12/31/2010. Digital Operations and Computations Course Notes. 01-Number Systems Text: Unit 1. Overview. What is a Digital System?

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1 Digital Operations and Computations Course Notes 0-Number Systems Text: Unit Winter 20 Professor H. Louie Department of Electrical & Computer Engineering Seattle University ECEGR/ISSC 20 Digital Operations and Computations Overview Introduction to digital systems Decimal system Non-decimal systems Conversion to decimal Conversion from decimal Codes What is a Digital System? Digital: something taking on a discrete set of values True, False On, Off -5 Volts, 0 Volts, 5 Volts Analog: something that can assume a continuous range of values Temperature 0-5 Volts Dr. Louie Dr. Louie 4 What is a Digital System? s of Digital Systems A/D Converter Alphabet Abacus Morse Code Computer Light Switch time Analog Signal time Digital Signal Dr. Louie 5 Dr. Louie 6

2 Decimal System What are the characteristics of our (decimal) numbering system? Valid digits: 0,, 2,, 9 Positional notation 00 is ten times greater than 0, which is ten times greater than, which is ten times greater than 0., etc. Positive, negative and zero can be represented We use a base 0 (decimal) system 0 = = 0 x 0 = 00 Decimal System 0 = 0 x 0 x 0 = 000 What about: 0 0 =? 0 - =? 0-2 =? Dr. Louie 7 Dr. Louie 8 0 = = 00 Decimal System 0 = 000 What about: 0 0 = (any value to 0 th power = ) 0 - = /0 = = /0/0 = /00 = 0.0 Find a 2, a, a 0, and a - (values between 0 and 9) such that: = a 2 x a x 0 + a 0 x a - x 0 - Dr. Louie 9 Dr. Louie 0 Find a 2, a, a 0, and a - such that: = a 2 x a x 0 + a 0 x a - x 0 - Solution = 0 x x 0 + x x 0 - Decimal System Another example 95.7 = 9 x x 0 + x x 0 - Try a negative number -205 = -2 x x 0-5 x Dr. Louie Dr. Louie 2 2

3 Other Numbering Systems What other numbering systems do you know of? (base 2) Ternary (base ) Quintal (base 5) Octal (base 8) Hexadecimal (base 6) The base is also known as the radix Is there a base? base 0? No! We are especially interested in binary electronic systems digital logic (true, false) Dr. Louie Dr. Louie 4 Valid digits: 0, (called bits) Some examples 0 2 = 0 0 (subscript refers to the base) 2 = = = = = 4 0 What is the decimal equivalent of 0 2? Recall that 95 = 9 x x 0 + x 0 0 Similarly, 0 2 = x x 2 + x 2 0 = = 5 0 Dr. Louie 5 Dr. Louie 6 What is the decimal equivalent of 0. 2? What is the decimal equivalent of 0. 2? 0. 2 = x x x 2 + x x x 2-2 = =.75 0 Dr. Louie 7 Dr. Louie 8

4 In binary conversion, knowing powers of 2 becomes very important: 2 0 = 2 = = 4 2 = = = = = = 256 and so on Negative powers of two: 2 - = /2 = /2 = = /2 2 = /4 = = /2 = /8 = 0.25 and so on Dr. Louie 9 Dr. Louie 20 Conversion In general, when positional notation is used: N = (a 4 a a 2 a a 0.a - a -2 ) R = a 4 x R 4 + a x R + a 2 x R 2 + a x R + a 0 x R 0 + a - x R - + a -2 x R -2 Where a i is a digit between 0 and R- (inclusive) The result is the decimal equivalent of N Hexadecimal 0 6 = 0 0 (subscript refers to the base) 6 = = 2 0 and so on 9 6 = 9 0 A 6 = 0 0 B 6 = 0 and so on F 6 = 5 0 Dr. Louie 2 Dr. Louie 22 s Convert the following to decimal AF 6 s Convert the following to decimal 0. 2 = = = AF 6 = 75 Dr. Louie 2 Dr. Louie 24 4

