EE 3170 Microcontroller Applications

Transcription

1 EE 37 Microcontroller Applications Lecture 3 : Digital Computer Fundamentals - Number Representation (.) Based on slides for ECE37 by Profs. Sloan, Davis, Kieckhafer, Tan, and Cischke Number Representation Integers in different radices (bases) decimal, binary, octal, hexadecimal Unsigned Binary Numbers Twos Complement Signed Binary Numbers Definition, Properties, Arithmetic The ASCII Character Code EE37/CC/Lecture#3 EE37/CC/Lecture#3 2 Preliminaries Three Systems of Number Representation Radix = the base of a number system Given radix k : one digit has one of k values need k symbols for values [ k-] Common Radices: decimal: k =, symbols = [ 9] binary: k = 2, symbols = [,] octal: k = 8, symbols = [...7] hexadecimal: k = 6, symbols = [ 9, a, b, c, d, e, f ] Three systems of representing unsigned numbers, all of them positional, but each based on a different radix, or base: Decimal (radix-): This is the standard representation that you re used to. Binary (radix-2): This is the actual representation used in all digital systems at the hardware level. Hexadecimal, or hex, (radix-6): This is closely related to binary, often used to represent values in digital systems using fewer digits than required by binary. The octal (radix-8) representation is also common, but we won t use it in this course. EE37/CC/Lecture#3 3 EE37/CC/Lecture#3 4

2 Binary Numbers Binary to Decimal Exercise Binary (base 2) numbers are arranged like decimal numbers in positions. Each position stands for a power of 2. Example: 2 is x2 5 + x2 4 + x2 3 + x2 2 + x2 + x2 = = 53 Changing from binary to decimal just means adding the powers of 2 for which the position has a. 2 A. 239 B. 247 C. 25 D. 253 E. 255 is EE37/CC/Lecture#3 5 EE37/CC/Lecture#3 6 How Do We Change from Decimal to Binary? Divide the decimal number repeatedly by 2. Put the remainders from right to left. 53/2 = 26 R 26/2 = 3 R 3/2 = 6 R 6/2 = 3 R 3/2 = R /2 = R Decimal to Binary Exercise 99 is A. B. C. D. E. 53 = 2 EE37/CC/Lecture#3 7 EE37/CC/Lecture#3 8 2

3 How Many Bits Do We Need? The range of unsigned eight-bit binary numbers is: The range of unsigned 6-bit binary numbers is: The range of unsigned n-bit binary numbers is: range = [, 2 n -] How Many Bits Do We Need? The number of binary digits (bits) we need depends on the size of the number we want to represent. 8 bits gives 256 numbers 6 bits gives 65.5k numbers 24 bits gives 6.7M numbers 32 bits gives 4.3 G numbers EE37/CC/Lecture#3 9 EE37/CC/Lecture#3 Observations Computer Prefixes Modern computers use multiples of 8 bits. Sometimes these have special names. 8 bits 6 bits 32 bits byte word Standard computer prefixes are based on powers of 2 rather than powers of and some are pronounced differently. Symbol K M G T Pronounced kay meg gig tera Power of bits long word P peta 5 We will mainly use 8 and 6 bits in EE37. E exa 6 Know at least the first four of these. EE37/CC/Lecture#3 EE37/CC/Lecture#3 2 3

4 Grouping Bits into Hex Long bit strings are hard to remember recognize work with Hence we group four bits together in a hexadecimal (base 6) digit or nibble. Hex Decimal Binary A B C 2 D 3 E 4 F 5 EE37/CC/Lecture#3 3 How Do We Change Binary to Hex? Starting at right, group bits by four. Change each group into its hex number. Put $ (Motorola s hex designator) in front. % is Motorola s binary designator. Decimal is default (no designator.) Note: Just pad on the left with enough s to get a multiple of four bits. Example: % D E 9 B $DE9B EE37/CC/Lecture#3 4 Binary to Hex Exercise % is A. $B586 B. $BD8E C. $E8DB How Do We Change Hex to Binary? Replace each hex digit by its 4-bit binary equivalent. Put a % (Motorola s binary designator) in front. Example: $A476 A % EE37/CC/Lecture#3 5 EE37/CC/Lecture#3 6 4

5 Hex to Binary Exercise Summary: Conversion Between Radices $28A7 is A. % B. % C. % From: binary or hexadecimal To: decimal use the radix expansions Binary to or from Hex Each hex digit = exactly 4 binary bits. So, one (8-bit) byte can hold 2 hex digits Simply collect bits into groups of 4 (starting at right): EE37/CC/Lecture#3 7 EE37/CC/Lecture#3 8 Boolean Values Unsigned Numbers Two states ON OFF +5 V V open closed clockwise counterclockwise all s any nonzero value EE37/CC/Lecture#3 9 Unsigned numbers Use all bits for magnitude Can be thought of as positive. b7 b6 b5 b4 b3 b2 b b Each bit represents 2 n Range is to 255 (2 8 -) Observations If the right bit is, the number is even. If the right n bits are, the number is divisible by 2 n If the left bit of an 8-bit number is, the number is at least 28. EE37/CC/Lecture#3 2 5

