# 1 Number System (Lecture 1 and 2 supplement)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 1 Number System (Lecture 1 and 2 supplement) By Dr. Taek Kwon Many different number systems perhaps from the prehistoric era have been developed and evolved. Among them, binary number system is one of the simplest and effective number systems, and has been extensively used in digital systems. Studying number systems can help you understand the basic computing processes by digital systems. 1.1 Positional Number Systems A good example of positional number system is the decimal number system in which we use them almost everywhere number is needed. Another example is the binary system that is used as the basic number system for all computers. In positional number systems, a number is represented by a string of digits where the position of each digit is associated with a weight. In general, a positional number is expressed as: d d d d. d d d m 1 m n where d m 1 is referred to as the most significant digit (MSD) and d n as the least significant digit (LSD). Each digit position has an associated weight b i where b is called the base or radix. The point in the middle is referred to as a radix point and is used to separate the integer and fractional part of a number. Integer part is in the left side of the radix point; fraction part is in the right side of the radix point. Fraction is a portion of magnitude of a number which is less than unit (e.g. fraction < 1) and thus it is called a fraction. Let D denote the value (or magnitude) of a positional number, then D can be always calculated by: m i D = 1 di b (1) i= n Example 1.1.1: Find the magnitude of D = = ECE 1315, Number System Supplement 1

2 A binary (base=2) number system is a special case of the positional number system in which the allowable digits are 0 and 1 that are called bits. The leftmost digit of a binary number is called the most significant bit (MSB) and the rightmost is called the least significant bit (LSB). Because the base of binary numbers is two, bit b i is associated with weight 2 i. Example 1.1.2: Magnitude of Binary number 10 = = If the base of a number system is larger than ten, the digits exceeding 9 are expressed using alphabet letters as a convention. For example, hexadecimal number system uses 1-9 and A-F; base-32 number system uses 1-9 and A-V. This example is shown in Table 1. One may then wonder how large-base number systems such as a base-64 are expressed. Fortunately, we rarely use such a high-base number system because we find no real advantages of using them in applications. Moreover, we can always convert them from any high-base number system to a lower base number system, which is the subject of the next section. 1.2 Conversion between 2 k bases Number systems with 2 k bases have an interesting property in that the conversion between them can be achieved without the computation of Eq. (1). Such number systems include binary, octal, hex, and base-32 number systems. Note that since these number systems possess base 2 k, all numbers within these systems can be uniquely represented by k binary bits. For example, octal numbers can be represented by three bits; hex numbers can be represented by four bits, etc. This relation allows us to easily convert between them by simply grouping their binary representation with k bits. Two examples are given in Example Since the binary representation of 2 k base numbers can be directly associated by simple grouping of k digits, the conversion from octal to hex or vice versa can be easily achieved through intermediate step of binary conversion. Example illustrates this conversion step. ECE 1315, Number System Supplement 2

3 Table 1. Decimal, binary, hexadecimal, and base-32 Number Systems Decimal Binary Octal Hexadecimal Base A A B B C C D D E E F F G H I J K L M N O P A Q B R C S D T E U F V ECE 1315, Number System Supplement 3

4 Example 1.2.1: Binary to hexadecimal or octal conversion 11 = = = 11 2 = D = = = = D2. B 2 Example 1.2.2: Octal to hexadecimal or vice versa 273 = = = BB We have seen that the conversion between numbers with power of radix 2 can be readily achieved through binary expression and regrouping of bits. This convenience led to utilization of hexadecimal (or octal) numbers in representing binary numbers for many computer architecture related issues. For example, the instruction LDAA (Load Accumulator A) of 68HC11 is encoded as the binary number , but for convenience of writing and reading it is usually expressed in hexadecimal 86, from which we save time and spaces. Very often, hexadecimal, octal, and binary numbers are interchangeably used in the computer architecture or microprocessor related fields. ECE 1315, Number System Supplement 4

