EE 261 Introduction to Logic Circuits. Module #2 Number Systems


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1 EE 261 Introduction to Logic Circuits Module #2 Number Systems Topics A. Number System Formation B. Base Conversions C. Binary Arithmetic D. Signed Numbers E. Signed Arithmetic F. Binary Codes Textbook Reading Assignments Practice Problems 2.1, 2.5, 2.6, 2.7, 2.8, 2.11, 2.12 Graded Components of this Module 2 homeworks, 2 discussions, 1 quiz (all online) Page 1
2 EE 261 Introduction to Logic Circuits Module #2 Number Systems What you should be able to do after this module Convert numbers between bases (decimal, binary, octal, hexadecimal) Perform arithmetic (both signed & unsigned) using other bases Create and Decode various binary codes (BCD, Gray, ASCII, Parity) Page 2
3 Number Systems Number System  a system that contains a set of numbers/symbols/characters and at least one operation (+, , ) BASE or RADIX  the number of symbols in the number system  humans use Base10 (why?)  computers use Base2 (why?) Positional Number System  we can represent 10 unique quantities w/ our Base10 system  what if we want more?  we can add a "leading" digit to our number that has increased "weight ex) 0 1 : : Page 3
4 Number Systems Radix Point  place in the string of digits at which numbers represent either whole or fractional. ex) p  Number of digits to the LEFT of the radix point. n  Number of digits to the RIGHT of the radix point. d  Digits in the system, described with a positional subscript (NOTE: if the radix point is missing, we assume it is to the right of the #) r  Radix or Base (Base10, r=10) i  Position  starting at 0  increasing to the left of the radix point  negative to the right of the radix point Weight  each digit has a weight based on its position = r i  we multiply the digit by its weight to find how much value that digit represents Page 4
5 Number Systems Value  the value of a number is the sum of each digit multiplied by its corresponding weight. D i p 1 n i d i r Example: Given the BASE 10 (r=10) number What is p? p=3 What is n? n=1 Write the digit notation? d 2 d 1 d 0. d 1 What is the weight for each position? Show the expanded decimal equivalent? = 3(10 2 ) + 2(10 1 ) +7(10 0 ) + 2(101 ) = Why all this framework? Because this generic format works for all Bases. Page 5
6 Number Systems Binary  A number system with 2 symbols (BASE=2 or r=2).  The symbols are 0 and 1  each symbol is called a "bit"  we ll give a subscript to the number to indicate its base ex) (decimal) (binary)  4 bits are called a "Nibble"  8 bits are called a "Byte"  the leftmost bit in a string is called the Most Significant (MSB) or High Order  the rightmost bit in a string is called the Least Significant (LSB) or Low Order Page 6
7 Number Systems Binary Systems B p 1 i b i r i n Example: Given the Binary number What is p? p=5 What is n? n=0 What is r? r=2 Write the bit notation? b 4 b 3 b 2 b 1 b 0 What is the weight for each position? Show the expanded decimal equivalent? B = 1(2 4 ) + 0(2 3 ) +0(2 2 ) + 1(2 1 ) + 1(2 0 ) = 1(16) + 0(8) + 0(4) + 1(2) +1(1) = Page 7
8 Base Conversions Base  the number of symbols in a number system  we instinctively know decimal  we've talked about Binary (two symbol)  there are other bases of interest in digital systems  the bases we typically care about are powers of 2 (or associated with Binary)  these bases are typically used to represent a lot of "bits" ex) it's hard to describe the value of a 64bit bus in 1's and 0's Octal  A number system with 8 symbols  0,1,2,3,4,5,6,7  each digit in this system is equivalent to 3bits  it is a positional number system ex) 0 1 : : Page 8
9 Base Conversions Hexadecimal  A number system with 16 symbols  we use alphabetic characters as symbols in the set above 90,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F  each digit in this system is equivalent to 4bits  it is a positional number system ex) 0 1 : 9 A B C D E F : 1F 20 : Page 9
10 Base Conversions How the Bases Relate to Each Other Decimal Binary Octal Hexadecimal Base10 Base2 Base8 Base A B C D E F  these are the commonly used bases in digital systems, we'd like to be able to convert between them Page 10
11 Base Conversions Converting to Decimal  we sum the products of each digit value with its positional weight D i p 1 n i d i r  this works for whole and fractional digits  this works for Binary to Decimal  this works for Octal to Decimal  this works for Hex to Decimal Page 11
