Chapter 2 Number System

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1 Chapter 2 Number System Embedded Systems with ARM Cortext-M Updated: Tuesday, January 16, 2018

2 What you should know.. Before coming to this class Decimal Binary Octal Hex x x x x x x x x x x xA xB xC xD xE xF See the review notes!

3 What you should know.. Before coming to this class See the review notes!

4 Terminologies Bit (1,0) BYTE 8 bits is equivalent to one byte NIBBLE 4 bits is equivalent to one nibble WORD 16 bits is equivalent to one word 8-BIT ME! bits is equivalent to how many bytes? 128/8 = What is the maximum number that can be represented by 1 bytes? = 255

5 Bit, Byte, Half-word, Word, Double-Word 5

6 Switch State and Digital Representation In each case we have 16 switches. 1- What Binary/Decimal/Hexadecimal number does each switch represent? 2- What is the maximum binary number we can represent using these switches? =291= =17767= =35243=89AB =52719=CDEF Maximum number: = =65535=64K in Computer terms!

7 Data Types One Word Double-Word

8 Data Representation Data representation can be Digital or Analog In Analog representation values are represented over a continuous range In Digital representation values are represented over a discrete range Digital representation can be Decimal Binary Octal Hexadecimal We need to know how to use and convert from one to another!

9 Digital Data Representations: Decimal (base 10), Binary (base 2), Hex(base 16) Decimal Binary Octal Hex x x x x x x x x x x xA xB xC xD xE xF

10 Digital Data Representations: Power of 2

11 Using Binary Representation Digital systems are binary-based All symbols are represented in binary format Each symbol is represented using Bits A bit can be one or zero (on or off state) Comparing Binary and Decimal systems: In Decimal a digit is [0-9] base-10 In Binary a digit is [0-1] base-2 In Decimal two digits can represent [0-99] à In Binary two digits can represent [0-3] à 2 2-1

12 Binary Counting Binary Representation Decimal Representation Binary Point

13 Counting in Different Numbering Systems Decimal 0,1,2,3,4,5,6,7,8,9,10,11,12,19,20,21,,29,30,,39. Binary 0,1,10,11,100,101,110,111,1000,. Octal 0,1,2,3,4,5,6,7,10,11,12,17,20,21,22,23,27,30, Hexadecimal 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,,1F,20,,2F,30,..

14 Learning Number Conversion Decimal Hexasdecimal Octal Binary BCD/ ASCII

15 Binary-to-Decimal Conversions à Decimal Hexasdecimal Decimal Octal = = Binary ( n Convert N - 1 nn n3n2n1n 0) b ¾¾ ¾ n0 b + n1 b + n2 b + n3 b... + nn -1 BCD b N -1 In the above example: Binary is base-2 (b=2) n 0 =1 n 1 =1 n 2 =0 n 3 =1 n 4 =1 Q: What is In Decimal? Ans: =27+(1x x2-2 ) = =27.75

16 Decimal-to-Binary Conversions 65 / 2 = 32 + Re mainder _ of _1 65 à Binary 32 / 2 = 16 + Re mainder _ of _ 0 16 / 2 = 8 + Re mainder _ of _ 0 8 / 2 = 4 + Re mainder _ of _ 0 4 / 2 = 2 + Re mainder _ of _ 0 MSB 2 / 2 = 1+ Re mainder _ of _ 0 1/ 2 = 0 + Re mainder _ of _ LSB Hexasdecimal Decimal Binary BCD Octal Last one should be 0 LSB = Least Significant Bit MSB= Most Significant Bit What if you are using a calculator? 65/2 = x 2 (base-2) = 1

