# Number Systems Richard E. Haskell

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 NUMBER SYSTEMS D Number Systems Richard E. Haskell Data inside a computer are represented by binary digits or bits. The logical values of these binary digits are denoted by and, while the corresponding physical values can be any two-state property such as a high (5 volts) or low ( volts) voltage or two different directions of magnetization. It is therefore convenient to represent numbers inside the computer in a binary number system. Hexadecimal and octal numbers are often used as a shorthand for representing binary numbers. In this appendix you will learn: How to count in binary and hexadecimal How integers and fractional numbers are represented in any base How to convert numbers from one base to another How negative numbers are represented in the computer By the end of this appendix you should be able to Convert any binary, hexadecimal or octal number to the corresponding decimal number Convert any decimal number to the corresponding binary, hexadecimal or octal number Take the two's complement of any binary or hexadecimal integer. D. COUNTING IN BINARY AND HEXADECIMAL Consider a box containing one marble. If the marble is in the box, we will say that the box is full and associate the digit with the box. If we take the marble out of the box, the box will be empty, and we will then associate the digit with the box. The two binary digits and are called bits and with one bit we can count from zero (box empty) to one (box full) as shown in Fig. D.. = empty box = full box no. of marbles = no. of marbles = Figure D. You can count from to with bit. Consider now a second box that can also be full () or empty (). However, when this box is full, it will contain two marbles as shown in Fig. D.2. With these two boxes (2 bits) we can count from zero to three, as shown in Fig. D.3. Note that the value of each

2 2 APPENDIX D 2-bit binary number shown in Fig. D.3 is equal to the total number of marbles in the two boxes. = empty box = full box Figure D.2 This box can contain either two marbles (full) or no marbles. Total no. of marble 2 3 Figure D.3 You can count from to 3 with two bits We can add a third bit to the binary number by adding a third box that is full (bit = ) when it contains four marbles and is empty (bit = ) when it contains no marbles. It must be either full (bit = ) or empty (bit = ). With this third box (3 bits), we can count from to 7, as shown in Fig. D Figure D.4 You can count from to 7 with 3 bits. If you want to count beyond 7, you must add another box. How many marbles should this fourth box contain when it is full (bit = )? It should be clear that this box

3 NUMBER SYSTEMS 3 must contain eight marbles. The binary number 8 would then be written as. Remember that a in a binary number means that the corresponding box is full of marbles, and the number of marbles that constitutes a full box varies as, 2, 4, 8, starting at the right. This means that with 4 bits we can count from to 5, as shown in Fig. D.5. It is convenient to represent the total number of marbles in the four boxes represented by the 4-bit binary numbers shown in Fig. D.5 by a single digit. We call this a hexadecimal digit, and the 6 hexadecimal digits are shown in the right column in Fig. D.5. The hexadecimal digits to 9 are the same as the decimal digits to 9. However, the decimal numbers to 5 are represented by the hexadecimal digits A to F. Thus, for example, the hexadecimal digit D is equivalent to the decimal number 3. No. of marbles in each full box (bit = ) Total no. of marbles Hex digit A B 2 C 3 D 4 E 5 F Figure D.5 You can count from to 5 with 4 bits. To count beyond 5 in binary, you must add more boxes. Each full box you add must contain twice as many marbles as the previous full box. With 8 bits you can count from to 255. A few examples are shown in Fig. D.6. The decimal number that corresponds to a given binary number is equal to the total number of marbles in all the boxes. To find this number, just add up all the marbles in the full boxes (the ones with binary digits equal to ).