12 Base Conversions Base Conversion Binary to Decimal  each digit has a weight of r i that depends on the position of the digit  multiply each digit by its weight  sum the resultant products ex) Convert 1011 to decimal (weight) = 1 (2 3 ) + 0 (2 2 ) + 1 (2 1 ) + 1 (2 0 ) = 1 (8) + 0 (4) + 1 (2) + 1 (1) = = 11 Decimal Page 12
13 Base Conversions Base Conversion Binary to Decimal with Fractions  the weight of the binary digits have negative positions ex) Convert to decimal = 1 (2 3 ) + 0 (2 2 ) + 1 (2 1 ) + 1 (2 0 ) + 1 (21 ) + 0 (22 ) + 1 (23 ) = 1 (8) + 0 (4) + 1 (2) + 1 (1) + 1 (0.5) + 0 (0.25) + 1 (0.125) = = Decimal Page 13
14 Base Conversions Base Conversion Hex to Decimal  the same process as binary to decimal except the weights are now BASE 16  NOTE (A=10, B=11, C=12, D=13, E=14, F=15) ex) Convert 2BC from Hex to decimal (weight) 2 B C = 2 (16 2 ) + B (16 1 ) + C (16 0 ) = 2 (256) + 11 (16) + 12 (1) = = 700 Decimal Page 14
15 Base Conversions Base Conversion Hex to Decimal with Fractions  the fractional digits have negative weights (BASE 16)  NOTE (A=10, B=11, C=12, D=13, E=14, F=15) ex) Convert 2BC.F to decimal (weight) 2 B C. F = 2 (16 2 ) + B (16 1 ) + C (16 0 ) + F (161 ) = 2 (256) + 11 (16) + 12 (1) + 15 (0.0625) = = Decimal Page 15
16 Base Conversions Base Conversion Decimal to Binary  the decimal number is divided by 2, the remainder is recorded  the quotient is then divided by 2, the remainder is recorded  the process is repeated until the quotient is zero ex) Convert 11 decimal to binary Quotient Remainder LSB MSB = 1011 binary Page 16
17 Base Conversions Base Conversion Decimal to Binary with Fractions  the fraction is converted to binary separately  the fraction is multiplied by 2, the 0 th position digit is recorded  the remaining fraction is multiplied by 2, the 0 th digit is recorded  the process is repeated until the fractional part is zero ex) Convert decimal to binary Product 0 th Digit MSB LSB finished decimal = binary Page 17
18 Base Conversions Base Conversion Decimal to Hex  the same procedure is used as before but with BASE 16 as the divisor/multiplier ex) Convert decimal to hex 1 st, convert the integer part Quotient Remainder LSB MSB 2 nd, convert the fractional part = 1A4 Product 0 th Digit MSB decimal = 1A4.A hexadecimal = 0.A Page 18
19 Base Conversions Base Conversion Octal to Decimal / Decimal to Octal  the same procedure is used as before but with BASE 8 as the divisor/multiplier Page 19
20 Base Conversions Base Conversion of "Powers of 2"  converting bases that are powers of 2 are simple due to straight forward mapping  An Octal digital represents 3 bits  A Hex digit represents 4 bits Page 20
21 Base Conversions Converting Binary to Hexadecimal  every 4 binary bits represents one HEX digit  begin the groups of four at the LSB  if necessary, fill with leading 0 s to form the groups of four ex) Convert from Binary to Hex D 4 Hex, notice that we had to fill with MSB 0's to get groups of four  This works for fractions too. The only difference is that the grouping starts at the Radix Point ex) Convert 0.11 from Binary to Octal C Page 21
22 Base Conversions Converting Binary to Octal  every 3 binary bits represents one OCTAL digit  begin the groups of three at the LSB  if necessary, fill with leading 0 s to form the groups of three ex) Convert from Binary to Octal Octal ex) Convert from Binary to Octal Octal, note we had to fill with 0's on the MSB side Page 22
23 Base Conversions Converting Hex to Binary  each HEX digit is made up of four binary bits ex) Convert ABC Hex to Binary A B C = = Page 23
24 Base Conversions Converting Octal to Binary  each Octal digit is made up of three binary bits ex) Convert 567 Octal to Binary = = Page 24
25 Base Conversions Terminology NIBBLE BYTE = 4 bits = 8 bits  you should be familiar with converting Binary Nibbles to Hex & Dec  there is a table on page 28 of your textbook which lists the basic conversions Page 25
26 Binary Arithmetic Addition and Subtraction  same as BASE 10 math, remember borrows and carries Addition Table Carry Subtraction Table Need to "Borrow" from a more significant bit Page 26
27 Binary Arithmetic Binary Addition  same as BASE 10 addition  need to keep track of the carry bit for a given system size (n), i.e., 4bit, 8bit, 16bit, ex) Add the binary numbers 1011 and Carry Bit Page 27
28 Binary Arithmetic Binary Subtraction  same as BASE 10 subtraction  need to keep track of the borrow bit for a given system size (n), i.e., 4bit, 8bit, 16bit, ex) Subtract the binary numbers 1010 and Borrow bits : if necessary, we could assume a borrow from an even higher significant bit Page 28