17 Octal/Decimal Conversions Binary is base-8 (b=8) 372 à Decimal Hexasdecimal Decimal Octal = = Binary ( n Convert N - 1 nn n3n2n1n 0) b ¾¾ ¾ n0 b + n1 b + n2 b + n3 b... + nn -1 BCD b N -1 What about ? Ans: =250+(2x8-1 )= Decimal-to-Octal: 266 à Octal Decimal 266 / 8 = 33 + Re mainder _ of _ 2 33 / 8 = 4 + Re mainder _ of _1 4 / 8 = 0 + Re mainder _ of _ MSB LSB Hexasdecimal Binary BCD Octal

18 Hex-to-Decimal Conversions Binary is base-16 (b=16) 2AF à Decimal Hexasdecimal Decimal Octal = = Binary ( n Convert N - 1 nn n3n2n1n 0) b ¾¾ ¾ n0 b + n1 b + n2 b + n3 b... + nn -1 BCD b N -1 Remember A=10, F=15 In the above example: Binary is base-16 (b=16) n 0 =F which is 15 n 1 =A which is 10 n 2 =2

19 Decimal-to-Hex Conversions 423à Hex Hexasdecimal Decimal Octal 423/16 = 26 + Re mainder _ of _ 7 26 /16 = 1+ Re mainder _ of _10 1/16 = 0 + Re mainder _ of _1 214 /16 = 13 + Re mainder _ of _ 6 13 /16 = 0 + Re mainder _ of _13 214à Hex 1A7 MSB LSB D6 Binary Remember 10 in Hex is A BCD Last one should be 0 Remember 13 in Hex is D

20 Converting from Hex-to-Octal 124 Hex à?? Oct Hexasdecimal Octal Decimal Binary Always convert to Binary first and then from binary to Oct. BCD Decimal Ans: =124 Hex = = = =444 Oct Hexasdecimal Binary Octal BCD

21 Counting Decimal 0,1,2,3,4,5,6,7,8,9,10,11,12,19,20,21,,29,30,,39. Binary 0,1,10,11,100,101,110,111,1000,. Octal 0,1,2,3,4,5,6,7,10,11,12,17,20,21,22,23,27,30, Hexadecimal 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,,1F,20,,2F,30,..

22 BCD Addition Any symbol can be represented by a code Example: Binary-Coded-decimal (BCD) Each digit has its own binary code Example: 6 10 =0110, = (In binary 16 is?) BDC numbers They are between 0 and 9 Hence each decimal number is represented by 4 bits Add each number between 0-9 individually n ß BCD for ß BCD for ß 13 but invalid! ß Add 6 to correct ß BDC for 13!!

23 ASCII Code Example: American Standard Code for Information Interchange (ASCII) Each symbol is represented by a sevenbit code (How many symbols can be represented? 127) Example: A= = 41 in Hex, 1= = 31 in Hex, $= = 24 in Hex (What is DAD in ASCII?) American Standard Code for Information Interchange Look at the ASCII code listing Don t memorize! Dec Hex Char Dec Hex Char Dec Hex Char Dec Hex Char 0 00 NUL SP SOH 33 21! A a 2 02 STX " B b 3 03 ETX # C c 4 04 EOT $ D d 5 05 ENQ % E e 6 06 ACK & F f 7 07 BEL G g 8 08 BS ( H h 9 09 HT ) I i 10 0A LF 42 2A * 74 4A J 106 6A j 11 0B VT 43 2B B K 107 6B k 12 0C FF 44 2C, 76 4C L 108 6C l 13 0D CR 45 2D D M 109 6D m 14 0E SO 46 2E. 78 4E N 110 6E n 15 0F SI 47 2F / 79 4F O 111 6F o DLE P p DC Q q DC R r DC S s DC T t NAK U u SYN V v ETB W w CAN X x EM Y y 26 1A SUB 58 3A : 90 5A Z 122 7A z 27 1B ESC 59 3B ; 91 5B [ 123 7B { 28 1C FS 60 3C < 92 5C \ 124 7C 29 1D GS 61 3D = 93 5D ] 125 7D } 30 1E RS 62 3E > 94 5E ^ 126 7E ~ 31 1F US 63 3F? 95 5F _ 127 7F DEL