4 4 APPENDIX D No. of marbles in each full box (bit = ) Total no. of marbles Figure D.6 You can count from to 255 with 8 bits. As the length of a binary number increases, it becomes more cumbersome to work with. We then use the corresponding hexadecimal number as a shorthand method of representing the binary number. This is very easy to do. You just divide the binary number into groups of 4 bits starting at the right and then represent each 4-bit group by its corresponding hexadecimal digit given in Fig. D.5. For example, the binary number 9 A is equivalent to the hexadecimal number 9AH. We will often use the letter H following a number to indicate a hexadecimal number. You should verify that the total number of marbles represented by this binary number is 54. However, instead of counting the marbles in the binary boxes you can count the marbles in hexadecimal boxes where the first box contains A x = marbles and the second box contains 9 x 6 = 44 marbles. Therefore, the total number of marbles is equal to 44 + = 54. A third hexadecimal box would contain a multiple of 6 2 = 256 marbles, and a fourth hexadecimal number would contain a multiple of 6 3 = 4,96 marbles. As an example, the 6-bit binary number 8 7 C 9 is equivalent to the decimal number 34,76 (that is, it represents 34,76 marbles). This can be seen by expanding the hexadecimal number as follows: 8 x 6 3 = 8 x 4,96 = 32,768 7 x 6 2 = 7 x 256 =,792 C x 6 = 2 x 6 = 92 9 x 6 = 9 x = 9 34,76 You can see that by working with hexadecimal numbers you can reduce by a factor of 4 the number of digits that you have to work with.

5 NUMBER SYSTEMS 5 D.2 POSITIONAL NOTATION Binary numbers are numbers to the base 2 and hexadecimal numbers are numbers to the base 6. An integer number N can be written in any base b using the following positional notation: N = P 4 P 3 P 2 P P = P 4 b 4 + P 3 b 3 + P 2 b 2 + P b + P b where the number always starts with the least significant digit on the right. For example, the decimal number 584 is a base number and can be expressed as 584 = 5x 2 + 8x + 4x = = 584 A number to the base b must have b different digits. Thus, decimal numbers (base ) use the digits to 9. A binary number is a base 2 number and therefore uses only the two digits and. For example, the binary number is the base 2 number 2 = x2 5 + x2 4 + x2 3 + x2 2 + x2 + x2 = = 52 This is the same as the first example in Fig. D.6 where the total number of marbles is 52 ( ). A hexadecimal number is a base 6 number and therefore needs 6 different digits to represent the number. We use the ten digits to 9 plus the six letters A to F as shown in Fig. D.5. For example, the hexadecimal number 3AF can be written as the base 6 number 3AF 6 = 3x6 2 + Ax6 + Fx6 = 3x256 + x6 + 5x = = 943 Microcomputers move data around in groups of 8-bit binary bytes. Therefore, it is natural to describe the data in the computer as binary, or base 2, numbers. As we have seen, this is simplified by using hexadecimal numbers where each hex digit represents 4 binary digits. Some larger computers represent binary numbers in groups of 3 bits rather than 4. The resulting number is an octal, or base 8, number. Octal numbers use only the 8 digits to 7. For example, the octal number 457 can be written as the base 8 number = 4x x8 + 8 = = 33

6 6 APPENDIX D Fractional Numbers The positional notation given above for integer numbers can be generalized for numbers involving fractions as follows: N = P 2 P P.P - P -2 P -3 = + P 2 b 2 + P b + P b + P - b - + P -2 b -2 + P -3 b -3 + As an example, consider the base number Using the above definition, this is equal to N = 3 x x + 5 x + x x -2 = = In this case the radix, or base, is and the radix point (decimal point) separates the integer part of the number from the fractional part. Consider now the binary number.. This is equivalent to what decimal number? Using the above definition, we can write. 2 = x2 3 + x2 2 + x2 + x2 + x2 - + x2-2 = = 3.75 Following the same technique, we can write the hexadecimal number AB.6 as AB.6 6 = x6 2 + x6 + x6 + 6x6 - = = As a final example consider the octal number We can find the equivalent decimal number by expanding the octal number as follows = x x8 + 3x8 + 2x x8-2 = = The examples in this section show how you can convert a number in any base to a decimal number. In the following section we will look at how to convert a decimal number to any other base and how to convert among binary, hexadecimal, and octal.