5 1.3 General Positional Number System Conversion This section discusses conversion of numbers from any base to any other base. Due to our familiarity and representation of decimal, a convenient way of base-conversion is conversion through the use of decimal. That is, for the conversion from base-k to base-p, we first convert a base-k number to a decimal, and then convert the decimal to a base-p number. Using Eq. (1) we can easily convert from any base to decimal by simply expressing the digits and weights using decimal as shown in Example Therefore, this is the simple case. Example 1.3.1: Base-k to decimal conversion 1bE 8 = = = = When a decimal number is converted to a base-p number system, thing get little bit more complicated. This process usually requires more computation. Let a general position number be denoted as an addition of integer and fractional part: D = I + F. (2) Then, with some manipulation we can express the integer part as I = (( (( d ) b + d ) b + ) b + d ) b + d (3) p 1 p Although this formula looks complicated at a first glance, its structure is exactly the same as the integer part of Eq. (2) except that the weights b k is now expressed as b b b k 1 2. Example illustrates conversion of a decimal number 5432 into the form given by Eq. (3). Example 1.3.2: Integer expressions of positional numbers 54 = = ( 5 + 4) = (( 5 + 4) + 3) + 2 ECE 1315, Number System Supplement 5

6 From Example 1.3.2, notice that if the last expression is divided by the remainder is the least significant digit 2 and the quotient is (( 5 + 4) + 3). The next significant digit can be obtained by dividing 535 again by. Due to this relation, the conversion to an arbitrary base number can be obtained by repeated division of quotient and collection of remainders. A simple hand-calculation method can be devised using the above relation. Let's express the integer division by the following form. Divisor )Dividend Quotient...Remainder Using this expression, Example shows conversion from a decimal to a binary. Example 1.3.3: Convert 179 to a binary. 2 )179 2 ) LSB 2 ) ) ) ) ) MSB The final conversion result reads 179 = It should be noted that the above method can be extended to conversion of any other base. For example, consider that we wish to convert a hexadecimal number to a base-5 number. Then, the base-5 number can be directly converted by repeated division by 5 and collecting remainders. However, this direct division means, you must divide the base- number by 5, which is not simple because we are only used to decimal numbers. Thus, it is essentially wise to first convert the hexadecimal to a decimal, and then convert it to base-5. Similarly to the expression of integer part in Eq. (3), the fractional part can be written in the following form: F = ( d + ( d + ( b d ) b ) b ) b (4) n Note that multiplying b to F in Eq. (4) produces d 1 as a part of the product. This representation of number system is illustrated using Example ECE 1315, Number System Supplement 6

7 Example 1.3.3: Fraction expression of positional numbers 012. = ( 1+ 2 ) = ( 1+ ( ) ) = ( 1+ ( 2 + ( ) ) ) F1 = ( ) F1A = ( ) ) F1AC = ( ( + 12 ) ) ) Due to the structure described above, a fractional number expressed by decimal can be converted into a base b number by collecting the integer part from left to right after each multiplication by b, i.e., see Example Example 1.3.4: Convert a decimal number to binary d 1 = 1 d = 0 d = Thus, = One should be careful to note that a closed form fraction in one number system does not always lead to a closed form in other number system. This case is illustrated in Example Example 1.3.4: Decimal to base-x conversion: Convert 0. 7 to a binary x 2 x 2 x 2 x 2 x 2 x d = 1 1 d = 2 0 d 3 = 1 d 4 = 1 d 5 = 0 d 6 = 0 ECE 1315, Number System Supplement 7

