# Course Schedule. CS 221 Computer Architecture. Week 3: Plan. I. Hexadecimals and Character Representations. Hexadecimal Representation

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1 Course Schedule CS 221 Computer Architecture Week 3: Information Representation (2) Fall 2001 W1 Sep 11- Sep 14 Introduction W2 Sep 18- Sep 21 Information Representation (1) (Chapter 3) W3 Sep 25- Sep 28 Information Representation (2) (Chapter 3) W4 Oct 2- Oct 5 Computer Architecture Pep/6 (Chapter 4) W5 Oct 9- Oct 12 Assembly Language (Chapter 5) W6* Oct 16- Oct 19 High-Level Languages (1) (Chapter 6) W7* Oct 23- Oct 26 High-Level Languages (2) (Chapter 6) Midterm* W8 Oct 30- Nov 2 Instruction Sets (Tanenbaum) W9 Nov 6 - Nov 9 Devices & Interrupts (Chapter 8) W10* Nov 13- Nov 16Storage Management (Chapter 9) W11 Nov 20- Nov 23 Combinational Logic (Chapter 10) W12 Nov 27- Nov 30 Sequential Logic & Review (Chapter 11) *Marked classes will be taught by Dave Dove, same place same slot CISC 221 Fall 2001 Week 3 1 CISC 221 Fall 2001 Week 3 2 Week 3: Plan Hexadecimal Representation Character Representation Floating Point Representation Introduction to PEP/6 Readings Chapter 3 of book Next week: Chapter 4 (please start reading!) I. Hexadecimals and Character Representations CISC 221 Fall 2001 Week 3 3 CISC 221 Fall 2001 Week 3 4 Hexadecimal Representation Binary numbers are hard to read for humans. Hexadecimal and octal notations are just two different choices of the base in a weighted radix position number system. Hexadecimal is base 16, and octal base 8. Remember the Octal System 0 10 (8) (7*8 = 56) 100 (8*8 = 64) (15) 77 This is called the octal system of counting or base = 1 x x x 8 0 = 65 CISC 221 Fall 2001 Week 3 5 CISC 221 Fall 2001 Week 3 6 1

2 Hexadecimal System 0 10 (16)... F0 (15*16 = 240) 100 (16*16 = 256) For decimal 10 through 15, we use A to F 9 19 This is called the hexadecimal system A 1A or base 16 B 1B C 1C 101 = 1 x x x 16 0 = 257 D 1D E 1E F 1F (31) CISC 221 Fall 2001 Week 3 7 So Why Hexadecimal? Hexadecimal and Octal are useful because They are easier for people to deal with than a strict binary representation: a shorthand. Because the bases are powers of two, the digits in hexadecimal and octal representations map straight onto groups of bits in the binary representation. Each hexadecimal digit corresponds to 4 binary bits or a nibble Each octal digit corresponds to 3 binary bits binary binary 7 F hex A 8 hex It is easy to remember the binary nibble patterns that correspond to a hexadecimal character! CISC 221 Fall 2001 Week 3 8 F00D is a hexadecimal number F00D 15 * * * * 16 0 = 15* * * = However, to convert hexadecimal into binary, just convert each hex digit into 4 bits: F00D is The sign of a hex constant in 2 s complement is indicated by the value of the most significant bit. FOOD is negative because 'f' represents a bit pattern greater than or equal to 8. CISC 221 Fall 2001 Week 3 9 From Decimal to Hex Divide the number by 16, the base, and write down the remainder / 16 remainder d by now we have divided by by now we have divided by by now we have divided by 16 4 Now read back in reverse order: 10d9! Use hex as intermediate when converting to binary 10d9 hex is in binary! CISC 221 Fall 2001 Week 3 10 Representing Characters Since all data are represented as bits, we need to encode characters as binary numbers. ASCII Table Assignment of values to the chars can be arbitrary But use of arithmetic comparisons of the codes should be possible to do alphabetical sorting. ASCII (American Standard Code for Information Interchange) is the most common code. Encodes letters used in English, punctuation marks, and some control characters into 7 bit numbers. However, typically a full byte is used for convenience, leaving the msb empty. CISC 221 Fall 2001 Week 3 11 CISC 221 Fall 2001 Week