6 How Can We Represent Signed Numbers? We need one bit for the sign. Use the leftmost bit. = +, = - Other bits are magnitude. Three possibilities shown with 4 bits. +5 =, -5 = sign-magnitude +5 =, -5 = s complement +5 =, -5 = 2 s complement Observations? Twos Complement Signed Numbers Definition: negation = complement all bits then add. The most-significant bit is the sign-bit = for positive numbers = for negative numbers Range = [-2 n-, 2 n- -] Uses the same arithmetic hardware as unsigned overflow bits are discarded Subtracting a number is equivalent to adding its twos complement negative number (pp.8) EE37/CC/Lecture#3 2 EE37/CC/Lecture# bit 2 s complement How Can We Find 2 s Complement Representations? 2 s complement is used in modern computers. Consider 3-bit 2 s complement. Range is -4 to +3. Number b3 b2 b For positive numbers Write binary equivalent Add s at left as needed for total of n bits. Example: 8-bit 2 s complement representation of = = in 2 s complement -4 EE37/CC/Lecture#3 23 EE37/CC/Lecture#3 24 6

7 How Can We Find 2 s Complement Representations? For negative numbers Write binary equivalent of positive number Add s at left as needed for n bits total. Complement each bit. ( ; ) Add. Example: 8-bit 2 s complement rep. of = = in 2 s complement -7 = + = EE37/CC/Lecture# s Complement Exercise -23 in 6-bit 2 s complement is A. B. C. D. EE37/CC/Lecture#3 26 Observations To find the magnitude of a negative 2 s complement number left bit = Complement each bit Add Example: + = or 7 2 s complement represents both positive and negative numbers. Observations We can negate a positive 2 s complement number with exactly the same procedure. Complement each bit. Add. Example: + = or -7 decimal EE37/CC/Lecture#3 27 EE37/CC/Lecture#3 28 7

8 Magnitude of 2 s Complement Number The magnitude of the 6-bit 2 s complement number is A. B. 2 C. 3 D. 4 E. 22 Range of 2 s Complement Numbers Range of n-bit 2 s complement numbers is -2 n- to 2 n- -. Examples: 4-bit to to +7 8-bit to to +27 EE37/CC/Lecture#3 29 EE37/CC/Lecture#3 3 Range of 6-bit 2 s Complement A. -64 to +63 B. -64 to +64 C. -32 to +32 D. -32 to +3 E. -3 to +32 How Do We Change 8-bit 2 s Complement to More Bits? Sign Extension Copy the sign bit to the left. Examples: 8-bit to 6-bit 8-bit to 32-bit EE37/CC/Lecture#3 3 EE37/CC/Lecture#3 32 8

9 How Do We Encode Decimal Numbers? Binary Coded Decimal (BCD) is handy for decimal numbers. Each digit is encoded in 4-bit binary. Why BCD? Convenient for certain I/O devices that work w/ decimal numbers Examples: 97 = BCD How Do We Know if Numbers Are Signed or Unsigned? You, the programmer, decide whether numbers are signed or unsigned. The computer doesn t know what a bit pattern means until you tell it. The bit pattern could be a character. 2 = BCD EE37/CC/Lecture#3 33 EE37/CC/Lecture#3 34 Character Representations Characters must be represented as binary numbers upper-case, lower-case, Numerals, Punctuation, control characters (non-printing) How many Bits? 7 bits : 28 Characters = EBCDIC & ASCII 8 bits : 256 Characters = Extended ASCII 6 bits : 26 Characters = Unicode How Do We Represent Characters? ASCII (American Standard Code for Information Interchange) Standard (7-bit) 27 characters Extended (8-bit) 255 characters Example: M = $4D = % from table EE37/CC/Lecture#3 35 EE37/CC/Lecture#3 36 9

10 ASCII Character Code Control Characters: First 32 values [ 2 6 ] Numerals: number X = 3X 6 i.e. [ ] very easy to recognize and decode Upper Case = [4 6 5A 6 ] = [,,] Lower Case = [6 6 7A 6 ] = [,,] Punctuation is scattered through remaining patterns ASCII EE37/CC/Lecture#3 37 EE37/CC/Lecture#3 38 Summary ASCII Extended Converting numbers among decimal, binary and hex (EE27) Two s complement numbers (EE27) Ranges of signed and unsigned numbers How to represent characters EE37/CC/Lecture#3 39 EE37/CC/Lecture#3 4