24 Saving ASCII Characters char str[13] = ARM Assembly ; // The length has to be at least 13 // even though it has 12 letters. The // NULL terminator should be included. Memory Memory Address Content Letter str x00 \0 str x79 y str x6C l str + 9 0x62 b str + 8 0x6D m str + 7 0x65 e str + 6 0x73 s str + 5 0x73 s str + 4 0x41 A str + 3 0x20 space str + 2 0x4D M str + 1 0x52 R str 0x41 A 24

25 Convert to Upper Case A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A 4B 4C 4D 4E 4F A a b c d e f g h i j k l m n o p q r s t u v w x y z A 6B 6C 6D 6E 6F A a A = 0x61 0x41 = 0x20 = 32 25

26 Digital Arithmetic

27 Binary Arithmetic Binary Numbers Binary Arithmetic Digital circuits are frequently used for arithmetic operations Fundamental arithmetic operations on binary numbers and digital circuits which perform arithmetic operations will be examined. 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. Addition is the basic arithmetic operation used by digital devices to perform subtraction, multiplication, and division. Representation Positive and Negative Numbers Operations (Adding, Subtracting, Inverting) If the numbers are unsigned and positive add them as follow: = 1 with carry of = 0 with carry of = 1 with carry of = 0 with borrow = 1 with borrow = 1 with borrow = 0 with borrow 0

28 Adding Unsigned Numbers Examples: X Y X+Y X Y X+Y X Y X+Y

29 Adding Unsigned Numbers Examples: X Y X+Y X Y X+Y X Y X+Y

30 Representing Signed Numbers Since it is only possible to show magnitude with a binary number, the sign (+ or -) is shown by adding an extra sign bit. A sign bit of 0 indicates a positive number. A sign bit of 1 indicates a negative number. The 2 s complement system is the most commonly used way to represent signed numbers. Magnitude bits +7 à à à à à Sign-bit Examples of 2 s complement representation ^3 +2^2 +2^1 +2^0

31 Comparison presenting +/- Numbers 00000/ 00000/ Signed magnitude representation 0 = positive 1 = negative One s complement representation Negative = invert all bits of a positive Two s Complement representation TC = invert all bits, then plus 1 31

32 Representing Signed Numbers Converting to 2 s Complement In order to change a binary number to 2 s complement it must first be changed to 1 s complement. To convert to 1 s complement, simply change each bit to its complement (opposite). To convert 1 s complement to 2 s complement add 1 to the 1 s complement. A positive number is true binary with 0 in the sign bit. A negative number is in 2 s complement form with 1 in the sign bit. A binary number can be negated by taking the 2 s complement of it. For example: +9 à (sign bit = 0, indicating +) 2 s complement of 9 à à à = -9 This is The sign BIT When the sign-bit is zero à Positive number If the sign-bit is set à Negative number Remember: 2 s complement is just a conventional way of representing signed numbers in digital arithmetic Don t ask why!

33 2 s Complement Representation Assuming N+1 bits representing a 2 s Complement (that is representing the number with N bits and one bit is dedicated to indicate the sign): Largest positive number will be 2 N -1 Smallest signed number (largest negative number) will be -2 N Total numbers (including zero) that can be represented will be 2 N+1 For example: Assume 3+1 bits Largest pos. number will be 0111= +7 Smallest number will be 1000 = -8 Remember: 1111= -1 Zero is represented by 0000 = Zero For example: Assume 6+1 bits Largest pos. number will be?? Smallest signed number will be?? Zero is represented by?? = = = 0

34 More Examples Integer 2's Complement Signed Signed Integer Unsigned 2's Complement Remember: Always know how many bits are provided!