8 Thus, 0. 7 = This example implies that the base conversion of fractional part can introduce errors by the conversion process itself. If a fractional has a repeating pattern, it is a repeating fraction. If a fractional part does not repeat but goes forever, it is called an irrational number. 1.4 Negative Numbers Signed Magnitude Number System Negative numbers can be represented in many ways. In our daily transactions, a signed magnitude system is used, where a number consists of a magnitude and a symbol indicating whether the magnitude is positive or negative. For example, 57, + 98, , In the above example, the symbols + and - were used to represent the sign of a number. An alternative is to use an extra digit to represent positive and negative instead of introducing a new symbol. This technique is frequently used in the binary number system, e.g., bit 1" is appended at MSB to represent negative and bit ``0'' appended for positive. Example illustrates this relation by 8-bit numbers with 7-bit magnitude and one sign-bit, which is called signed magnitude binary numbers. Example 1.4.1: Examples of signed magnitude binary numbers 0011 = + 2D 10 = 2D = + 7F = 7F sign bit Complement Number System In the complement number system, a negative number is determined by taking its complement as defined by the system. Radix complement and diminished-radix complement are the two basic methods in this system. ECE 1315, Number System Supplement 8

9 i) Radix complement: The complement of an n-digit number is obtained by subtracting it from b n. See Example Example 1.4.2: Radix complements 's complement: = 's complement: = 's complement: = 's complement: = 012 As in the above example, direct subtraction from b n is inconvenient or at least cumbersome to calculate because of borrows. A simpler and easier way is derived by modifying the subtraction as: n n b D = ( b 1) D + 1 Notice that b n 1 has the form that all digits are the highest digits in the number system. For 4 example, in decimal 1 = 9999, in octal = 7777, in binary = 1111, etc. This means that the computation never needs borrow, so it makes the 2 radix computation easier. ii) Diminished-Radix complement: The complement of an n-digit number D is obtained by substituting it from b n 1. This can be accomplished by complementing the individual digits of D without adding 1. Example 1.4.3: 9's complement In decimal, the diminished-radix complement is called the 9's complement because the complement is obtained by independently subtracting each digit from 9. Complement of = 8150 = 1849 Complement of = 2067 = 7932 Complement of = 9992 = 0007 Example 1.4.4: 1's complement Similarly to the decimal case, the diminished-radix complement of a binary number is called 1's complement because the complement is obtained by subtracting each digit from 1. Complement of = 002 = 112 ECE 1315, Number System Supplement 9

10 Complement of = 2 = 012 Complement of = = Note from Example that 1's complement is simply obtained by inverting each digit, i.e. 1 0 and 0 1. Thus, the main advantage of 1's-complement system is its simplicity of conversion and the symmetry of complements. However, this symmetry causes the existence of two zeros, i.e., a positive zero and a negative zero Hence implementing addition of 1's complement numbers to a digital computer system leads to significant inefficiency because the system must check for both representations of zeros or it must convert one to another zero. This is the main reason why 2's complement number system is used for all of today's digital computers, which has a unique zero ( ). Observe the differences between the different sign systems from Table 2. Table 2. 4-bit Numbers in Different Signed Systems Decimal 2's 1's Signed Complement complement Magnitude or or ECE 1315, Number System Supplement

12 ( 8 ) 11 ( +7 ) ( 8 ) ( + 7 ) All of the examples given in Example have the overflow error condition. What that means is that you cannot compute the given numbers with only four bits. You need more bit positions, if you wish to correct the error Signed subtraction Signed subtraction in most computers is done by taking 2's complement of the subtrahend and then adding it to the minuend following the normal rules of addition. Overflow condition must be checked after the addition in order to obtain the correct computational result. If no overflow condition is detected, the correct answer of the subtraction is obtained from the result by simply discarding the carry-out bit of the MSB if a carry-out bit exists. If an overflow condition is detected, there are two ways of dealing with this error. The first approach is simply reporting an error message that indicates the overflow condition. Most computers use this approach and leave the responsibility of handling the error to the user. The second approach is modifying the result to a correct one by allocating more bits to the addends. Whenever an overflow occurs, only one more bit extension to operands is needed to express the overflowed number. However, due to the fixed data width of computers, the data width is usually extended twice of the data width, i.e., if a single precision computation is overflowed, a double precision (twice the data width) is used to correct the error. Example 1.5.2: Signed subtraction with no overflow Compute =? Step 1) Compute the 2's complement of 0011, i.e., = 11 Step 2) Add the complemented number to the minuend: Step 3) Check overflow. Since the signs of the addends are different, there is no overflow. Simply discard the MSB carry-out bit and the correct answer of this computation is Example 1.5.3: Signed subtraction with overflow and the correction Compute =? Step 1) Compute the 2's complement of 11, i.e., = 0011 ECE 1315, Number System Supplement 12