5 Conversion from Decimal Convert 5 0 to Cannot directly apply the previous method First consider the case of a decimal integer Consider an integer N = (a n a n- a 2 a a 0 ) R = a n R n + a n- R n- + + a R + a 0 Dividing by R yields N =a R n- +a R +...+a R +a =Q n-2 R n n- 2 where the remainder is a 0 Dr. Louie 25 Dr. Louie 26 Next divide the quotient Q by R Q =a R +a R n-...a R +a =Q n-2 n n- 2 2 R where the remainder is a The process repeats until a n is left Convert 5 0 to R = remainder a 0 = 2 26 remainder 0 a = remainder a 2 = 26 remainder 0 a = 0 a 4 = a 5 = Dr. Louie 27 Dr. Louie 28 Putting the digits together 5 0 = 00 2 Now try converting 5 0 to hexadecimal (base 6) Putting the digits together 5 0 = 00 2 Now try converting 5 0 to hexadecimal (base 6) 6 5 remainder 5=a 0 6 remainder =a = 5 6 Dr. Louie 29 Dr. Louie 0 5

6 F = (.a - a -2 a -m ) R = a - R - + a -2 R n a -m R -m We know what R (base converting to) and F (decimal number) are Need to isolate a - F = (.a - a -2 a -m ) R = a - R - + a -2 R n a -m R -m We know what R (base converting to) and F (decimal number) are Need to isolate a - Multiply by R FR a a R a R... a R a F 2 m 2 m Where a - is the integer part of the result and F is the fractional part Dr. Louie Dr. Louie 2 Repeat by multiplying F with R FR=a +a R...+a R =a +F -2 -m m -2 2 The process may not terminate, in which case the result is a repeating fraction Convert to binary R = 2, F =0.75 Dr. Louie Dr. Louie 4 Convert to binary R = 2, F = FR=a +a R +a R...+a R -m+ =a +F.5=a +F m - - =>F =0.5 Convert to binary R = 2, F = FR=a +a R +a R...+a R -m+ =a +F.5=a +F m =>F =0.5 Next step - -m+2 F R=a +a R +...+a R =a +F m -2 2 =a +F =>F 2=0 stop Dr. Louie 5 Dr. Louie 6 6

7 Putting it together: = 0. 2 Now try converting to binary Dr. Louie 7 Dr. Louie 8 Now try converting to binary ) RF=.4=a +F - =>F =0.4-2) RF =0.8=a +F =>a =0-2 =>F = ) RF =.6=a +F =>F =0.6-4) RF =.2=a +F -4 =>F = ) RF =0.4=a +F =>a =0-5 =>F = ) RF =0.8=a +F =>a =0 =>F = Numbers If converting.75, do the integer division method for, and the decimal method for.75 and combine the result It is possible to use either method when converting from bases other than decimal, but this is difficult because it involves non-decimal based math Better idea is to convert to decimal, then convert to the desired base Repeating result: Dr. Louie 9 Dr. Louie 40 to Hexadecimal Conversion A few shortcuts exist Consider in hexadecimal Regroup as Convert each word D 5 C Codes codes are a way of representing decimal numbers that are intended to be easier than converting the entire decimal number to binary The result is 4D.5C 6 Dr. Louie 4 Dr. Louie 42 7

8 Codes codes are a way of representing decimal numbers that are intended to be easier than converting the entire decimal number to binary : This is known as BCD ( Coded Decimal) Decimal Digit Codes Code 6--- Code Gray Code Dr. Louie 4 Dr. Louie 44 Codes Codes and 6--- are weighted codes N = w a + w 2 a 2 + w a + w 0 a 0 Decimal 8 in is: 000 8() + 4(0) + 2(0) + (0)=8 In 6--- it is: 0 6() + (0) + () + ()=8 Gray Code: only one digit changes Useful in analog to digital conversion More reliable than if two digits change (it is unlikely that two or more switches would occur simultaneously) Gray Code Dr. Louie 45 Dr. Louie 46 ASCII Code ASCII (American Standard Code for Information Interchange) Alphanumeric code 7-bits =>28 symbols See text page 22 for a partial list Bits and Bytes In binary, the digits are known as bits (binarydigits) Collection of 8 bits is commonly known as a byte (or more specifically, as a octet) Able to represent keyboard characters Dr. Louie 47 Dr. Louie 48 8

9 Bits and Bytes kilobyte: 2 0 bytes (,024) megabyte: 2 20 bytes (,048,576) gigabyte: 2 0 bytes (,07,74,824) Dr. Louie 49 9

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