3 Other Coding Schemes There are standards to extend ASCII to include characters from more alphabets: Characters are commonly stored one per 8-bit byte, which gives you another bit to play with. ISO-8859-x series codes the first 128 characters the same as ASCII, and the upper 128 characters are different sets of special symbols and accented characters from various Roman alphabets. ISO is called "Latin One". Unicode is a 16-bit encoding that represents scripts for all the alphabets in the world. Char types in Java use Unicode. The least significant 8 bits of Unicode encodings are compatible with ISO-8859 and thus ASCII. CISC 221 Fall 2001 Week 3 13 II. Floating Point Representations CISC 221 Fall 2001 Week 3 14 Announcements There will be an extra tutorial on PEP/6 this Friday Details will follow So we haven t discussed fractions! So how do we represent fractions in binary? We can t use any of the schemes we ve used so far. Because they can only represent powers of 2. But instead of interpreting binary numbers using positive exponents like 2 n We could interpret them as negative exponents: 2 -n!! 1010 = 1* * * *2-4 1 * 1/2 + 0* 1/4 + 1*1/8 + 0*1/16 =.625 CISC 221 Fall 2001 Week 3 15 CISC 221 Fall 2001 Week 3 16 Fractions in Decimal Note that this works like decimal scientific notation (decimal) means 1* * * * * *10-4 = 1*10 + 2*1 + 4/10 + 5/ / /10000 In scientific notation: = * 10-5 = * 10-4 = * 10-2 = * 10-1 = * 10 1 (this is called normalized notation) Negative Fractions in Binary So how do we represent negative numbers? Simple: use 2 0 (=1). It s the MSB, and we can use it as a sign bit. So our original bit pattern should have been: = - 0* * * * *2-4 or * 1/2 + 0* 1/4 + 1*1/8 + 0*1/16 = = * 1/2 + 0* 1/4 + 1*1/8 + 0*1/16 =.375 Our notation now is called Fixed Point notation CISC 221 Fall 2001 Week 3 17 CISC 221 Fall 2001 Week

4 Floating Point Notation Note that we could have put the decimal point at any arbitrary location and make it a standard. However, it would be fixed after that. Which is why it is called fixed point However, scientific calculations often involve very small as well as very large numbers. Would be neat if one number could represent The mass of the sun: 2 * grams The mass of an electron: 9 * grams We would need a lot of bits for this in fixed point. To write them both out as complete decimal fractions that you could line up for addition would require 62 decimal digits (34 before the decimal point and 28 after it). Yet each number has only one significant digit, so the 64 digit representation would store a great many 0s. CISC 221 Fall 2001 Week 3 19 Floating Point Notation (2) The alternative would be to have a limited precision: we could not represent all numbers. Note that we can never represent all fractions! Or have a floating point, where we could adjust precision according to our needs. Scientific decimal notation actually does this. A number is represented by a fraction or mantissa which corresponds to the precision of the value. The fraction is multiplied by ten raised to the power of a specified exponent (e below): x = f * 10 e where f is the fraction or mantissa and e is the exponent CISC 221 Fall 2001 Week 3 20 So what exponent do we use? Where do we put the decimal point? In scientific notation, fractions are normalized the exponent is adjusted so that there is one digit to the left of the decimal point. Floating Point in Binary (2) Floating Points in binary work the same as in decimal, except the base is 2 rather then 10 x = f * 2 e where f is the fraction and e is the exponent = ( * 10 4 ) decimal corresponds to * 10-5 = * 10-4 = * 10-2 = * 2 0 = * 10-1 = * 10 1 (normalized notation) and then we normalize by shifting right * 2 14 = So write * 10 4 rather than * *(0* * * 2-27) CISC 221 Fall 2001 Week 3 22 CISC 221 Fall 2001 Week 3 21 Conversion to Binary: Whole part How do you convert from decimal fractions to binary ones? Convert the whole number part using familiar method of repeated divisions & remainders Conversion to Binary: Fraction How do you convert the fractional part? Repeatedly multiply the fraction by two. When the result is greater than or equal to one, subtract one from the product and record a one in the corresponding bit position. CISC 221 Fall 2001 Week 3 23 CISC 221 Fall 2001 Week

5 Conversion to Binary: Fraction (2) This counts how many many halves are in the fraction, then how many quarters, how many eighths, how many sixteenths, etc. You stop until you have reached your precision (run out of bits). Or get exactly zero after subtracting one. So What Do Floating Points Look Like? Remember x= f*2 e So how do bits represent the fraction & exponent? IEEE 754 Standard Floating Point Format. One for 32 bit precision and 64 bit precision. Corresponds to float and double in java. The resulting is the closest binary approximation for that fraction. 32-bit single precision 64-bit double precision CISC 221 Fall 2001 Week 3 25 CISC 221 Fall 2001 Week 3 26 What Do Floating Points Look Like (2) Sign bit different from integers or fixed point Simply 1 for negative numbers and 0 for positive numbers: thus 2 representations of zero (-0, +0) The fractional part thus always represents an unsigned fraction The exponent bits need also represent negative. Do not use two s complement because we want to be the smallest and the largest. Thus, we can more easily compare small and large numbers only by comparing their exponents. Exponent = value of bits - (2 n-1-1) If exponent has 8 bits, value is bits (range -127 to +127 ) If exponent has 11 bits, value is bits (range to +1023) CISC 221 Fall 2001 Week 3 27 The Fraction, Mantissa, Significant 23 bits in single precision, 52 in double precision. When a binary fraction is normalized, there will always be a 1 to the left of the binary point. It therefore isn't represented in the IEEE format The word significand is used to refer to the combination of the implied 1, the binary point, and the explicit 23 or 52 bits. So we can actually represent 24 or 53 bits x = sign * 1. fraction * 2 exponent CISC 221 Fall 2001 Week 3 28 s Decimal fraction that is easy to convert to binary is a quarter plus one eighth 0.011(binary) * 2-0 or 1.1(binary) * 2-2. The sign bit is 0 for a positive number. The exponent bits equivalent to 7D(hex) or 125(dec) represent an exponent of = -2. The remaining bits represent the significand 1.1(binary): remember that the leading 1 is not stored. Floating Point Overflow & Underflow With integers we have the problem of overflow when the absolute value of a number is too great to fit into the number of bits With floating point, there is the possibility of underflow as well: an absolute value too small to be represented. Take single-precision values. When the fraction is normalized, we can't express negative values which are less than: (binary) * (~ -2 * = , or -1.7 * ) or positive values greater than (binary) * (~ 2 * = or 1.7 * ). IEEE 754 Single-precision floating point representing CISC 221 Fall 2001 Week 3 29 Numbers whose absolute values are greater than these limits constitute an overflow. CISC 221 Fall 2001 Week