29 Signed Numbers Negative Numbers  So far, we've dealt with Positive numbers  The real world has Negative numbers  We need a method to represent negatives in binary  However, our number system doesn't have a "", just 0's and 1's Sign Bit  We will use the MSB to represent a "+" or "" 0 = Positive 1 = Negative Page 29
30 Signed Numbers Signed Magnitude Representation  a negative number system (there is more than one)  uses the MSB as the sign bit  the remaining LSB's represent the number ex) 85 dec = dec = (additional sign bit adds one bit to the number) 85 dec = Is this Efficient?  The number of unique combinations that a number can represents is given by: N = r x N = # of unique combinations x = # of digits in the number (p + n) r = base ex) 2bits: N=2 2 = 4 (00, 01, 10, 11) 7bits: N=2 7 = bits: N=2 8 = 256 Page 30
31 Signed Numbers Signed Magnitude Range  The Range of Signed Magnitude is given by: (2 x11) < N SM < (2 x11) ex) if we use 8bits, the range is: ( ) < N SM < ( ) 127 < N SM < This is only 255 unique numbers  But, if we are using 8bits, there should be 2 8 unique numbers, or 256?  This is because in Signed Magnitude, there are two representations for Zero = = 0 Page 31
32 Signed Numbers Advantages of Signed Magnitude  Very easy to understand and use  the binary number is simply represented in binary, then a sign bit is added Disadvantages of Signed Magnitude Representation  We loose a possible number by having +0, and This also creates a gap in our number list, which makes simple math harder  Addition/Subtraction in general are more difficult Algorithm : if ( signs are the same)  add and give same sign else if (signs are different)  compare magnitudes  subtract smaller from larger  give the result the sign of larger number Page 32
33 Signed Numbers Complement Numbers  Complementing a Binary Number means: 0 changes to 1 1 changes to 0  using this technique, number can be a "complemented" to find the negative representation, then the arithmetic becomes much simpler  the ones we care about are 1) "Radix Complement" (2's complement for binary) 2) "1's Complement Page 33
34 Signed Numbers Radix Complement  i.e., 10's complement, 2's complement  Technique to get the Radix Complement:  subtract current number from r x D rcomp = r n  D orig  Rules:  The MSB is still the sign bit (0="+", 1="")  Now that we're subtracting, we can't have arbitrary number of bits x is predetermined and fixed  A simpler method is : 1) Complement all digits in D orig 2) Add 1 to the result, ignore any carryout Page 34
35 Signed Numbers 2's Complement  this is Radix Complement on a BASE2 number system  straight forward to use the "complement and add 1" technique ex) Give the 8bit, 2's complement representation of = (Notice that the MSB is the sign bit, 0=+) Step 1: (we first complement all bits) Step 2: + 1 (we then add 1) (Notice the Sign Bit is negative)  these 8bits represent in 2's complement Page 35
36 Signed Numbers 2's Complement Checking  We can convert back to find Original magnitude and do checking ex) Give the 8bit, 2's complement representation of = 's Comp (Notice that the MSB is the sign bit, 1="") Step 1: (we first complement all bits) Step 2: + 1 (we then add 1) (Notice the Sign Bit is positive)  these 8bits represent ,Which is what we originally started with, Page 36
37 Signed Numbers 2's Complement Range  We need to know how many numbers we can represent using this system  Notice that we are still using a bit for the sign  BUT, we don't duplicate Zero in this system ex) = = The Range of 2's Comp is given by: (2 x1 ) < N 2Comp < (2 x11) ex) if we use 8bits, the range is: (2 81 ) < N 2Comp < ( ) 128 < N 2Comp < There are now 256 unique numbers, this is a more efficient use of bits Page 37
38 Signed Numbers 1's Complement  In 2's complement, the codes are asymmetric Lowest 2's Comp # = Comp = Highest 2's Comp # = Comp = This is good because we have all 256 possible numbers that 8bits can give us  1's Comp is similar, but it gives us symmetry around Zero  To find the 1's Comp, we subtract current number from (r x 1) D 1Comp = (r n 1)  D orig Page 38
39 Signed Numbers 1's Complement  Or we can use the simple way 1) Complement all the #'s (Don't add 1) ex) Give the 8bit, 1's complement representation of = (Notice that the MSB is the sign bit, 0=+) Step 1: Comp (complement all bits) Page 39
40 Signed Numbers 1's Complement Range  The Range of 1's Complement is given by: (2 x11) < N SM < (2 x11)  Once again, we have two values for Zero = = 0 Page 40