15 1.6.2 Unsigned Subtraction In unsigned subtraction, the minuend must be larger than the subtrahend. Otherwise, the result would become negative, which violates the definition of unsigned computation. If the subtracted result is actually negative, an occurrence of error should be indicated. In a computer implementation, this error condition is shown through a borrow bit. If the borrow bit is set, it means that the minuend is smaller than the subtrahend and indicates an error condition for unsigned computation. References [1] J. F. Wakerly, Microcomputer Architecture and Programming, John Wiley \& Sons, Inc., [2] D. Knuth, Seminumerical Algorithms, Addison-Wesley, 1969 ECE 1315, Number System Supplement 15

### Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8

ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: -Our standard number system is base, also

### The string of digits 101101 in the binary number system represents the quantity

Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for

### Chapter 1: Digital Systems and Binary Numbers

Chapter 1: Digital Systems and Binary Numbers Digital age and information age Digital computers general purposes many scientific, industrial and commercial applications Digital systems telephone switching

### Chapter 6 Digital Arithmetic: Operations & Circuits

Chapter 6 Digital Arithmetic: Operations & Circuits Chapter 6 Objectives Selected areas covered in this chapter: Binary addition, subtraction, multiplication, division. Differences between binary addition

### Lecture 11: Number Systems

Lecture 11: Number Systems Numeric Data Fixed point Integers (12, 345, 20567 etc) Real fractions (23.45, 23., 0.145 etc.) Floating point such as 23. 45 e 12 Basically an exponent representation Any number

### Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

### NUMBER SYSTEMS. 1.1 Introduction

NUMBER SYSTEMS 1.1 Introduction There are several number systems which we normally use, such as decimal, binary, octal, hexadecimal, etc. Amongst them we are most familiar with the decimal number system.

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

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

### Binary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal.

Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems

### Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University

Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent

### Lecture 2. Binary and Hexadecimal Numbers

Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations

### Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 INTRODUCTION TO DIGITAL LOGIC

Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n

### CPEN 214 - Digital Logic Design Binary Systems

CPEN 4 - Digital Logic Design Binary Systems C. Gerousis Digital Design 3 rd Ed., Mano Prentice Hall Digital vs. Analog An analog system has continuous range of values A mercury thermometer Vinyl records

### CSI 333 Lecture 1 Number Systems

CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...

### Lecture 8: Binary Multiplication & Division

Lecture 8: Binary Multiplication & Division Today s topics: Addition/Subtraction Multiplication Division Reminder: get started early on assignment 3 1 2 s Complement Signed Numbers two = 0 ten 0001 two

### CS201: Architecture and Assembly Language

CS201: Architecture and Assembly Language Lecture Three Brendan Burns CS201: Lecture Three p.1/27 Arithmetic for computers Previously we saw how we could represent unsigned numbers in binary and how binary

### Systems I: Computer Organization and Architecture

Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems - Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x

### Useful Number Systems

Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2

### Binary Representation

Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must tbe able to handle more than just values for real world problems

### Decimal Numbers: Base 10 Integer Numbers & Arithmetic

Decimal Numbers: Base 10 Integer Numbers & Arithmetic Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 )+(1x10 0 ) Ward 1 Ward 2 Numbers: positional notation Number

### LSN 2 Number Systems. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology

LSN 2 Number Systems Department of Engineering Technology LSN 2 Decimal Number System Decimal number system has 10 digits (0-9) Base 10 weighting system... 10 5 10 4 10 3 10 2 10 1 10 0. 10-1 10-2 10-3

### Divide: Paper & Pencil. Computer Architecture ALU Design : Division and Floating Point. Divide algorithm. DIVIDE HARDWARE Version 1

