 1 1 X 1 0 1 = ??

1 1 +1 1 Program Studi T. Elektro FT - UHAMKA 1 0 0 1 Slide - 2 17

Binary Division

Division in binary follows the same procedure as division in decimal.

1 1 0 ÷ 11 = ??

1 0 11 1 1 0 1 1 0 0 0 Try This: 1 1 0 ÷ 10 = ??

Program Studi T. Elektro FT - UHAMKA Slide - 2 18

1’s and 2’s Complements of Binary Numbers

  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 Program Studi T. Elektro FT - UHAMKA Slide - 2 19

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

20 Program Studi T. Elektro FT - UHAMKA Slide - 2

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 Program Studi T. Elektro FT - UHAMKA Find the 2’s complement of 10110010 + 10110010 01001101 (Binary number) (1’s complement) 1 (Add 1) 21

Alternative Method to find 2’s Complement

  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 1011 1000 0100 1000 (Binary Number) (2’s Complement) These bits stay the same Slide - 2 Try This: 10010001

01101111

22 FT - UHAMKA

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 Program Studi T. Elektro FT - UHAMKA Slide - 2 23

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 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.

Program Studi T. Elektro FT - UHAMKA

Sign Bit

Decimal number, +25 is expressed as an 8-bit signed binary number using sign magnitude form as:

Magnitude Bit

Slide - 2 00011001 24

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 Program Studi T. Elektro FT - UHAMKA Slide - 2 25

Express the decimal number -39 in sign-magnitude, 1 ’s complement and 2 ’s complement 00100111 00100111 >>> 1 0100111 00100111 >>> 11011000 00100111 >>> 11011001 Slide - 2 26 Program Studi T. Elektro FT - UHAMKA

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.

Determine the decimal value of this signed binary number expressed in sign magnitude: 1 0 0 1 0 1 0 1 2 6 2 5 2 4 2 3 2 2 2 1 2 0 0 0 1 0 1 0 1 >> 16 + 4 + 1 = 21 Program Studi T. Elektro FT - UHAMKA Slide - 2 The sign bit is 1: Therefore, the decimal number is

-21

27

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 -2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 0 0 0 1 0 1 1 1 16 + 4 + 2 + 1 =

+23

FT - UHAMKA Slide - 2 -2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 1 1 1 0 1 0 0 0 -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 -2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 0 1 0 1 0 1 1 0 64 + 16 + 4 + 2 =

+86

FT - UHAMKA Slide - 2 -2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 1 0 1 0 1 0 1 0 -128 + 32 + 8 + 2 =

-86

29

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 .

Program Studi T. Elektro FT - UHAMKA Slide - 2 30

Program Studi T. Elektro FT - UHAMKA

Addition

0 0 0 0 0 1 1 1 +0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 7 + 4 The Sum is Positive and is therefore in true binary Discard Carry 1 0 0 0 0 1 1 1 1 +1 1 1 1 1 0 1 0 0 0 0 0 1 0 0 1 The Final Carry is Discarded. 15 + (-6) 31

Program Studi T. Elektro FT - UHAMKA

Addition

0 0 0 1 0 0 0 0 +1 1 1 0 1 0 0 0 1 1 1 1 1 0 0 0 16 + (-24) The Sum is Negative and is therefore in 2’s complement form Discard Carry 1 1 1 1 1 1 0 1 1 + 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 -5 + (-9) The Final Carry is Discarded.

The Sum is Negative and is therefore in 2’s complement form 32

Subtraction

To subtract two signed Complement numbers, take the of the subtrahend and ADD .

Discard final carry bit 2 ’s any 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 8 – 3 = 8 + (-3) = 5 Discard Cary Program Studi T. Elektro FT - UHAMKA + 1 0 0 0 0 1 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 Slide - 2 2’s Complement Difference 33

Multiplication

The numbers in a multiplication are the multiplicand , the multiplier Products and the product .

Direct Addition and Partial are two basic methods for performing multiplication using addition.

Program Studi T. Elektro FT - UHAMKA 8 X 3 = 24 8 + 8 + 8 = 24 (Decimal) + + 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 Standard Procedure Slide - 2 34

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 Program Studi T. Elektro complete. Step 2 or If ZERO or NEGATIVE the division is Slide - 2 35

Hexadecimal Numbers

 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  16 possible symbols  0-9 and A-F  Allows for convenient handling of long binary strings.

Program Studi T. Elektro FT - UHAMKA Slide - 2 36

Hexadecimal Numbers

 Convert from hex to decimal by multiplying each hex digit by its positional weight.

Example: 163 16  1  ( 16 2 )  6  ( 16 1 )  3  ( 16 0 )  1   256 355 10  6  16  3  1 Program Studi T. Elektro FT - UHAMKA Slide - 2 37

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.

Program Studi T. Elektro FT - UHAMKA Slide - 2 38

Hexadecimal Numbers

 Example of hex to binary conversion: Program Studi T. Elektro FT - UHAMKA Slide - 2 39

Hexadecimal Numbers

Program Studi T. Elektro FT - UHAMKA Slide - 2 40

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.

41 Program Studi T. Elektro FT - UHAMKA Slide - 2

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.

Program Studi T. Elektro FT - UHAMKA Slide - 2 42

BCD

 To convert the number 874 10 to BCD: 8 7 4 1000 0111 0100 = 100001110100 BCD  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 Slide - 2 FT - UHAMKA 43

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.

Program Studi T. Elektro FT - UHAMKA Slide - 2 44

Alphanumeric Codes

  Represents characters and functions found on a computer keyboard.

ASCII – American Standard Code for Information Interchange.

 Seven bit code: 2 7 = 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 “ Program Studi T. Elektro FT - UHAMKA Slide - 2 46