41 Signed Numbers Negative Representation  We have covered 3 different "signed" codes  You need to KNOW THE CODE you are using ex) Represent ) Signed Magnitude : SM 2) 2's Complement : Comp 3) 1's Complement : Comp Page 41
42 Signed Arithmetic Two s Compliment Arithmetic  Two's complement has advantages when going into Hardware  Two's complement addition is straight forward because the numbers are in sequential order (+1) from their least significant ( ) to their most significant ( )  There is only one value for Zero, so "avoiding the gap" isn't necessary Page 42
43 Signed Arithmetic Two s Compliment Addition  Addition of two s compliment numbers is performed just like standard binary addition.  However, the carry bit is ignored Page 43
44 Signed Arithmetic Two s Compliment Subtraction  We can build a subtraction circuit out of the same hardware as an adder  2's Comp inherently adds negative numbers so  to subtract, we can just complement one number, and add it. ex) Subtract 8dec from 15dec 15dec = dec = > two s compliment > invert add = 8dec Now Add: 15 + (8) = % % = 7dec Disregard Carry Page 44
45 Signed Arithmetic Two s Compliment Overflow  If a two s compliment arithmetic operation results in a number that is outside the range of representation (i.e., 8bits : 128 < N < +127), an overflow has occurred. ex) 100dec 100dec = 200dec (can t represent)  There are three cases when overflow occurs 1) Sum of like signs results in answer with opposite sign 2) Negative Positive = Positive 3) Positive Negative = Negative  Boolean logic can be used to detect these situations. Page 45
46 Signed Arithmetic Remember 2's Comp Range  The Range of 2's Comp is given by: (2 x1 ) < N 2Comp < (2 x11) ex) if we use 8bits, the range is: (2 81 ) < N 2Comp < ( ) 128 < N 2Comp < +127 What goes on in real Hardware?  A generic, Binary adder circuit is created  the user must be aware of when 2's complement is being used  additional circuitry checks for "Overflow" Page 46
47 Binary Codes Codes  a string of xbits that represent information  we've seen codes already using the binary # system 1) Signed Magnitude 2) 2's Complement 3) 1's Complement  the same information can be encoded differently  it's up to the engineer to KNOW THE CODE that is being used Code Word  the term to represent the discrete string of bits that make up the information ex) 4bit code words in a stream of information word Page 47
48 Binary Codes Binary Coded Decimal (BCD)  Sometimes we wish to represent an individual decimal digit as a binary representation (i.e., 7segment display to eliminate a decoder)  We do this by using 4 binary digits. Decimal BCD ex) Represent 17dec Binary = BCD = Page 48
49 Binary Codes BCD Addition  If we are using BCD, traditional addition doesn t work. 12 = correct answer in BCD is 20. Traditional addition only considers bit addition carry. In BCD we need to consider 4bit addition carry. 1A  The solution is to add 6 to every sum greater than 9 12 = When a nibble addition results in A or greater, we add A Page 49
50 Binary Codes Gray Code  There are applications where we want to count, but we only want one bit to transition each time we increment counts  Reduce Power & Noise in Digital Electronics  Electromechanical devices  Such a is called a "Gray Code" ex) 3bit Gray Code Decimal Binary Gray Page 50
51 Binary Codes ASCII  American Standard Code for Information Interchange  English Alphanumeric characters are represented with a 7bit code  See Table 211 in text ex) A = $40 a = $61 Page 51
52 Binary Codes Parity Codes  there are times when information can be corrupted during transmission  we can include an "Error Checking" bit along with the original data  this "Error Checking" bit contains additional information about the original data  this can be used by the receiver to monitor whether an error in the data occurred PARITY  the number of 1's in the information are counted,  the parity bit represents whether there are an EVEN or ODD number of 1's EVEN PARITY = 0, if there are an EVEN number of 1's in the information = 1, otherwise ODD PARITY = 0, if there are an ODD number of 1's in the information = 1, otherwise Page 52
53 Binary Codes Parity Example Information EVEN Parity ODD Parity Page 53
54 Module Overview Topics  # systems & bases  # system conversions  to & from decimal  to and from binary/oct/hex  binary arithmetic  addition (carries)  subtraction (borrows)  negative #'s in binary  Signed Magnitude  2's Complement  1's Complement  2's complement arithmetic  complement & add  range & overflow  Codes  BCD, Gray, ASCII, Parity Page 54
Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
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