Divide: Paper & Pencil Computer Architecture ALU Design : Division and Floating Point 1001 Quotient Divisor 1000 1001010 Dividend 1000 10 101 1010 1000 10 (or Modulo result) See how big a number can be

### ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-470/570: Microprocessor-Based System Design Fall 2014.

REVIEW OF NUMBER SYSTEMS Notes Unit 2 BINARY NUMBER SYSTEM In the decimal system, a decimal digit can take values from to 9. For the binary system, the counterpart of the decimal digit is the binary digit,

### Goals. Unary Numbers. Decimal Numbers. 3,148 is. 1000 s 100 s 10 s 1 s. Number Bases 1/12/2009. COMP370 Intro to Computer Architecture 1

Number Bases //9 Goals Numbers Understand binary and hexadecimal numbers Be able to convert between number bases Understand binary fractions COMP37 Introduction to Computer Architecture Unary Numbers Decimal

### 6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

### 3. Convert a number from one number system to another

3. Convert a number from one number system to another Conversion between number bases: Hexa (16) Decimal (10) Binary (2) Octal (8) More Interest Way we need conversion? We need decimal system for real

### CDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012

CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Data Representation Binary Codes Why 6-3-1-1 and Excess-3? Data Representation (1/2) Each numbering

### Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 04 Digital Logic II May, I before starting the today s lecture

### Numeral Systems. The number twenty-five can be represented in many ways: Decimal system (base 10): 25 Roman numerals:

Numeral Systems Which number is larger? 25 8 We need to distinguish between numbers and the symbols that represent them, called numerals. The number 25 is larger than 8, but the numeral 8 above is larger

### 198:211 Computer Architecture

198:211 Computer Architecture Topics: Lecture 8 (W5) Fall 2012 Data representation 2.1 and 2.2 of the book Floating point 2.4 of the book 1 Computer Architecture What do computers do? Manipulate stored

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic

Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1

### 2011, The McGraw-Hill Companies, Inc. Chapter 3

Chapter 3 3.1 Decimal System The radix or base of a number system determines the total number of different symbols or digits used by that system. The decimal system has a base of 10 with the digits 0 through

### Solution for Homework 2

Solution for Homework 2 Problem 1 a. What is the minimum number of bits that are required to uniquely represent the characters of English alphabet? (Consider upper case characters alone) The number of

### Binary Numbers. Binary Octal Hexadecimal

Binary Numbers Binary Octal Hexadecimal Binary Numbers COUNTING SYSTEMS UNLIMITED... Since you have been using the 10 different digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 all your life, you may wonder how

### COMPSCI 210. Binary Fractions. Agenda & Reading

COMPSCI 21 Binary Fractions Agenda & Reading Topics: Fractions Binary Octal Hexadecimal Binary -> Octal, Hex Octal -> Binary, Hex Decimal -> Octal, Hex Hex -> Binary, Octal Animation: BinFrac.htm Example

### Numbering Systems. InThisAppendix...

G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### Number Systems I. CIS008-2 Logic and Foundations of Mathematics. David Goodwin. 11:00, Tuesday 18 th October

Number Systems I CIS008-2 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 11:00, Tuesday 18 th October 2011 Outline 1 Number systems Numbers Natural numbers Integers Rational

### Number and codes in digital systems

Number and codes in digital systems Decimal Numbers You are familiar with the decimal number system because you use them everyday. But their weighted structure is not understood. In the decimal number

### Chapter 2. Binary Values and Number Systems

Chapter 2 Binary Values and Number Systems Numbers Natural numbers, a.k.a. positive integers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative numbers A

### Section 1.4 Place Value Systems of Numeration in Other Bases

Section.4 Place Value Systems of Numeration in Other Bases Other Bases The Hindu-Arabic system that is used in most of the world today is a positional value system with a base of ten. The simplest reason

### Number Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi)

Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi) INTRODUCTION System- A number system defines a set of values to represent quantity. We talk about the number of people