35 2 s Complement Representation Integer Signed 2's Complement

36 Flags Carry, Borrow, Overflow The result of a Arithmetic Operation in a microprocessor can often be expressed by flags Carry: = 1 carry 1 Borrow 4-7 = 7 Borrow 1 Overflow = 7 overflow 1 (if only one digit can be used)

37 Flags When adding two unsigned numbers in an n-bit system, a carry occurs if the result is larger than the maximum unsigned integer that can be represented (i.e. 2 " 1). When subtracting two unsigned numbers, borrow occurs if the result is negative, smaller than the smallest unsigned integer that can be represented (i.e. 0). On ARM Cortex-M3 processors, the carry flag and the borrow flag are physically the same flag bit in the status register. For an unsigned subtraction, Carry = NOT Borrow

38 Carry/borrow flag bit for unsigned numbers If the traverse crosses the boundary between 0 and 2 n 1, the carry flag is set on addition and is cleared on subtraction Carry A carry occurs when adding 28 and 6 Carry + Extra bit is discarded bit result Carry flag = 1, indicating carry has occurred on unsigned addition. Carry flag is 1 because the result crosses the boundary between 31 and NOT 32! 38

39 Carry/borrow flag bit for unsigned numbers If the traverse crosses the boundary between 0 and 2 n 1, the carry flag is set on addition and is cleared on subtraction Carry Borrow bit result Carry flag = 0, indicating borrow has occurred on unsigned subtraction. For subtraction, carry = NOT borrow. A borrow occurs when subtracting 5 from 3. 39

40 Arithmetic Operations using 2 s Complement Inverting A positive number to a negative number A negative number to a positive number Either case just take the 2 s complement Adding A and Bà A+B (assuming N bits represent the magnitude and one bit is dedicated as the signbit) Magnitude Overflow: (max unsigned number that can be represented using 8 bits is 255) The carry will be ignored X Y X+Y X Y X-Y Subtracting B from A à A-B Just take the 2 s Complement of B Add A and B (A+B) NOTE: When a Positive and a Negative number are added together, the overflow will be a signed overflow and it is ok! Note the value is negative Overflow: (max signed number that can be represented using 4 bits is -8) X Y X-Y X Y X+Y X Y X+Y The overflow can be discarded If Carry into and out of the sign bit are the same Overflow: (max signed number that can be represented using 4 bits is 7) X Y X+Y

41 BCD Addition BDC numbers They are between 0 and 9 Hence each decimal number is represented by 4 bits Add each number between 0-9 individually n ß BCD for ß BCD for ß 13 but invalid! ß Add 6 to correct ß BDC for 13!!

42 BCD Addition Another Example BDC numbers They are between 0 and 9 Hence each decimal number is represented by 4 bits Add each number between 0-9 individually ß BCD for ß BCD for ß 91 but invalid! ß Add ß BDC for 97!!

43 Hex Arithmetic Addition of Hex numbers is similar to decimal addition Remember the largest value is 16 (F) and NOT 9! Carry a 1 if the number is larger than F Use the following steps Add the hex digits in decimal. If the sum is 15 or less express it directly in hex digits. If the sum is greater than 15, subtract 16 and carry 1 to the next position In Hex=11 3AF + 23C = 5EB 8+B=8+(11)= 19à19-16=3, Carry In Decimal = =1 In Hex = B = A3

44 Hex Arithmetic Subtraction of hex numbers follows a similar method as binary numbers Make sure you take the 2 s complement of the negative hex number Method 1: Subtract the number from FFFF Write the results in hex Add one to the final answer For example: 2 s complement of 3A5 is: FFF-3A5=C5A à C5A+1 = C5B Method 2: Convert the number to Binary 592-3A5 à592 + C5B = 11ED 67F - 2A4 à67f + D5C = 3DB Take the 2 s complement Convert back the results into hex value Simply add the values together 5+C=2+(12)= 17à17-16=1, Carry 1 Must be discarded! FFF- 3A5= (15-3),(15-10),(15-5) = 12, 5, 10 à C5A+1=C5B F+C=(15)+(12)=27 à27-16=11 or B, Carry 1

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