8 8 APPENDIX D There is an easier way to take the two's complement of a binary number. You just start at the rightmost bit and copy down all bits until you have copied down the first. Then complement (that is, change from to or to ) all the remaining bits. For example, complement remaining bits copy this first As a second example, two's complement = complement remaining bits two's complement = copy all bits to first Verify that if you add the 8-bit binary numbers given in the examples above to their two's complement value you obtain. An 8-bit byte can contain positive values between and ; that is, between H and 7FH. This corresponds to decimal values between and 27. A byte in which bit 7 is a is interpreted as a negative number, whose magnitude can be found by taking the two's complement. For example, how is -75 stored in the computer? First, write down the binary or hex value of the number, 4BH, as shown in Fig. D.8. Then take its two's complement. The value 75 is therefore stored in the computer as B5H. Note that if you take the two's complement of a positive number between and 7FH the result will always have bit 7 (the most significant bit) set to. 75 = 4BH = Two's complement = -75 = B5H = Two's complement of B5H = 4BH = Figure D.8 The negative of a binary number is found by taking the two's complement. Given a negative number (with bit 7 set), you can always find the magnitude of this number by taking the two's complement. For example, the two's complement of B5H (-75 ) is 4BH (+75 ), as shown in Fig. D.8. Note that the two's complement of H is FFH and the two's complement of 8H is 8H, as shown in Fig. D.9. This last example shows that signed 8-bit binary numbers "wrap around" at 8H. That is, the largest positive number is 7FH = 27 and the smallest negative number (largest magnitude) is 8H = -28. This is shown in Table D.2.

9 NUMBER SYSTEMS 9 = H = Two's complement = - = FFH = 28 = 8H = Two's complement = -28 = 8H = Figure D.9 Negative 8-bit numbers can range between FFH (-) and 8H (-28). Table D.2 also shows that the hex values between 8H and FFH can be interpreted either as negative numbers between -28 and - or as positive numbers between 28 and 255. Sometimes you will treat them as negative numbers and sometimes you will treat them as positive values. It is up to you as the programmer to make sure you know whether a particular byte is being treated as a signed or as an unsigned number. Whereas bit 7 is the sign bit in an 8-bit byte, bit 5 is the sign bit in a 6-bit word. A 6-bit signed word can have values ranging from 8H = -32,768 to 7FFFH = +32,767. Similarly, bit 3 is the sign bit in a 32-bit double word. Such a double word can have values ranging from 8H = -2,47,483,648 to 7FFFFFFFH = +2,47,483,647. Table D.2 Positive and Negative Binary Numbers Signed decimal Hex Binary Unsigned decimal FD FE FF D E F 27

10 APPENDIX D D.4 NUMBER SYSTEM CONVERSIONS Binary p fhex Converting a binary number to hex is trivial. You simply partition the binary number in groups of 4 bits, starting at the radix point, and read the hex digits by inspection using the hex digit definitions in Fig. D.5. For example, the binary number. can be partitioned as follows:. 6 A 8. F 5 C Therefore,. 2 = 6A8.F5C 6. Note that leading zeros can be added to the integer part of the binary number, and trailing zeros can be added to the fractional part to produce a 4-bit hex digit. Going from hex to binary is just as easy. You just write down the 4 binary digits corresponding to each hex digit by inspection (using the table in Fig. D.5). Binary p f Octal Converting a binary number to octal is just as easy as converting it to hex. In this case you just partition the binary number in groups of 3 bits rather than 4 and read the octal digits ( to 7) by inspection. Again the grouping is done starting at the radix point. Using as an example, the same binary number. that we just converted to hex we would convert it to octal as follows: Therefore,. 2 = Note again that leading zeros must be added to the integer part of the binary number, and trailing zeros must be added to the fractional part to produce 3-bit octal digits. You reverse the process to go from octal to binary. Just write down the 3 binary digits corresponding to each octal digit by inspection. Hex p f Octal When converting from hex to octal or from octal to hex, it is easiest to go through binary. Thus, for example, to convert 6A8.F5C 6 to octal, you would first convert it to the binary number. 2 by inspection, as shown in the example above. Then you would convert this binary number to , as we just did in the previous example.

12 2 APPENDIX D read down.7825 x 6 = 2.5 integer part = 2 = C.5 x 6 = 8. integer part = 8 Therefore,.7825 =.C8 6. Combining the integer and fractional parts, we have found that = F3D.C8 6. This rule for converting the fractional part of a decimal number will work for any base. Sometimes the remainder may never become zero and you will have a continuing fraction. This means that there is no exact conversion of the decimal fraction. For example, suppose you want to represent the decimal value. as a binary number. Following our rule, we would write read down. x 2 =.2 integer part =.2 x 2 =.4 integer part =.4 x 2 =.8 integer part =.8 x 2 =.6 integer part =.6 x 2 =.2 integer part =.2 x 2 =.4 integer part =.4 x 2 =.8 integer part =.8 x 2 =.6 integer part =.6 x 2 =.2 integer part = It is clear that the remainder will never go to zero and that the pattern will go on forever. Thus,. can only be approximated as. =. 2 This means that. cannot be represented exactly in a computer as a binary number of any size!

