Number and codes in digital systems


 Lydia Bruce
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1 Number and codes in digital systems Decimal Numbers You are familiar with the decimal number system because you use them everyday. But their weighted structure is not understood. In the decimal number system each of the ten digits, 0 through 9, represents a certain quantity (or weight ). These are base TEN numbers. Consider x x x x 1 5x10 + 6x10 + 7x10 + 8x10 and for x10 + 8x1 + 2 x x x10 + 8x10 + 2x10 + 5x10 Problem : Express the following as the sum of values of each digit = = = Binary Numbers The binary number system is another way of counting and it is simpler than the decimal system, since it has only two digits( 0 and 1). These are base TWO numbers. Consider 1011 (binary) = and for (binary) = x8 + 0x4 + 1x2 + 1x1 1x2 + 0x2 + 1x2 + 1x2 B431 Principles of Digital Systems : Hassan Parchizadeh Page 1
2 1x2 + 0x1 + 0x x x x2 + 0x2 + 0x2 + 0x2 + 1x2 Problem : Express the following as the sum of values of each digit = = = Decimal To Binary Conversion We can convert a decimal number into a binary equivalent by dividing the decimal number successively by 2 and noting the remainders. The equvalent number is then found by writing these remainders in the REVERSE order. Example : Convert 37 decimal into binary. 37 / 2 = 18 remainder 1 LSB 18 / 2 = 9 remainder 0! 9 / 2 = 4 remainder 1!! 4 / 2 = 2 remainder 0!!! 2 / 2 = 1 remainder 0!!!! 1 / 2 = 0 remainder 1!!!!! MSB!!!!!! Binary equivalent of 37 Example : Convert decimal into binary x 2 = Carry 0 MSB x 2 = 1.25 Carry 1! 0.25 x 2 = 0.50 Carry 0!! 0.5 x 2 = 1.00 Carry 1!!! LSB!!!! equivalent of Problem : Convert the following decimals to binary form. : i. 23 ii. 49 iii) B431 Principles of Digital Systems : Hassan Parchizadeh Page 2
3 Binary Arithmetic All the digital computers and microprocessors work on a binary basis, processing numbers and instructions coded in 1 and 0 patterns. Binary arithmetic is very similar to normal decimal arithmetic, perhaps even easir, since we are only dealing with 0's and 1's. The four operations are shown below : Addition Substraction Multiplication Division = = 0 0 x 0 = = = 1 C=1 0 x 1 = 0 0 / 1 = = = 1 1 x 0 = = 0 C=1 11 = 0 1 x 1 = 1 1 / 1 = 1 C = Carry / Borrow Example : Perform binary i. Addition ii. Subtraction iii. Multiplication and iv. Division on the 110 and 10 binary numbers. 11 i) ii) iii) 110 x iv) 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. There are three ways in which signed numbers can be represented. Signmagnitude numbers : =  10 In the signmagnitude system, the negative and positive numbers have the same magnitude, but the sign bit(msb) is 1 for negative numbers. 1'S Complements of Binary Numbers =  10 B431 Principles of Digital Systems : Hassan Parchizadeh Page 3
4 positive numbers are represented in the same way as signmagnitude numbers, but the negative numbers are 1's complement of the positive number.. The 1'S complement is found by simply changing all 1s to 0s and all 0s to 1s. 2'S Complements of Binary Numbers 's complement = 10 2's complement The 2'S complement is found by adding 1 to 1'S complement. In computers, the 2's complement system is the most widely used for handling signed numbers. Problem :Determine the 2'S complement for each 8bit binary number. i ii iii Hexadecimal Numbers The hexadecimal system has a base of 16, that is composed of 16 digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F) where A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15. Hexadecimal to Decimal conversion Consider 1A34 1x x16 + 3x16 + 4x Problem : Express i and ii. ABCD as the sum of values of each digit. Hexadecimal to Binary conversion Consider 1A Problem : Determine the binary numbers for i and ii. CDEF 16. B431 Principles of Digital Systems : Hassan Parchizadeh Page 4
5 Binary to Hexadecimal conversion Consider 100,1110,0110, E 6 4 4E64 16 Problem : Determine the hexadecimal numbers for the following. i ii Decimal To Hexadecimal Conversion We can convert a decimal number into a hexadecimal equivalent by dividing the decimal number succesively by 16 and noting the remainders. The equvalent number is then found by writing these remainders in the REVERSE order. Example : Convert 37 decimal into binary. 650 / 16 = 40 remainder = A 16 LSB 40 / 16 = 2 remainder 8 10 = / 16= 0 remainder 2 10 = 2 16 MSB 28A 16 hexadecimal equivalent of Problem : Determine the hexadecimal numbers for i and ii Note to Students : Also look at BCD (Binary Coded Decimal) and Gray Codes. B431 Principles of Digital Systems : Hassan Parchizadeh Page 5
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