### Binary Division. Decimal Division. Hardware for Binary Division. Simple 16-bit Divider Circuit

Decimal Division Remember 4th grade long division? 43 // quotient 12 521 // divisor dividend -480 41-36 5 // remainder Shift divisor left (multiply by 10) until MSB lines up with dividend s Repeat until

### Unsigned Conversions from Decimal or to Decimal and other Number Systems

Page 1 of 5 Unsigned Conversions from Decimal or to Decimal and other Number Systems In all digital design, analysis, troubleshooting, and repair you will be working with binary numbers (or base 2). It

### Simulation & Synthesis Using VHDL

Floating Point Multipliers: Simulation & Synthesis Using VHDL By: Raj Kumar Singh - B.E. (Hons.) Electrical & Electronics Shivananda Reddy - B.E. (Hons.) Electrical & Electronics BITS, PILANI Outline Introduction

### CS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011

CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS

### Digital Design. Assoc. Prof. Dr. Berna Örs Yalçın

Digital Design Assoc. Prof. Dr. Berna Örs Yalçın Istanbul Technical University Faculty of Electrical and Electronics Engineering Office Number: 2318 E-mail: siddika.ors@itu.edu.tr Grading 1st Midterm -

### Binary, Hexadecimal, Octal, and BCD Numbers

23CH_PHCalter_TMSETE_949118 23/2/2007 1:37 PM Page 1 Binary, Hexadecimal, Octal, and BCD Numbers OBJECTIVES When you have completed this chapter, you should be able to: Convert between binary and decimal

### Signed Binary Arithmetic

Signed Binary Arithmetic In the real world of mathematics, computers must represent both positive and negative binary numbers. For example, even when dealing with positive arguments, mathematical operations

### A Step towards an Easy Interconversion of Various Number Systems

A towards an Easy Interconversion of Various Number Systems Shahid Latif, Rahat Ullah, Hamid Jan Department of Computer Science and Information Technology Sarhad University of Science and Information Technology

### Number Representation

Number Representation CS10001: Programming & Data Structures Pallab Dasgupta Professor, Dept. of Computer Sc. & Engg., Indian Institute of Technology Kharagpur Topics to be Discussed How are numeric data

### 2010/9/19. Binary number system. Binary numbers. Outline. Binary to decimal

2/9/9 Binary number system Computer (electronic) systems prefer binary numbers Binary number: represent a number in base-2 Binary numbers 2 3 + 7 + 5 Some terminology Bit: a binary digit ( or ) Hexadecimal

### COMBINATIONAL CIRCUITS

COMBINATIONAL CIRCUITS http://www.tutorialspoint.com/computer_logical_organization/combinational_circuits.htm Copyright tutorialspoint.com Combinational circuit is a circuit in which we combine the different

### Binary. ! You are probably familiar with decimal

Arithmetic operations in assembly language Prof. Gustavo Alonso Computer Science Department ETH Zürich alonso@inf.ethz.ch http://www.inf.ethz.ch/department/is/iks/ Binary! You are probably familiar with

### 2 Number Systems 2.1. Foundations of Computer Science Cengage Learning

2 Number Systems 2.1 Foundations of Computer Science Cengage Learning 2.2 Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish between

### plc numbers - 13.1 Encoded values; BCD and ASCII Error detection; parity, gray code and checksums

plc numbers - 3. Topics: Number bases; binary, octal, decimal, hexadecimal Binary calculations; s compliments, addition, subtraction and Boolean operations Encoded values; BCD and ASCII Error detection;

### Number Systems and Radix Conversion

Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will

### YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

DETAILED SOLUTIONS AND CONCEPTS - DECIMALS AND WHOLE NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

### CS321. Introduction to Numerical Methods

CS3 Introduction to Numerical Methods Lecture Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 August 7, 05 Number in

### Number Systems, Base Conversions, and Computer Data Representation

, Base Conversions, and Computer Data Representation Decimal and Binary Numbers When we write decimal (base 10) numbers, we use a positional notation system. Each digit is multiplied by an appropriate