### 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

### 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

### 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

### 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

### 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,

### 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

### 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

### 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,

### 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

### 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...

### 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

### 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

### 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

### = 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

### 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

### 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,

### Decimal to Binary Conversion

Decimal to Binary Conversion A tool that makes the conversion of decimal values to binary values simple is the following table. The first row is created by counting right to left from one to eight, for

### 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.

### 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

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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)

### 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

### COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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;

### Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6

Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6 Number Systems No course on programming would be complete without a discussion of the Hexadecimal (Hex) number

### 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

### Introduction to Fractions

Introduction to Fractions Fractions represent parts of a whole. The top part of a fraction is called the numerator, while the bottom part of a fraction is called the denominator. The denominator states

### 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

### 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

### 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

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### 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

### 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

### 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

### 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

### NUMBER SYSTEMS APPENDIX D. You will learn about the following in this appendix:

APPENDIX D NUMBER SYSTEMS You will learn about the following in this appendix: The four important number systems in computing binary, octal, decimal, and hexadecimal. A number system converter program

### 4 Operations On Data

4 Operations On Data 4.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, students should be able to: List the three categories of operations performed on

### 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,

### CHAPTER TWO. 2.1 Unsigned Binary Counting. Numbering Systems

CHAPTER TWO Numbering Systems Chapter one discussed how computers remember numbers using transistors, tiny devices that act like switches with only two positions, on or off. A single transistor, therefore,

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Basic Logic Gates Richard E. Haskell

BASIC LOGIC GATES 1 E Basic Logic Gates Richard E. Haskell All digital systems are made from a few basic digital circuits that we call logic gates. These circuits perform the basic logic functions that

### 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

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Basic review Writing fractions in simplest form Comparing fractions Converting between Improper fractions and whole/mixed numbers Operations

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### 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

### 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

### 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

### 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 -

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### BINARY CODED DECIMAL: B.C.D.

BINARY CODED DECIMAL: B.C.D. ANOTHER METHOD TO REPRESENT DECIMAL NUMBERS USEFUL BECAUSE MANY DIGITAL DEVICES PROCESS + DISPLAY NUMBERS IN TENS IN BCD EACH NUMBER IS DEFINED BY A BINARY CODE OF 4 BITS.

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

APPENDIX C The Hexadecimal Number System and Memory Addressing U nderstanding the number system and the coding system that computers use to store data and communicate with each other is fundamental to

### 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

### 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 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

### 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

### 1.3 Order of Operations

1.3 Order of Operations As it turns out, there are more than just 4 basic operations. There are five. The fifth basic operation is that of repeated multiplication. We call these exponents. There is a bit

### Data Representation. What is a number? Decimal Representation. Interpreting bits to give them meaning. Part 1: Numbers. six seis

Data Representation Interpreting bits to give them meaning Part 1: Numbers Notes for CSC 100 - The Beauty and Joy of Computing The University of North Carolina at Greensboro What is a number? Question:

### > 2. Error and Computer Arithmetic

> 2. Error and Computer Arithmetic Numerical analysis is concerned with how to solve a problem numerically, i.e., how to develop a sequence of numerical calculations to get a satisfactory answer. Part

### 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

### 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

### 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,

### INTEGER DIVISION BY CONSTANTS

12/30/03 CHAPTER 10 INTEGER DIVISION BY CONSTANTS Insert this material at the end of page 201, just before the poem on page 202. 10 17 Methods Not Using Multiply High In this section we consider some methods

### Lecture N -1- PHYS 3330. Microcontrollers

Lecture N -1- PHYS 3330 Microcontrollers If you need more than a handful of logic gates to accomplish the task at hand, you likely should use a microcontroller instead of discrete logic gates 1. Microcontrollers

### Activity 1: Bits and Bytes

ICS3U (Java): Introduction to Computer Science, Grade 11, University Preparation Activity 1: Bits and Bytes The Binary Number System Computers use electrical circuits that include many transistors and

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### 3.3 Addition and Subtraction of Rational Numbers

3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.

### 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

### 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