Digital Electronics- Number systems & codes
- 1. F.Y.B.SC.(C.SC.) PRINCIPLES OF DIGITAL ELECTRONICS- NUMBER SYSTEMS -Prof. Vandana Pagar Assistant professor MIT ACSC.
- 2. TOPICS TO BE COVERED ➢Number Systems: 1. Decimal Number System 2. Binary Number System 3. Hexadecimal Number System ➢Representation of Numbers of Different Radix ➢Conversion of Numbers from one Radix to Another Radix
- 3. WHAT IS NUMBER SYSTEM ? •system for representing number of certain type. • Example: –There are several systems for representing the –counting numbers. – These include the usual base “10” or decimal system : 1,2,3 ,…..10,11,12,..99,100,… ,…..10,11,12,..99,100,…
- 4. System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- decimal 16 0, 1, … 9, A, B, … F No No COMMON NUMBER SYSTEM
- 5. Decimal Binary Hexa- decimal 0 0 0 1 1 1 2 10 2 3 11 3 4 100 4 5 101 5 6 110 6 7 111 7 COUNTING
- 6. DECIMAL NUMBER SYSTEM In Decimal number system, an ordered set of ten symbols 0,1,2,3,4,5,6,7,8 and 9 are used to specify the quantities. Symbols used are known as digits. Radix or base of decimal number system is 10
- 8. BINARY NUMBER SYSTEM The binary number system is a code that uses only two basic symbols i.e. 0 and 1 Generally in digital electronics 0 represents low level and 1 represents high level. Symbols in binary number system are called as bits.
- 10. HEXADECIMAL NUMBER SYSTEM Hexadecimal Number System has a base of sixteen. It is extensively used in Microprocessor work It uses 16 distinct symbols 0 to 9 and Ato F 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, F
- 11. BINARY TO DECIMAL CONVERSION: (i) Get the decimal number (ii) Write the weights in exponential form (iii) Place 1 and a 0 at appropriate positions i.e. place a 1 if weight is taken during the sum and 0 otherwise. iv.) Add all the products
- 12. HEXADECIMAL TO DECIMAL CONVERSION: In a hexadecimal number system each digit position corresponds to power of 16. The weight of the digit position in a hexadecimal number is as follows: ▪ In hexadecimal number system each digit position corresponds to appropriate power of 16. The weights of the digit positions in hexadecimal number are as follows: ………163 162 161 160 . 16-1 16-2 16-3 ▪ To convert Hexadecimal number to decimal number, multiply each hexadecimal digit by its weight and add the resulting product.
- 13. DECIMAL TO ANY BASE Steps: 1. Convert integer part ( Successive Division Method ) 2. Convert fractional part ( Successive Multiplication Method )
- 14. DECIMAL TO BINARY CONVERSION Steps in Successive Division Method 1. Divide the integer part of decimal number by desired base number, store quotient (Q) and remainder (R) 2. Consider quotient as a new decimal number and repeat step1 until quotient becomes 0 3. Note down the remainders in the reverse order
- 15. DECIMAL FRACTIONS TO BINARY Multiply Rule is used Multiply bit by two and record the carry in the integer position Note down the integers in forward direction(top to bottom). (0.364) 10 = ( ?)2 0.364*2 = 0.728 0 0.728*2 = 1.456 1 0.456*2 = 0.912 0 0.912*2 =1.824 1 0.824*2= 1.648 1 Ans:(0.364) 10 = ( 0.01011)2
- 17. HEXADECIMAL TO BINARY To convert a hexadecimal number to a binary number, first write each hexadecimal digit to its 4-bit equivalent using the binary codes, and then write the binary numbers without a gap. Example : Convert (9A)16 to binary equivalent. Solution :- (9 A) (1001 1010) 2 Thus (9A)16 = (1001 1010) 2
- 18. BINARY TO HEXADECIMAL CONVERSION (BIN TO HEX) : Converting a binary to hexadecimal is a straight forward procedure. Simply break the binary number into four-bit groups starting at binary point, and replace each group with the equivalent hexadecimal symbol. Example : Convert (101101)2 to Hexadecimal equivalent 10 1101 2 D (101101)2=(2D)16
- 19. BINARY ADDITIONS Binary numbers are added like decimal numbers. In decimal, when numbers sum more than 9 a carry results. In binary when numbers sum more than 1 a carry takes place. 0 + 0 = 0 1 + 0 = 1 1 + 1 = 0 + carry 1 1 + 1 + 1 = 1 + carry 1
- 20. BINARY CODED DECIMAL (BCD) CODE:- The binary coded decimal (BCD) code is a weighted code. This code is found very convenient for representing digits. Each group of four bits is used to represent one decimal digit. It is also called as 8421 code. This code consists of four bits which have the weights as 8 4 2 1 . The four bit combination that represents the decimal digits 0 to 9 are shown in table 1.4.
- 21. BCD CODE
- 22. BINARY SUBTRACTION: To perform a binary subtraction you first have to represent the number to be subtracted in its negative form. This is known as its two's complement. The two's complement of a binary number is obtained by: 1. Replacing all the 1s with 0s and the 0s with 1s. This is known as its one's complement. 2. Adding 1 to this number by the rules of binary addition. Now you have the two's complement. Example: The decimal subtraction 29 - 7 = 22 is the same as adding (29) + (-7) = 22 1. Convert the number to be subtracted to its two's complement:
- 23. 2’S COMPLEMENT SUBTRACTION 00000111 (decimal 7) 11111000 (one's complement) + 00000001 (add 1) 11111001 (two's complement) 29 00011101 +-7 11111001 22 (1)00010110
- 24. NON-WEIGHTED CODES:- Non-weighted binary codes do not follow the positional weight principle. The example of non- weighted codes is gray codes, excess-3 code and alphanumeric code etc. Alphanumeric Code:- In communication along with ordinary number, we need some characters, punctuation marks and also control signals. The code we have studied so far is not enough for data communication. For overcoming this problem binary symbols are used for representing number, alphabetic character and special symbols. The binary codes which are uses to represents number, alphabetic character and special symbol are called alphanumeric code. ASCII EBCDIC
- 25. ASCII Code The ASCII code is extensively used for data communication and in computers. It is a 7-bit code, so it can represents 27 or 128 different characters. It can represent decimal number 0-9, letters of alphabets in lower case and upper case, special symbol and control instructions. Each symbol has 7- bit code which is made up of a 3-bit group followed by a 4-bit group. The number is represented by 8421 code.
- 27. GRAY CODE: Binary to Gray Converter: In this conversion the most significant bit (MSB) is copied as it is, so these represents MSB of gray code, then going from left to right add each adjacent pair of binary digits to get the next gray code. After the addition forget the about carry.
- 28. GRAY CODE
- 29. GRAY TO BINARY CODE CONVERSION In this conversion first write MSB bit as it is of a gray code, this one is an MSB bit of binary code, and then add each MSB of binary code to the next bit of gray code. If carry is generated then neglect the carry.
- 30. REFERENCE BOOKS: 1. Digital Fundamentals: Floyd T.M., Jain R.P., Pearson Education 2. Digital Electronics: Jain R.P., Tata McGraw Hill 3. Digital Principles and Applications: Malvino Leach, Tata McGraw-Hill 4. M.Morris Mano, “ Digital Design “ 3rdEdition, PHI, NewDelhi.