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

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

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

Sistem Bilangan Desimal Biner Oktal Hexadesimal 0 ~ 9 0 ~ 1 0 ~ 7 0 ~ F Program Studi T. Elektro FT - UHAMKA Slide - 2

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

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

Contoh : Konversikan seluruh bilangan biner 1101101 ke desimal Coba ini!! 1111001 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 Program Studi T. Elektro FT - UHAMKA Slide - 2

Bilangan Desimal Bilangan Biner 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 22 21 20 0 0 0 0 0 1 0 2 0 0 2 1 4 0 0 4 0 1 4 2 0 4 2 1 23 22 21 20 8 0 0 0 8 0 0 1 8 0 2 0 8 0 2 1 8 4 0 0 8 4 0 1 8 4 2 0 8 4 2 1 Program Studi T. Elektro FT - UHAMKA Slide - 2

Ilustrasi sebuah penggunaan hitungan Aplikasi Digital Ilustrasi sebuah penggunaan hitungan biner sederhana. Program Studi T. Elektro FT - UHAMKA Slide - 2

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

Metode Sum-of-Weight 1 0 0 1 Example: 1 0 0 1 Example: Convert the following decimal numbers to binary: a) 12 b) 25 c) 58 d) 82 Bilangan desimal 9 sebagai The decimal number 9, for example, can be expressed as the sum of binary weight of: 1100 11001 111010 1010010 Program Studi T. Elektro FT - UHAMKA Slide - 2

Repeated Division by 2 Method A systematic method of converting whole numbers from decimal to binary is the repeated division-by-2 process. Remainder Example Convert the decimal number 12 to binary MSB LSB Stop when the whole-number quotient is 0 Convert decimal number 39 to binary? Program Studi T. Elektro FT - UHAMKA Slide - 2

Converting Decimal Fractions to Binary Sum-of-Weight 0.625 = 0.5 + 0.125 = 2-1 + 2-3 = 0.101 Repeated Multiplication by 2 MSB LSB Carry . 1 0 1 0.625 x 2 = 1.25 0.25 x 2 = 0.50 0.50 x 2 = 1.00 1 Stop when the fractional part is all zeros Program Studi T. Elektro FT - UHAMKA Slide - 2

Binary Arithmetic Binary arithmetic is essential in all digital computers and in many other types of digital systems. Addition, Subtraction, Multiplication, and Division Program Studi T. Elektro FT - UHAMKA Slide - 2

Binary Addition Example: 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 1 1 0 1 1 + 0 0 1 1 0 0 Carry Example: Try This: 11 + 11 = ?? Program Studi T. Elektro FT - UHAMKA Slide - 2

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

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

Binary Division Example: 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 Example: Try This: 1 1 0 ÷ 10 = ?? Program Studi T. Elektro FT - UHAMKA Slide - 2

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

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 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 2's Complement = (1's Complement) + 1 Find the 2’s complement of 10110010 10110010 (Binary number) + 01001101 (1’s complement) 1 (Add 1) 01001110 Example: Program Studi T. Elektro FT - UHAMKA Slide - 2

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 10111000 (Binary Number) 01001000 (2’s Complement) Try This: 10010001 Example: These bits stay the same 1’s Complements of original bits 01101111 Program Studi T. Elektro FT - UHAMKA Slide - 2

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 Program Studi T. Elektro FT - UHAMKA Slide - 2

0 = Positive Number and 1 = Negative Number 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: 00011001 Example: Magnitude Bit Program Studi T. Elektro FT - UHAMKA Slide - 2 Sign Bit

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

Example: Sign-Magnitude: 1's Complement: 2's Complement: 00100111 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 Program Studi T. Elektro FT - UHAMKA Slide - 2

The Decimal Value of Signed Numbers 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. Sign-Magnitude: Determine the decimal value of this signed binary number expressed in sign magnitude: 1 0 0 1 0 1 0 1 Example: 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 Program Studi T. Elektro FT - UHAMKA Slide - 2

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 Example: 00010111 11101000 -27 26 25 24 23 22 21 20 -27 26 25 24 23 22 21 20 0 0 0 1 0 1 1 1 1 1 1 0 1 0 0 0 Program Studi T. Elektro FT - UHAMKA 16 + 4 + 2 + 1 = +23 Slide - 2 -128 + 64 + 32 + 8 = -24 + 1 = -23

The Decimal Value of Signed Numbers The weight of the sign bit in a negative number is given a negative value 2’s Complement: Determine the decimal values of this signed binary numbers expressed in 1’s complement Example: 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 Program Studi T. Elektro FT - UHAMKA 64 + 16 + 4 + 2 = +86 Slide - 2 -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 . Program Studi T. Elektro FT - UHAMKA Slide - 2

Addition 0 0 0 0 0 1 1 1 +0 0 0 0 0 1 0 0 Both Number Positive: 7 + 4 0 0 0 0 1 0 1 1 The Sum is Positive and is therefore in true binary Positive Number with Magnitude Larger than Negative Number: Discard Carry 0 0 0 0 1 1 1 1 +1 1 1 1 1 0 1 0 15 + (-6) 1 0 0 0 0 1 0 0 1 The Final Carry is Discarded. The Sum is Positive and is therefore in true binary Program Studi T. Elektro FT - UHAMKA Slide - 2

Addition 0 0 0 1 0 0 0 0 +1 1 1 0 1 0 0 0 Negative Number with +1 1 1 0 1 0 0 0 Negative Number with Magnitude Larger than Positive Number: 16 + (-24) 1 1 1 1 1 0 0 0 The Sum is Negative and is therefore in 2’s complement form Discard Carry 1 1 1 1 1 0 1 1 + 1 1 1 1 0 1 1 1 Both Number Negative: -5 + (-9) 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 Program Studi T. Elektro FT - UHAMKA Slide - 2

Subtraction Example: Solution: 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 0 0 0 0 0 1 0 1 + 2’s Complement 1 Discard Cary Difference Program Studi T. Elektro FT - UHAMKA Slide - 2

Multiplication Direct Addition: Partial Product: 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: 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 8 X 3 = 24 + 8 + 8 + 8 = 24 (Decimal) + Partial Product: Standard Procedure Program Studi T. Elektro FT - UHAMKA Slide - 2

Division See Example 2-23 Page: 71 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. See Example 2-23 Page: 71 Program Studi T. Elektro FT - UHAMKA Slide - 2

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

Hexadecimal Numbers Convert from hex to decimal by multiplying each hex digit by its positional weight. Example: Program Studi T. Elektro FT - UHAMKA Slide - 2

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

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

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

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. 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

BCD To convert the number 87410 to BCD: 8 7 4 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. Program Studi T. Elektro FT - UHAMKA Slide - 2

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

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

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