### 2 Number Systems. Source: Foundations of Computer Science Cengage Learning. Objectives After studying this chapter, the student should be able to:

2 Number Systems 2.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish

### The Essentials of Computer Organization and Architecture. Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003

The Essentials of Computer Organization and Architecture Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003 Chapter 2 Instructor's Manual Chapter Objectives Chapter 2, Data Representation,

### Binary Numbering Systems

Binary Numbering Systems April 1997, ver. 1 Application Note 83 Introduction Binary numbering systems are used in virtually all digital systems, including digital signal processing (DSP), networking, and

### Integer multiplication

Integer multiplication Suppose we have two unsigned integers, A and B, and we wish to compute their product. Let A be the multiplicand and B the multiplier: A n 1... A 1 A 0 multiplicand B n 1... B 1 B

### Binary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

### = Chapter 1. The Binary Number System. 1.1 Why Binary?

Chapter The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base-0 system. When you

### Fractional Numbers. Fractional Number Notations. Fixed-point Notation. Fixed-point Notation

2 Fractional Numbers Fractional Number Notations 2010 - Claudio Fornaro Ver. 1.4 Fractional numbers have the form: xxxxxxxxx.yyyyyyyyy where the x es constitute the integer part of the value and the y

### 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:

Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of

### Binary Numbers The Computer Number System

Binary Numbers The Computer Number System Number systems are simply ways to count things. Ours is the base-0 or radix-0 system. Note that there is no symbol for 0 or for the base of any system. We count,2,3,4,5,6,7,8,9,

### Base Conversion written by Cathy Saxton

Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Chapter 4. Arithmetic for Computers

Chapter 4 Arithmetic for Computers Arithmetic Where we've been: Performance (seconds, cycles, instructions) What's up ahead: Implementing the Architecture operation a b 32 32 ALU 32 result 2 Constructing

### Lab 4 Large Number Arithmetic. ECE 375 Oregon State University Page 29

Lab 4 Large Number Arithmetic ECE 375 Oregon State University Page 29 Objectives Understand and use arithmetic and ALU operations Manipulate and handle large numbers Create and handle functions and subroutines

### MACHINE INSTRUCTIONS AND PROGRAMS

CHAPTER 2 MACHINE INSTRUCTIONS AND PROGRAMS CHAPTER OBJECTIVES In this chapter you will learn about: Machine instructions and program execution, including branching and subroutine call and return operations

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### This 3-digit ASCII string could also be calculated as n = (Data[2]-0x30) +10*((Data[1]-0x30)+10*(Data[0]-0x30));

Introduction to Embedded Microcomputer Systems Lecture 5.1 2.9. Conversions ASCII to binary n = 100*(Data[0]-0x30) + 10*(Data[1]-0x30) + (Data[2]-0x30); This 3-digit ASCII string could also be calculated

### CS 16: Assembly Language Programming for the IBM PC and Compatibles

CS 16: Assembly Language Programming for the IBM PC and Compatibles First, a little about you Your name Have you ever worked with/used/played with assembly language? If so, talk about it Why are you taking

### Chapter 7 Lab - Decimal, Binary, Octal, Hexadecimal Numbering Systems

Chapter 7 Lab - Decimal, Binary, Octal, Hexadecimal Numbering Systems This assignment is designed to familiarize you with different numbering systems, specifically: binary, octal, hexadecimal (and decimal)

### Borland C++ Compiler: Operators

Introduction Borland C++ Compiler: Operators An operator is a symbol that specifies which operation to perform in a statement or expression. An operand is one of the inputs of an operator. For example,

### COMPUTER ARCHITECTURE. ALU (2) - Integer Arithmetic

HUMBOLDT-UNIVERSITÄT ZU BERLIN INSTITUT FÜR INFORMATIK COMPUTER ARCHITECTURE Lecture 11 ALU (2) - Integer Arithmetic Sommersemester 21 Leitung: Prof. Dr. Miroslaw Malek www.informatik.hu-berlin.de/rok/ca