6 Floating Point Overflow & Underflow We also are unable to represent non-zero negative numbers greater than: (binary) * (-2-126, approx * ) i.e. negative fractions with tiny absolute values, or non-zero positive numbers less than: (binary) * (2-126, approx 1.2 * ) Numbers whose absolute values are less than these limits constitute an underflow. CISC 221 Fall 2001 Week 3 31 Special Values In addition to the normalized numbers, there are 4 special cases: If the exponent is 0 and the fraction is 0: the number represents + or - 0. If the exponent bits are all 1s and the fraction bits are all 0s: + or - infinity (e.g., division by zero) If the exponents are all 1s and the fractional bits are anything but all zeros this is not a number (NaN), the value used to represent undefined results like infinity divided by infinity. Also + or - If the exponent is 0 and the fractional bits are anything but all zeros this is a 'denormalized' number, with 0 instead of 1 as the implicit bit before the binary point. Used to represent smaller numbers, up to CISC 221 Fall 2001 Week 3 32 What is PEP/6 PEP/6 is a virtual computer running on your real computer as a software program II. Introduction to PEP/6 It was designed to teach you the essence of programming machine code and assembly Without becoming too hardware dependent. Without having to crash a unix box. We will use the PEP/6 simulator throughout the course as an instructional tool It s up to you to relate this to real processors. CISC 221 Fall 2001 Week 3 33 CISC 221 Fall 2001 Week 3 34 Simulated Hardware PEP/6 Hardware consists of four major components at the machine level: The central processing unit (CPU) The main memory The input device The output device Communicate with each other via a bus. Input device CPU Bus Main Memory Output device CISC 221 Fall 2001 Week 3 35 The CPU The CPU contains seven direct onboard memory locations called registers The 4-bit status register (NZVC) Negative Zero overflow Carry The 16-bit accumulator (A) Contains result of operation The 16-bit index register (X) Contains result of operation The 16-bit base register (B) Contains start address of array The 16-bit program counter (PC) Contains address of current instruction The 16-bit stack pointer (SP) Contains address of a stack The 24-bit instruction register (IR) Holds instruction after fetch from memory The CPU also holds all the circuitry to execute instructions CISC 221 Fall 2001 Week

9 ADD Add Operand to Register R Adds one word from main memory to either the index register or accumulator Affects all status bits Add A direct operand address 004A (h) in main memory Before A: FFB6 (h) After A: FFd9 (h) Mem[004A]: 0023 (h) Mem[004A]: 0023 (h) NZVC: 1000 result is negative Subtract Subtract Operand to Register R Subtracts one word from main memory from contents of either the index register or accumulator Affects all status bits Subtract X direct operand address 004A (h) in main memory Before X: 000d (h) 13 (d) After X: FFFE (h) -2 (d) Mem[004A]: 00 (h) 15 (d) Mem[004A]: 00 (h) 15 (d) NZVC: 1001 result is negative and carry because we borrowed into the msb CISC 221 Fall 2001 Week 3 49 CISC 221 Fall 2001 Week 3 50 AND AND Operand to Register R Performs logical AND operation to either the index register or accumulator with a word from memory Affects N and Z bits Handy for stripping away bits: masking AND with 0 strips bits, AND with ones keeps bits : Strip most significant 12 bits (3 nibbles) And X direct operand address 004A (h) in main memory OR OR Operand to Register R Performs logical OR operation to either the index register or accumulator with a word from memory Affects N and Z bits Handy for inserting one bits into bit patterns: fill OR with 1 always sets bit : Set most significant 12 bits to ones (3 nibbles) Or X direct operand address 004A (h) in main memory Before X: 6B5D (h) After X: 000D (h) Mem[004A]: 00 (h) Mem[004A]: 00 (h) NZVC: 0000 CISC 221 Fall 2001 Week 3 NZVC: 1000 because register turned negative 52 CISC 221 Fall 2001 Week 3 51 Before X: OC4D (h) After X: FFFD (h) Mem[004A]: FFF0 (h) Mem[004A]: FFF0 (h) Questions?? CISC 221 Fall 2001 Week

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