### Positional Numbering System

APPENDIX B Positional Numbering System A positional numbering system uses a set of symbols. The value that each symbol represents, however, depends on its face value and its place value, the value associated

### Sistemas Digitais I LESI - 2º ano

Sistemas Digitais I LESI - 2º ano Lesson 6 - Combinational Design Practices Prof. João Miguel Fernandes (miguel@di.uminho.pt) Dept. Informática UNIVERSIDADE DO MINHO ESCOLA DE ENGENHARIA - PLDs (1) - The

### This Unit: Floating Point Arithmetic. CIS 371 Computer Organization and Design. Readings. Floating Point (FP) Numbers

This Unit: Floating Point Arithmetic CIS 371 Computer Organization and Design Unit 7: Floating Point App App App System software Mem CPU I/O Formats Precision and range IEEE 754 standard Operations Addition

### 4 Combinational Components

Chapter 4 Combinational Components Page of 8 4 Combinational Components In constructing large digital circuits, instead of starting with the basic gates as building blocks, we often start with larger building

### To convert an arbitrary power of 2 into its English equivalent, remember the rules of exponential arithmetic:

Binary Numbers In computer science we deal almost exclusively with binary numbers. it will be very helpful to memorize some binary constants and their decimal and English equivalents. By English equivalents

### Fixed-Point Arithmetic

Fixed-Point Arithmetic Fixed-Point Notation A K-bit fixed-point number can be interpreted as either: an integer (i.e., 20645) a fractional number (i.e., 0.75) 2 1 Integer Fixed-Point Representation N-bit

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### Chapter 4: Computer Codes

Slide 1/30 Learning Objectives In this chapter you will learn about: Computer data Computer codes: representation of data in binary Most commonly used computer codes Collating sequence 36 Slide 2/30 Data

### Recall the process used for adding decimal numbers. 1. Place the numbers to be added in vertical format, aligning the decimal points.

2 MODULE 4. DECIMALS 4a Decimal Arithmetic Adding Decimals Recall the process used for adding decimal numbers. Adding Decimals. To add decimal numbers, proceed as follows: 1. Place the numbers to be added

### Lecture 2: Number Representation

Lecture 2: Number Representation CSE 30: Computer Organization and Systems Programming Summer Session II 2011 Dr. Ali Irturk Dept. of Computer Science and Engineering University of California, San Diego

### Number Systems. Introduction / Number Systems

Number Systems Introduction / Number Systems Data Representation Data representation can be Digital or Analog In Analog representation values are represented over a continuous range In Digital representation

### quotient = dividend / divisor, with a remainder Given dividend and divisor, we want to obtain quotient (Q) and remainder (R)

Binary division quotient = dividend / divisor, with a remainder dividend = divisor quotient + remainder Given dividend and divisor, we want to obtain quotient (Q) and remainder (R) We will start from our

### ECE 0142 Computer Organization. Lecture 3 Floating Point Representations

ECE 0142 Computer Organization Lecture 3 Floating Point Representations 1 Floating-point arithmetic We often incur floating-point programming. Floating point greatly simplifies working with large (e.g.,

### A Short Guide to Significant Figures

A Short Guide to Significant Figures Quick Reference Section Here are the basic rules for significant figures - read the full text of this guide to gain a complete understanding of what these rules really

### A NEW REPRESENTATION OF THE RATIONAL NUMBERS FOR FAST EASY ARITHMETIC. E. C. R. HEHNER and R. N. S. HORSPOOL

A NEW REPRESENTATION OF THE RATIONAL NUMBERS FOR FAST EASY ARITHMETIC E. C. R. HEHNER and R. N. S. HORSPOOL Abstract. A novel system for representing the rational numbers based on Hensel's p-adic arithmetic

### (Refer Slide Time: 00:01:16 min)

Digital Computer Organization Prof. P. K. Biswas Department of Electronic & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture No. # 04 CPU Design: Tirning & Control